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THE CLASSICAL STRUCTUREOF BLOOD BIOCHEMISTRY-A MATHEMATICAL MODEL
E, C. DeLand
C -DDC
L LPRPASM FOR:L
UNITED STATES AIR FORCE PROJECT RAND
SAW !C AI
MEMORANDUM
RM-4962-PRJULY 1966
S
THE CLASSICAL STRUCTURE
OF BLOOD BIOCHEMISTRY-A MATHEMATICAL MODEL
E. C. DeLand
This research is sponsored by the United State- Air Force under Project RAND--Con-tract No. AF 49(6381-1700 -monitored by the Directorate of Operational Requirementsand Development Plans, Deputy Chief of Staff. Research and Development. 11q USAF.Views or conclusions contained in this Memorandum should not be interpreted asrepresenting the official opinion or policy of the United States Air Force.
D)ISTIRIIIUTION STATEMENTDistrilbution of this document is unlimited.
S7OV ,AN S' )AN. O'ONK 1..# ..A*IIO4IINI. . $
-*"pop
o-iii-
PREFACE
This is the third in a continuing series of Memoranda
on the development of a mathematical simulation of human
blood biochemistry. The first--Dantzig, et al. [li--was
a feasibility study of the methods for mathematical anal-
ysis and computer simulation; the spcond--DeHaven and
DeLand [21--added some details of hemoglobin chemistry
and carefully described the behavior of that system under
various conditions.
The subsequent body of critical conmment and addi-
tional laboratory data enabled further elaboration and
definition of the model. Professor F.J.W. Roughton, F.R.S.,
Cambridge University, particularly improved the hemoglobin
chemistry. In addition, the quantitative aspects of Lhie
plasma proteins have been reviewed and their buffering
properties incorporated.
The present model therefore approaches more closely
the complexity of the real blood system. Properties of
the mathematical model such, as gas exchange, buffering,
and response to chemical stress in the steady state are
practically indistinguishable from those -roperties of
real blood within the limits of our current validation
program.
,i~ ~- . -
-iv,-
The program consists of two parts: detailed chemical
analysis of human blood under a variety of chemical stresses;
and further elaboration of the model, e.g., by including
the phosphate system, double-valent ion binding by proteins,
and red cell organic constituents ocher than hemoglobin.
Laboratory experiment3 vali.datiLng the model will be pub-
lished in the doctoral thesis of Eugene Magnier, M.D.,
Temple University Medical School.
The research done with Dr. F.J.W. Roughton was sup-
ported by the Department of Health, Education, and Welfare
Grant HE-08665 (National Inhtitutes of Health).
-v-
SUMMARY
This Memorandum, one of a series on the modeling of
blood biochemistry, describes, and examines the consequences
of, the classical analysis of human blood. Classically, the
macroscopic features of healthy blood--the principal f1 xid
and electrolyte distributions--can be said to be in a
hemostatic steady state under the influence of several
Interacting constraints, e.g., the osmotic effects of the
fixed proteins, the (fixed) charge of the proteins and
.he neutral electrostatic charge condition, and the active
cation pumps. Chapter I examines these constraints from the
vantage of a theoretical model, and shows them to be suf-
ficient explanation for the steady state; whether they are
also necessary has not been demonstrated.
Chapter II considers the incorporation of the micro-
scopic properties o1 the proteins in the model, particularly
thtir bufCering behavior. This work is ot complete, owing
principally to the lack of firm data en hemoglobin.
Chapter III discusses the va!idation of the mathematical
model and the consequences of the previous biochemical struc-
tural detail, testing the model under various conditions--
changes in gas pressure and pH, and additions of a few
-vi-
che•,ical stresses. It compares several validation experi-
ments with the literature. The Memorandum shows that,
except for certain interesting discrepancies, the model
satisfactorily agrees with the literature.
-vii-
CONTENTS
PREFACE ................................. ......... Lii
SUMMARY .......................................... v
FIGURES .............. .... ................... . . . ix
TABLES ................................ xi
ChapterI. A PRELIMINARY MODEL OF BLOOL BIOCHEMISTRY 1
1. Introduction ............2. IsolatIng and Defining the Physio-
logical System .................... 43. Development of the Blood Model ......... 11
General Considerations ............... 11Statement of the Prelirtinary Problem . 12First Approximations ................. 13Model A--Primitive System, Plasma .... 14Model B--Osmolarity .................. 15Model C--Electrical Neutrality ....... 17Model D--Gibbs-Donnan and
Hamburger Shift ..................... 19Model E--The Sodium Pump ............. 25Model F--Simultaneous Condition. 31Model G--Hill's Equation, 4yoglobin 33
I1. CONCEPTUAL COMPARTthENTS ..................... 431. Introduction ........................... 432. Conceptual Compartments ................ 48
Protein Ionization--Example .......... 483. Blood Proteins: Plasma ................ 564. Blood Proteins: Red Cell .............. 61
III. VALIDATION OF THE MODEL ..................... 781. Iitroduction ........................... 782. The Standard State ...................... 79
The Hatrr.x of Partiai Derivatives 0.... 10Oxygen Dissociation Curves ........... 108The "Astrup" Experiments ............. 114
REFERENCES ....................................... 121
- ix-
FIGURES
1. Experimental Hemoglobin SaturationHypotheses ..... .........................
2. Oxygen Dissociation Curves for HumanBlood ..................................... 110
3. Ast-up- type Experiments .......... 115
"-xi-
TABLES
I Standard Blood Composition 'for Resting
Young Adult Male .................... 7II DistriLution of Species in Liter of
Standard Artetial Blood (Adult Male)and Alveolar Sac Gas ................ 8
III Elementary Oswolarity--Model B ........ 18
IV Gibbs-Donnan Fhenomenon--Mo.el. D ...... 20
V Input Data--Elementary Model .......... 28
VI Sodium Pump--Model E .................. 29
VII Nonlinearity--Hodel F ................. 32
VIII Input Data--Model C ................... 36
IX Completed Distribution--Model G ....... 37
X Titratable Groups per 65,000 GramsSerum Albumin ....................... 53
XI Hemoglobin Reactions and HemoglobinConceptual Compartments ............. 73
XII Standard Model Inputs ................. 80
XIII Expected Output Species ....... 81
XIV Standard Arterial Blood ............... 84
XV Srond-rd Venous Blood ................. 85
XVI Comuparison ef Literature and ComputedValue Model BFFRol ......... ......... 86
XVII Modified Arterial Blood ............... 90
XVIII Venous Blood Standard Model tFFI-R,Hole Fraction Chang@e 1rom Atterial 97
XLXA Output Species Increments per Unit
Increase in Input Species ........... 103-105
XIXB Increments in Total Moles i1 EachCopsrtment per Unit Increase inInput Specips ........................ 1•!%.
Chapter I
A PRELIMINARY MODEL OF BLOOD BIOCHEMISTRY
1. INTRODUCTION
This is the third in a continuing series of Memoranda
detailing the development of a mathematical simulation of
human blood biochemistry. The first--Dantzig, et al. [1]--
was a feasibility study of a method for mathematical formu-
lation and computer solution of a biochemical simulation
problem. An example of a "simple" problem is, given the
basic molecular components (reactant species) of a chemical
milieu in a single phase, compute the concentration of all
S4species in the equilibrium mixture for which the chemical
reaction coefficients and mass action constants are also
given.
Such a system may, however, become complex if many re-
actions are interrelated, if some of the molecules in the
milieu (such as proteins) are themselves complex, or if
the system is multiphasic, multicompartmented, or alive.
These complexities complicate the comput ition; in particular,
for viable systems, one speaks of the computations for a
supposed "steady state" of the living organism, rather than
the "equilibrium state."
-2-
The Dantzig Memorandum introduced the computer method
for miimizing the Gibbs free energy function to compute
the (unique) steady state of a multicompartmented blood
system. The procedures are described in Ref. 1, and de-
veloped further in Shapiro and Shapley [3], Shapiro [4],
and Clasen [51. DeHaven and DeLand [21 (the second Memo-
randum on blood simulation) explore the computer method for
determining a steady state in the three-conmpartment model
(lung gas, plasma, and red cell), emphasizing the properties
of the chemical systems themselves. Even that rudimentary
model had many important characteristics of a blood system,
e.g., the proper distribution of water and electrolytes,
and a reasonable representation of normal physiologic
functions.
This Memorandum extends the previous results, docu-
menting them in more detail, and initiates the validation
of an improved model of the blood chemistry. Chapter I
analyzes the biochemical structure of the two-compartment
blood system. The purpose of this chapter is to elucidate,
with a rudimentary blood model such as that used in
Dantzig [I], the conventional roles of the fixed proteins,
the neutral electrostatic charge constraints, and the
active cation pumps in d-termining the major physicjl
characteristics of hemostatic blood. While these con-
ventional roles, as is well known, yield a sufficient
general explanation for the fluid and electrolyte distri-
bution of normal blood, a mathematical model, which permits
experiments impossible in the laboratory, shows their
detailed interplay.
Chapter II describes a special mathematical procedure
for representing large chemical systems and discusses the
representation of the essential oroteins. Finally, Chap.
III discusses a model of the respiratory biochemistry of
the blood, using the previously developed details, and
shows some of the model's initial validation procedures.
-4-
2. ISOLATING AND DEFINING THE PHYSIOLOGICAL SYSTEM
The blood, contained in the vascular system, may be
regarded as a uniform and continuous subsystem of the body.
Of course, it changes chemically in the circuit around
the body, visibly changing color. However, conceptually
isolating the blood as a subsystem and considering only
the biochemistry of the respiratory function, this sub-
system exhibits the following properties:
a) The subsystem has natural boundaries; its extentis well-defined. It: is, for practical purposes,contained within the vascular system. Importantand interesting exceptions to this sometimesoccur--e.g., protein may pass through the capil-lary walls to be returned to the vascular systemby lymph flow.
b) The function of the bounding membranes can bedefined and the substances crossing these mem-branes analyzed--i.e., the inputs to and lossesfrom the subsystem are measurable. The actualmechanisms and biochemical processes within themembranes themselves are, for the most part,still hypothetical, but only the net results oftheir functions are necessary to simulationsof the steady-state. Transport of substancesacross these bounding membranes occurs prin-cipally--and perhaps only--in capillary bedseither at the lung or other body tissues.
c) Biochemical analyses of the blood can be madeeasily and quickly and, with care, thoroughne3sand accuracy is possible. Furthermore, samplesfrom this subruystem are available. Thus, forexample, conservation of mass equations, funda-mental to the simulation, can be substantiated.
-5-
d) The subdivisions of the subsysLem on the grosslevel are clear, e.g., the red cells taken to-gether as a compartment constitute a subdivisionof whole blood. However, the exterior environ-ment of the vascular system must also be examinedfor a model, since this environment largely de-termines the transfer of substances across thebounding membranes. Except in the interior ofcertain organs, the environment of the vascularsystem is interstitial fluid. However, in thelung (and perhaps the brain) the layer of inter-stitial fluid is negligibly thin. In the lungwe may consider the environment of the capillarybed to be alveolar sac gas.
e) Lhe physiological function of respirati-n forthe subsystem "blood" is well-defined; itspurpose is clear. The biochemistry involved inaccomplishing this purpose is, ion the grosslevel, relatively well-understood. However, onthe level of chemical detail, many unsolvedproblems still exist. For example, the explana-tions of the "chloride shift" phenomenon, orthe "Bohr shift" phenomenon, or the "sodium pump"phenomena are, among many other such problems,still subjects of research.
The chemical composition of blood varies, of course,
according to the point in the body, the age and sex of
the individual, his response to disease, etc. An average,
resting, young, adult male human is the standard for this
exposition. The relevant data come from man-. sources, but
particularly Dittmer (Ref. 6, Blood and Other r..dy Fluids)
and Spector (P f. 7, Handbook of Biological Data).
-6-
Table I summarizes these dlata for three standard
types of blood for the average, young, adult male. Table
II shows the distribution of the substanca from Table I
for standard or average artericl blood between plasma and
red cell compartments. These -.omposition tables are
generally concerned with electrolytes, water, and gases.
This chipter does not detail the amino acids, lipids,
carbohydrates, organic phosphates, or other secondary
constituents. It mentions only the constituents which
play a direct role in respiration.
The plasma proteins and impermeable macromolecules,
principally albumin and the various globulins, amount
to about 8.7 x 3.0 moles per liter of blood. At pH 7.4,
the average chn:•ge per molecule for the proteins is from
-1.0 to -20, as can he d&rcermined by examining the titra-
tion curves for albumin and globulin [81. For this first
model where the proteins are not yet buffering, the value
of negative equivalents per mole will be fixed at -10.
Thus, the number of negative equivalents of plasma pro-
teins, used to determine that the solution is electrically-3
neutral, is 8.7 x 10
The rea cell proteins present a much more complicated
situation. Of course, hemoglobin act!%'tty is crucial to
-7-
Table I
STANDARD BLOOD COMPOSITION FOR RESTING YOUNG ADULT MALE a
Mixed-Venous, Arterial,Venous Blood, Pulmonary Pulmonary
Item Large Vein Artery Vein
pH 7.37 7.3E 7.39
p0 2 65.0 mm 40.0 mm 100.0 mm
pCO 2 46.0 mm 46.0 mm 40.0 mm
pN2 573.C mm 573.0 mm 573.0 mm
HCO3 22.3 meq/L 22.0 meq/L. 21.0 meq/L
% Sat. 91.0 73.0 97.0
H 20 46.50 moles/L (b) (b)
Cl" 80.05 meq/L
Na+ 84.82 "
K + 45 05
Ca+ + 3.210 "
Mg++ 3.225
SO 4 1.050 "
HPO4 2.860
Lactic" 2.033 "NH+ 0.869N4
Urea 3.142 mmoles/L
Glucose 3.666-10
PL PR 0.870
PC PRKx 10.57
Hb4 2.2725
aSee notes in text.
bAll the following items in this column do noL change
with the type of blood, or Lheir change has noE been measured,or tht change is not signlfirar-: at this stage of the slmulatio-
"-8-
Table II
DISTRIBUTION OF SPECIES IN LITER OF STANDARD ARTERIAL
BLOOD (Adult Male) AND ALVEOLAR SAC GAS
AlveolarItem Sac Plasma Red Cell
pH 7.39 7.19-5 -5
02 13.15 6.33x10 5holes 6.43xi0 moles
CO2 5.26 7.01xlO0 " 4.76x10 4 "
N2 75,4 2.20xlO- 4 2.20xlO" 4
H 20 6.11 28.50 " 18.0
C1 56.65 meq 23.40 meq
Na+ 76.45 " 8.37 "
K+ 2.310 " 42.75 "
Ca ++ 2.86 " 0.344 "
Mg++ 0.935 •' 2.295 "
so 0.680 " a04
HPO4 1.88 "
Lactic 1.30 "
NH4 +0.054
HCO 3 13.836 " 6.025
Glucose 2.00 ,wmoles
Urea 1.73 " 1.40 nnoles
Misc PR 0.87 " 10.568
Hb4 ..... 2. 2725
a,, ...... indicates the value is eitter not reported
or is reported in a form not pertinent to this represent~tion.
-9-
respiration, carryinS CO2 as well as oxygen in combina-
tion. In more complex models (Chap. III), the buffering
activity of hemoglobin also pertains. This preliminary,
illustrative model treats hemoglobin simply as a procein
combining with oxygen and contributing to the osmolarity
and, hence, hematocrit of the blood.
However, the red cells also contain miscellaneous
proteins, complex phosphates, and other anionic species
(excluding hemoglobin) generally represented in the liter-
ature [7] by X . This symbol indicates an undeterminea
anion residue necessary to make the red cell milieu
neutrally charged and to make the osmolarity of the red
cell milieu equal to that of plasma. Chapter III examines
this assumption of equal osmolarity, but the preliminary
model uses the undetermined anion residue to obtain the
proper hematocrit and a neutral eT'ctrical charge.
Neither the distributions of glucose and urea nor of
the various sulphates and phosphates between plasma and
red cell have been carefully detailed for this preliminary
model. The diffusion of glucose through the red cell
membrane is presumably an intricate process, while urea
is thought to pass freely through the membrane. Table Iin .
r7eflects neither of these facts. The species represented
"/
-10-
by SO and HPO4 are the acid-soluble sulphates and phos-
phates reported in the literature. Table If does not
indicate that these substances may also be intricately
bou,,d to the proteins of red cell and plasma. Similarly,
Ca++ and Mg++ may be bound by protein. Chapter III will
discuss these conditions.
Most of the Na+ and a larg, ?art of the Cf" are in
plasma, while most of the K is in the red cell compart-
ment, a consequen.2e of the "sodium pump" and the Gibbs-
Donnan phenomenon. The distribution of the species H2 0
in Table II requires some exi aation. When large proteins
and other aggregated molecules are dissolved in water, the
volume of the solution is usually larger than the volume
of solvent. This "volume factor" has been found to be 94
percent for normal plasma; that is, in Table II, the 28.50
moles of H20, which is 516 cc at 38 0 C, represent 516/
0.94 - 549 cc of plasma per liter of blood. The red ceil
volume is, thus, 451 cc per liter of blood and the hamato-
crit is 45.1 percent. Since, in the red cells, there are
only 18.0 moles or 326 cc of H2 0, the volume factor is
72.3 percent, which includes the volume occupied by the
strome.
3. DEVELOPMENT OF THE BLOOD MODEL
GENERAL CONSIDERATIONS
The data of the previous section enoible the simula-
tion of the average arterial jlood of Table II, which
represents a liter of blo, . the steady, resting state.
We assume that metabolism continues at a steady rate
and, because metabolism in the red cells is principally
anaerobic, gives a net production of H+ ions and lactate
ion along with other minor products. We also assume d
complex but undefined activity (referred to as the "sodium
pump") for the red cell membrane. One result is that all
mobile ions, particularly Ne and K+, acquire across this
membrane steady concentration gradients different from
those without the active membrane. Also, the system has
a steady "membrane potential"--roughly, that electrical
potential which is measured by suitable electrodes suit-
ably placed (Chap. III analyzes this definition). A
steady pH exista in the plasma and, because of the activity
of the cell, a different, lover pH holds in the "interior"
of the red cell.
The structui e of the red cell in simWler than other
boJy cells since the red cell is not nucleated and contains
-12-
but one principal protein. In the model, the interiors
of all red cells in the liter of blood are taken together
as a single "'compartment," uniformly mixed and separated
from the plasma by an ideal semi-permeable, though active,
membrane.
Conceptually, than, two distinct compartments, with
chemical analyses given in Table II, communicate with
each othcr through the idealized membrane. Proteins and
certain other large molecules (e.g., complex phosphates)
cannot penetrate, but other ions and molecules pass
through regardless of the mechanism or of the time required.
The membrane is flexible and will not support a pressure
gradient, so the "red cell" may change volume by as much
as 5 percenlt between arterial and venous blood. Alsc,
the temperature is supposed uniform throughout.
STATEMENT OF THE PRELIMINARY PROBLEM
Given these basic hypotheses and assumptions (as well
as further assumptions carefully defined as A-quired), the
essential problem of this 3imulation is to croate Table II
from Table I by -icans of a mathematical model--that is,
given the total compoiition ot a liter of whole blood and
using the general mathematical program, distribute these
S. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
-13-
"input" substances into the compartment's plasma and red
cell so that the "output" duplicates Table II.
FIRST APPROXIMATIONS
The construction of a model of arterial blood could
begin with the simulation of ordinary "soda water." a
two-phase system of gas and liquid with both inter- and
intraphase reactions. The gases, 02) CO2, N2, have
determined solubility constants at a given temperature
and pressure, and the solvent water has known vapor pres-
sure. One could, therefore, compute the concentrations
of each in the liquid phase as well as the partial pres-
sures of H20 vapor in the gaseous phase. In the liquid
phase, the carbonate system reaction constants are well-
documented, so with fixed partial pressures of the gases
one could compute the concentrations of HCO3, CO;, H+Q3 3
Or , and H 20 with arbitrary accuracy.
If the gas volume is finite, the formation of the
sp2cies in solution will, by conservation of mass, alter
the gas composition. Therefore, tht preliminary model
uses a very Iarge gas phase, which also gives better
Reference 2 further details this model.
See, for example Fdscll and Wyman, Re., 8, Chap, 10.
-14-
control of independent variables. Using, say, 1000 moles
of gas phase at a determined composition versus one liter
of liquid phase cause3 the gas phase to remain almost
constant, thus holding the gases in solution at constant
concentration. The gas composition can be varied, of
course, as required.t
MODEL A--PRIMITIVE SYSTEM. PLASMA
The first model, Model A, only slightly enlarges the
soda water model which consists of the gases 02) Co2 , N2 ,
HLO, the solvent H 0, and two phases, gas and liquid.4 2
Adding more species to the liquid phase without adding
intraphase reactions will change the resilts for soda
water only slightly. Thus, adding Na+, ClI K+, HPO4 ,
Ga, , ,Mg. so urea, glucose, lactic acid, and protein
up to the approximate concentrations of these substances
in plasma, changes the concei rations of dissolved gases
only slightly, although the pH may require adjustment
with NaOH or HCI. The gas concentrations do change be-
cause the solubility constant for the gases differs in
Previous studies--Ref3. I through 5--describe thecomputation procedure employed here.
S... . .. . . . . . . .. . . . . . . . .
-15-
plasma and in water, and because the actual number of
moles of liquid phase, x, has increased. No new intra-
phase reactions are introduced yet. Later the protein
will buffer H+ ion and bind Ca++ Mg and CI.
Here, the protein makes the solution non-ideal and
increases the volume of plasma over the water volume
by 6 percent. Otherwise, in the normEl viable range,
this two-phase preliminary system can be assumed to be
well-behaved. Again, concentration curves could be
computed for this simple system, but it is not yet of
much interest.
MODEL B--OSMOLARITY
A second model, Model B, has a third compartment
called "red cells" and several additional conditions.
For this model, the total amounts of subýtances added
to the total system will be exactly those of Table I,
arterial blood. The problem is to determine how much
of each of these substances is distributed into the
red cell compartment under various conditions. Such
a determination is a solution of the problem under
those conditions.
-16-
First, for Model b, let the membrane between plasma
and red cells be permeable to all species--including
protein. In this case, the mathematical solution is not
unique, for the membrane could divide the two compartments
arbitrarily, and the mixed composition of the two com-
partments would be identical and uniform except for xpl
and x RC the total number of moles of all species in
each.
Second, Model B a) supposes the membrane impermeable
to proteins only, b) ignores the electrical charge on the
ions and proteins, and c) supposes (as do subsequent
models in Chap. I) that the gases have equal solubility
in red cells and plasma.
With the protein distributed and fixed in plasma and
red cells, and with the membrane flexible so that hydro-
stetic pressure gradients cannot arise, all species dis-
tribute between the compartments in proportion to the
amounts of impermeable proteins in each, P result of the
elementary "osmotic" phenomenon: i.e., each substance±.
including water, will distribute so that it has the same
concentration on eech side of t1e membrane, independently
of the other species. In addition, since the proteins
are impermeable, all other species must tove; i.e., ý:he
4 r . ... .. ..
-17-
membrane must move, so that the proteins or. each side are
equally dilute. Thus, the total moles in each compartment
have the ratio
xPLI/RC 8. 706x40- 10.568),10"3 -- 2.2725xi0") = 0.0678
where the numbers on the right are the moles of protein
on each side of the membrane. Table III, a reproduction
of the computer results for Model B, illustrates these
comments.
MODEL C-- ELECTRICAL NEUTRALITY
Model C involves an hypothetical experiment. Suppose
that, in addition to the above conditions, each protein
now has an electrical charge of -1 per molecule, and the
other ions have their normal chemical valerce. Supoose,
further, that the sum of the charged input substances is
such that the whole is neutrally charged. Now apply thc'
mathematical condition that each compartment, plasma and
red cells, be neutrally charged, i.e., that the sum over
all charged species in each compartment is zeco. Under
these conditions, no change at all will occur in the
computed distribution of. the species and the results will
be identical to Model B. That is, under the conditions
-18-
Table I1II
ELEMENTARY OSMOLARITY- -MODEL B
PREIM~INA*RY RLinO') MODEL %.F ?I T1 CHANGEF 04 %A ,Up 0 DL &NO
AIR OUT PL AS01 RFO C*LLS
A-URn 9.99160f 02 1.97125F 00 4.19209F 01
P94 -0. 1.14SI11 00 1.34SM7 00
02 KOL FS 1.)1%Of1 01 *.953'13-0& I.0OZS1F-04
C02 MOLES S.*1oflE 01 ?.l.5I G'e 11181II-01RFRAC 11.16460-02 2.401911-05 P.407911-05
Witt S491 ?.S4000~ 02 2.ZNlMOF-0S 3.2VSM1-04NFRAC 1.5%6)4f-Ot ?.49114F-06 7.49141-06
man0 xOt IS 6.10509f C1 2.9%S82F 00 f4.Wfi93 01PUFSAC 6.10221:k-02 9.949001-01 9.946001-01
M# pOLES -nl. 1.416471-09 I.S6&SNI-011PFRAC -.a *.l195st1- .3940
OH- WMQ9f -0. 2.113914f-9S4.0510NFRAC -0. 9.S5&0qf-0q 9.SS&0%F-09
Ck- MOLE% -0). S,003091-01 t.496691-0?
MAO. lu.0915 -0. *.2A91%1-0) A.1?40?1-0214F 0AC -0. 1 .41-Ul1-0) .42-0
1K# Monfs -C. 2.U640b31-01 .140PISAC -0. ;.?49-* .62169#-04
CA*" 94439.1 -0. 1.01916f-0* I.SOlO"f-01"MFmIC -0. 1.4i10041-os 1.41006f-os
MG.. MtS -0. 1.0114?f-04 l.Sto111-01NFRAC -0. ).44 091-0% 1.44009f-Os
S,04. Plo~ttS -0. i.)W1OE-ON 4.l141f -04
P4034. Wx tS %0 9.0SOISI-11% 1.119?91-01of A~ a~ 9. 10% 60p f- 01 1. b60540fl-0S,
9*9* 0011% -'1 .44$41-0l4 1.9414tI-01wooet -0. 7.1&009-"¶ C'. 4804-0 %
Gi tLL n 09)1 t' -ý 1.3) 1511-04 i .. -4 %1 - 0ofsUac -r. 1.016WV-0% ?. atta%1 -0%
L A(Y f 0* tcWt ,%C,. 1.1,0409-04 '.001*11-01*0 1At f 4.I4Pl-0 .94%-01
W4* I39 %'I Ili*0*0 lISW-04
noLC a AIL -n. I.011w,1'# ICW- n
694 1Wf % ". a*. :000 1.I6~0106
ota 40* .. - o6 4F-44.104,1844-04
S40A C -0 *. 1 1644 -04
-19-
for B but with all protein uniformly charged, all sub-
stances are already distributed on both sides of the
membrane such that each compartment is electrically
neutral.
MUDEL D--GIBBS-DONNAN AND HAMBURGER SHIFT
Model D supposes that the average charge of the
protein molecules in plasma only is -10 per molecule, The
moles of input substances are identical to Table I, and
the average charges on hemoglobin and the miscellaneous
fixed species in red cells are such that the total inputs
are neutrally charged. Apply .g the neutral charge re.-
straint now to plasma and to red ce ls, if the average
charge per molecule of the red cell protein happens also
to be -10, the results are agaii as in Midel B or C. Any
other charge on the average, red cell, protein molecule
causes lifferent results.
The average charge for all red cell proteins in
Table I is -3.5, and a comparison of this with -10 for
the average charge of plasma protein indicates that posi-
tive ions should be displaced into plasmai and the permeable
negative ions displaced or repelled into red cells. Table
IV, a reproduction of the computer results for Kodel D,
-20-
Table IV
GIBBS-DONNAN PHENOMENON--MODEL D
PRELIMINARY BLOOD MODEL NO SODIUM PUMP, PL PR CHO -to DELANO
AIR OUT PLASMA RIO CELLS
X-BAR 9.991361 0? S.9S0291 00 4.08629E 01
PH -0. 7.780tSF 00 7.15342f 00
02 MOLES 1.115071 02 1.1889IE-05 9.5429IF-05
MFRAC 1.31620E-01 2.33422E-06 2.3342ZE-O6
CO? MOLES 5.26022E 01 1.43282E-04 9.64',56E-04MFRAC S.?6'.7?1-02 2.40799F-05 2.40799E-05
NW MOLES ?.S4000E 02 4.4S674F-05 3.06349E-04MFRAC 7.S4652E-01 7.4933.'F-06 ?.493)2f-06
H20 MOLES 6.102711 01 5.91722F 00 4.06557E 01OFRAC 6.10799E-07 9.944'IF-OI 9.94442E-01
H+ MOLES -0. 5.61919E-09 3.26460E-08MEPAC -0. 9.4.41861-10 7.98523E1-0
ON- MOLES -0. 4.90004FE.08 3.98084E-07MFRAC -n. 8.23496E-09 9.73716F-09
ct- MOLrS, -0l. 6.7734G1-03 7.1276SE-02MFRAE -0. 1.474461-03 1.7434,3E-03
NA+ MOLFS -0. 1.245361-02 7.23662E-02MERAC -0. 2.09298E-03 1.7700OE-03
K# MOLFS -n. 6.6145)E-03 3.6435SE-02MFRAC -0. 1.11163F-03 9.4013SE-04
CA..# MOLES -0. 2.71376E-04 1.33362E-03MFRAC -0. 4.56072F-05 1.26206F-05
MG*# MOLES -0. 2.??6'441-04 i1339866-03mFRAC -n. 4.56?03F-05 3.277301-05
S04- MOLES -0. 4.950001-05 4.755001-04MFRAC -0. 8.3189?E-06 1.163061-0%
mPn4- MOLES -0. 1.14829F-04 1.295171-01MFRAC -,). ?.?659?F-05 1.168001-OS
UREA MOLES -0. 1.992001-04 ?.?4?SOE-03MFRAC -. 6.106911-0% 6.708911-05
GLUCOS MOLES -0. 4.bS7?bF-04 3.700221-03"MERAC -0. 7.A27781-0S 7.627761-05
LACTIC MOLES -0. 2.2261?f-04 1.810161-03MFRAC -0. 3.744.6~14-0 4.42?131-OS
NH4* MOLE% -0. 1.7?59?[-04 7.414061-04MFRAC -0. 2.1'430f-OS 1.41349F-05
HCOI- MOLE% -0. 2.20S2ZF-01 1.7939?1-02MFRAC -0. 3.711111E-04 4.388061-04
H2Co3 MOLES -0. 2.037721-07 1.40006E-06MFRAC -n. 3.474S71-06 3.424S71-08
Col. M1OLES -0. 2.64-0 2:2943141-05MFRAC -0. 4.011'897?1-076 'A.61 198-07
MISCPR MOLES -0. 6.706001-04 1.05680f-02MFRAC -0. 1.463121-04 2.58494E-04
464 MOLES -0. -0. 2.2?ZS01-03MFRAC -0. -0. 5.55956E-05
-21-
shows this actually occurs. Table IV demonstrates that
Na +, say, is now predominantly in plasma, and ClI is pre-
dominantly in red cells. The total mole ratio of plasma
and red ce'l compartments, XpL/KRCP has changed to 0.1455.
Note that this movement of the positive and negative
ions is anomalous. The directions of movement of the Cl
ion and K+ ion here are opposite to the standard blood
distribution. Thus, the zero-charge restraint plu.- the
impermeable fixed-protein distribution alone are in-
adequate to establish the desired (Table II) electrolyte
distribution. This observation is significant because
Model D contdins both of the essential elements for the
classical Gibbs-Donnan equilibrium [91, i.e., the fixed-
protein and the charge restraint.
The Gibbs-Donnan phenomenon is usually explained
with an hypothetical system where electrically charged
protein is on only one side of a rigid membrane in a
water solution containing other electrolytes. The zero-
charge constraint is then applied to a compartment on
one side of the membrane. Substances move across the
rigid membrane, in response to both the osmotic effect of
the protein and the cha-gie cozdition, until a hydrostatic
-?2
pressure builds up on the protein side sufficient to stop
further flow. Simultaneously, the electrolytes rearrange
to satisfy the charge condition. The hydrontatic pres-
sure thus acquired by the system is an empirical measure
of the "osmotic pressure" of the system.
In the present case, the membrane is flexible so
hydrostatic pressure cannot build up--the complementary
compartments simply change size. Replacing the hydro-
static pressure which balances the flow, a second protein
is dissolved in the second compartment. Osmotic re-
distribution will cease when the two proteins are equally
dilute, as in Model B. The charge restraint simultan-
Pousiy redistributes the electrolyte, as in Models C and D.
The Gibbs-Donnan phenomenon, however, is basically the
same, and the above results show it is insufficient in
itself to establish the des1i-ed (Table II) distribution
of electrolytes for arterial blood.
The Gibbs-Donnan equilibrium phenomenon does, how-
ever, contribute an important characteristic to the blood
model--namely, the so-cai~ed "chloride" or "Hamuurger
shift" which takes place reversibly a& the lung surface
and in the body tissue capillary beds r'l At the body
tissue level, CO2 dissolves in plasma and diffuses into
m- • 2
-23-
the red cell, where the en~yme czarbonic anhydrase comn-
binies it with A molecule of water Lo become H2 CO Y The
consequent ions LICO 3and H + are, of ctzrte1, subject to
t~he charge constraint of the Gibba-Donnan phenomenon,
just as any cither ion. Conventionally, the charge con-
straint affects the concentration of HCO exactly as it
does Cl ,yielding
[HCO 3J" [Ccl'3Pl k jPl (1)
fHCOi' [cCI]3 ,RC R
where square brackets indicate concentration. In thi.s
case k - 1; Model G below introduces a difference in
solubility for CO 2between plasma and red cells, so k 0 1.
As the concentration ý)f bicarbonate increases in the red
cell, the HCO 3 moves into the plasma under a diffusion
gradient. This movement of H%.0 would leave a net poai-
tive charge in the rehd cell except that Cl , as well as
other anions, then shtfts into the red cell nilitu (the
chloride shift). Thie solvent spccies H 20 MUSt also 3hift
to maint~in Eq. (1). Exactly the opposite set of re-
ac'[kons occurs at the lung surtaco.
-24-
Two important tentetive assumptions derive from this
discussion. First, since Eq. (1) is accepted in the
literature [101 and verified empirically [6,7V, netther
Cl nor HCO is likely to be subject to active transport
(see below, Model E) at or in the cell membrane. If the
concentration ratios of either were effected, Eq. (1)
could not hold over the range of the Hamburger shift un-
less both were affected exactly equally by an active
transport mecnanism, which is unlikely.
The second assumption is that Eq. (1) may be made
much stronger. Indeed, any ion not subject to active
transport must have the same chemical potential gradient
given by the zero-charge constraint, i.e.,
r - F1+ /EScf2 1-[C IHCO 3 PI [+RC fS4P
[Cl- 1 - k =- - . (2)
[Ci _RC - RC L Pi RC
where the hydrogen ratio is inverted because of its sign
and the SO4 ratio is reduced because of it. valence.
The dots indicate sLmilar ratios for other ions, For
example, in the present Model P Na' aid K are not subject
SSee Ref. 8, p. 227.
-25-
to active transport (see Model E) and therefore they, too,
satisfy Eq. (2), as shown in Table IV.
MODEL E--THE SODIUM PUMP
Model E, again a hypothetical experiment. ignort
the electrical charge of the species in order to simplify
a first simulation of the sodium pump. Because the species
subject to the sodium pump have electrical charge, the
effects of the pump and the effects of the 7ero-charge
restraint applied in D would otherwise int%.zdct. In
order, then, to isolate the function of the sodium pump,
the mathematical zerc-charge restraint of the Gibbs-
Donnan phenomena is te-norarily suspended. Ali other
condi.tions--semipermeable, flexible membrane, etc. -- still
apply.
"Sodium pump" is a generic name for the phenomena
whereby living cells, via a biochemical and/or biophysical
mechantsm within or at the surface of the membrane of
tiose cells, selectivety excrete or absorb cations. The
sodium pump is a subclass of the more general "active
transport" processes, the exact me:chaninms for which are
still gererally obscure 111,I2".
-26-
The present Model E concerns not the biochemical or
biophysical mechanism whereby the sodium pump is accom-
plished, but only the net result. As may be seen in
Table II, the sodium pump acting in the erythrocyte metn-
branes causes most of the Na+ and Ca++ ions to concentrate
in plasma, and most of the K+ -nd Mg++ ions to concentrate
!n the interiors of the red cells. The above examples
"rhow clearly that the Gibbs-Donnan forcing functions,
osmotic effects, and zerci-charge condition cannot account
for the disposition of cations in Table II. Additional
work on these cations is necessary.
In order to quantize the sodium pump phenomena, we
assuie that the species Na3+ and the species N +as~neP LASMA 'RED CELL
are not identical. It is convenient to say, instead, that
Na~ + Na +C b (3)P1 C
is a chemical reaction converting one species to the
other, an interphase reaction, even though Eq. (3) is
not a complete reaction in the zhermodynamic sense.
,urther, the increment of free energy, LF, is proportionaL
to the work done on the substance during conversion, al1-
though the actual work in vivo is difficult to measure
-27-
and may differ considerably from this theoretical minimum
vClue required to maintain a gradient." Thus, though we
do not represent the membrane cr the mechanism, we postu-
late a free energy parameter proportional to the dif-
ference in concentrations or mole fractions of the two
species Io Eq. (3) at steady state.
Tables V and VI respectively list the input data
and reproduce the computer results for Model E. Table V
lists the species to be expected in the three compartments,
their respective free energy parameters, and the input
com.nonents from which the output species originate, as in
a chemical reaction.
The "free energy parameters" in Table V have various
appropriate meanings. The 02 species has a free energy
parameter -10.94 in the "Air Out" compartment, and zero
in "Plasma." This number is the log base e of the solu-
bility constant in mole fractions for 02, in plasma,
0.02089 cc/cc/atm. The species FCO" in plasma has the3
free energy parameter -21.35, which is the log bae e
for the mass action con.stant in mole fractiors for the
t eaction"
OH + CO 2 ,- 3 , (4)
See Ref. 2. p. 15.
-28-
Table V
INPUT DATA--ELEMENTARY MODEL
FreeExpecLed Energy InputSpecies Parameter Components
AIR OUT02 -10.Q40000 1.000 02C,)2 -?.690000 1.000 COLN2 - .1•20000 1.000 N2H20 -16.600000 1.000 .4, 1.000 0N-
PL ASNiA02 1. . 00f 02CO2 0. 1.000 CO.N.2 9. 1.000 N2H* 0. 1.000 H*0"- 0. 1.00 O0N-CL- 1.. 0.00 CL-NA*. 1.000 NA*
*. 1.000 K.CAs* 0. 1.000 CA#+MG.- 0. 1.000 "..S04, 0. 1.000 S04-HP04- 0. 1.0W(,J HP04AURFA 0. 1.000 UIR f AGLU LOC 0. 1.000 GLUCOSLACTtC 0. 1.000 LACTICNH4* 1. 1.000 104*0*0- -2?1.1%0000 :1.000 CU2 1.000 0*4-HZCfl) -32.840000 1.000 C02 1.000 .* 1.000 ON-cO6. b.?60000 1.000 COz -1.00a .* 1.000 f)"-"J20 -19.3•0030 1.000 No 1.000 0*-MISCPM 0. 1.000 N1 ICPL
NlE CILLS0? -0. 1.OO, -)0(0.? 0. 1.000 CC?NJ -0. • OOfl 'wJ
*4.~~ .) I00:)N(.44- , 1.000 1)-CL- 0. 1.000 CL-
1.000 *4* •. II f, .00M0 !L A
nc, .1.000 M.W•'ll** ... :'•I ,*rO .0 0 4.
"n04. -0. .(00' "404.0t r, . 1.0W) uR4A
vc, 0 s1.0on GiUCOSLAC TIC I. ".0o LACTIC
"40 1.000 ws..fl~~. ~ 1.000 Coll.0~ 04
""-,,.#V000 1.000 C(0 1.o00 N, 1.000 ta-('01- *.14,0000 1.0 w C.01 -1.O00 ,- I.'000 0"-
' in . . no "* 1.000 0*-I %c v 4". 1.000 of5(31
*49 0.1.000 "64
-29-
Table VI
SODIUM PUl"!--MODET. E
PPELIMIW4ARY BLOOD MODEL W4 CHARGE EQUATION DELAMO
AIR out PLASMA RED CELLS
0-SAR 9.99116f 01 1.49531! 01 1.9819SE 01
PH4 -11. 1.144411 00 1.54481k 00
02 MOLFS I.JISOIE 01 6,9l9SEf-O '..b50331-0901ERAC I.1IA20*-og 2.))4221E-0 2.136221-06
C02 040. 1S ',.16022F 01 6.49042t-0*4 4.1*49?f-04PFRAe. %.1A6011-01 2.,0?99E-os 2.5079ve-os
142 MOLES Y.S4000t 02 2.01113F-04 1.46%441-04PFRAC ?.546S2E-Ot1 .49112F-06 7.493)2f-06
"020 Mlot ES 6.01071f 01 2.64039t 01 1.916901 01Mrfitac 6.10199t-02 9. 94"421 -01 9.945521(-0
If# M0ILES -0. 2.19s13E-00 1.41900#-06MUMA&C -0. 4.14401(-10 6.15407I-10
014- ACL FS -0. Z.ilS35E-01 l.8979SE-01"MFmAC -n. q.5412Z)[-09 q.95721Ts-09
CL- MoLts -0. 4.60?0#f-02 1.39792[-02MPIFAC -0. 1.709261-03 1.709264-03
MA*4 M4OLES -0. ?.)JE0 .447211-01MVUAC -0. 2.t107481-Q) 1.Z61141-0)4
K* Pot fS -0. ).004701-01 4. 20413H-0:MESAC -0. 1.11625F-04 Z.114901-03
CA## pot f S -0. 1.5695.Z#-0) 1.199?&E-04NFRAC -0. S.il,461-4s S.11114009-06
MGK M* oLFS -4). 7.44S§3f-04 11.679026-04ofadc -0. 2.7&21Es-0'1 4.0sisof9-os
S06.. MJUf s -0. .0I1-5 1.216591"-04MfmAc. -(0. 1.11100*-0% 4.111001-01
0004. 0401 fS -0. 6.21$00S-04 6.049901-04M#RAE -0. 1.01111411-0% .339-0s
UstA MOOLE -0. 1.606)01-01 1. MI1W-0)mfe&C -0. G.?08910-009 *.704091-ai
vL4CnM NMOt -0. 1.10,4641-01 l.%s4151-0)
LA #44CLI -0. 7.I0:772-011 1. 6) ?in#-0
N~vkc -m. 4.140941-010 .19516
" "45 I41$s -0. %.W1I~~0 $.S6066"90.
*TA(-4.1.0149#-044 4. VCl59-04k
URBAC~~ -0. G.1%I~ .Varstf-0
6*8AC -c. -0.""-0
" " Is -0. 1 4%0
-30-
or
H20 + CO -2 H2CO3 H+ + HCO3
plus
OH + H H H20
giving
(HCO 3 )= K (5)
(OH ) (CO2 )
Using the free energy parameters for Nap+ and Na+C
in Table V, we may compute from Table VI
In -N' P) VlRC In 9.08l/3.24340 ) 2.1934
(6)
That is, the cations have acquired the gradients given *y
their respective free energy parameters, c . proportional
to AF of Eq. (3). Taking their valence into consideration,
the -ther cations satisfy similar eqtations. References I
-31-
and 3 develop the detailed mathematical background for
the procedures and computations involved here.
Note that the distribution of species in Table VI
is not yet that of Table II. From Table VI, because of
the new disposition of the solute resulting from the cation
pilmps, the ratio of XPL/X/RC is now 26.95/19.88 - 1.355.
The pH - 7.34 is the same for each compartment since H+
ion, assumed to be not subject to an active pump, is
distributed uniformly throughout. Model F again dpplies
the charge constraint, causing tAhe H ion to shift nearer
to that 3f TAble II.
MODEL F--SiMULTANEOUS CONDITIONS
Model F finally applies the conditions of the previous
examples simultaneously, i.e., the cation pumps, the zero-
charge restraint, and the impermeable protein with average
charge -10 per molecule on the plasma proteins.
Ihe input data for this model are, therefore, es-
senitially the same as Table V, except for the eq.iation
which computes and imposes the charge constraint. Table
VII reproduces the computer results for Model F.
An anoinalous shift of electrolytes has impcr,:ant re-
sults. Compared with Table 111. the charge eq*ation
-32-
Table VII
NONLINEARITY- -MODEL F
psEleIqm fillo POART RO.OIH F 61 Ct.*iGr ANDO No "U0 MEA110
its )UT p SPLA$A POE. ILL S
1-444 9.991)6f 02 Of3~I 03 .47vo 03
014 -0. 71400011 00 t.214961 co
02 MOILF, 1.11,019 02 6.11SW-?s01, &t.124691-05wFornc I.110201-01 21 1 $Qi -06 ?. 3142 n 06
co? POOLE$ S.160i./E 01 1'.012of6-04 4.?',ilf-o4
002 POOL S 1.%#40uo( 0? 2.18*61I-0% l.)Z4?SE-04NIBNAC YA'bilE1-01 1.a93.12E-(6 7.49112F-06
M20 0OWS 6.10111F 01 2119921F 01 I.7560o~f (itPFRAC 6..I07q9i-0. 9.9444IE-0l 9.94442t-0a
POO NOlakS -0, 2.9ll-A .15*42F-06EInfac -n. .3Jf-O 1.044911-09
0"- 440t fS -'j . 1.1604OF-07 1.)1Oa61-0v
CL- not~ F C'. S.0661F21- 2. 146)1E-oil
*Ott% -0. 1.1&47i01-01 1.614501-0'
NFAAC -0C. 2.&A0%)1-0, 4b. 110411F-0%
*Ott% -C. 24.1bl*41-oll 4.2s000t-0%
WO,4IC -0. 8.640NIF-OS 2.410506-0)
%0.. OliI' - 0. 4.0901*1i-04 1. 1%916#11621011C -0. f.402,4i-0% 6.%IQu0%t04
#*A -M.* I.411.*t -u I4 . ?966 it -a
USF :x %-0 t*4l' ).14160tS-07
00114C -1.10, 11 t - 0 Ok. PO O t -(A'
a#W -M W
-1-
4 o
At 04% r -1. *41F-1 .
-33-
effect of Table IV drives a small amount of Na+ into the
plasma compartment. On the other hand, compared with
Table VI, the same charge equation effect of Table VII
drives a small amount of Na+ into the red cell compart-
ment. That is, the charge equation applied to Model B
results in a different direction of ion movement than
would the same charge equation applied to Model E.
Thus, it would be convenient if a stress or forcing
constrai.nt applied to the system always gave the same re-
suits. But this wish is naive. The result of a stress
is clearly a function of the system to which it is applied.
The hematocrit of Table VII is now 44 percent, the
pH of plasma is nearly correct at 7.4, but the pH of red
cells is still alkaline (compared to Table II). The model
requires further modification.
MODEL G--HILL'S EQUATION. MYOCLOBIN
Model G, the final model in this preliminary series,
applies three small modifications tc Model F. The crude
outlines of a simulation of Table II are already apparent
in Model F. The general descriptive characteristicq of
live arterial blood at steady state have heen deduced
according to the classical principles of the I.ochemical
-34-
structure--i.e., using the 'ixed proteins and the charge
constraint (to obtain the Gibbs-Donnan phenomena), and
the sodium pump hypothesis. At this point, the model
resembles that of Ref. I which was derived from a dif-
ferent point of view.
The first kind of addition to be illustrated con-
cerns additional interphase reactions--or transport. As
an example, consider thr HPO and SO: anions. The total
of these ions, as acid-soluble phosphate and sulfate,
shown in Table I were computed from Edelman ý13"; the
distribution shown in Table II was estimated from Spector
[1i and f..elman 13Y. The anions considered here do not
include the pho3phates and sulfur bound, for exampl , in
the structure of the nucleoproteins and lipids
In Table VII, these acid-soluble anions, in cesponse
to the charge equation, have a• great[kr mole fraction 10
plasmi than in red cells, 1,,st as Lhe chl.,ride ion does.
in fact, however, most of r-r Iz 1 in whele blood is in
the red cell interior "i". and we presume the same :or
SO, '13. although the evidence for" this is neagei and
somewhat contradictory.
ion selectivity o(. the membrane may cause these
anc•', "o appear mainly in the cell int•.L.r; HPOQ in
-35-
particular may become involved in large complexes with
low permeability, giving a choice of methods to insure
the approximate distribution: An intraphase, red cell
reaction to combine the HPO4 and SO: with an impermeable
species; or interphase, transprt-forcing functions similar
to the scL'um pump. Either process would hold these red
cell anicns in a predetermined prooirtion--a proportion
which wcild vary, however, with the number of other anions
present because of the charge equation and Gibbs-Donnan
effects.
We choose for this example the interphase pump for
distributing the HFO4 and SO:- Thus, in Table VIII, the
free energy peruieteis, c ., for these anions differ by
2.0 between plasma and red cell. As in Eq. (6), thLs
causes a mole fraction gradient between the two compart-
ments: the mole fraction ratio is approximately three,
the mole fraction being greater tn red cells (Table IX).
More detail will be added in later models where the phos-
phate also innizes in a buffering reaction.
The second modification gives the atmosphric gases
a different solubility coefficient in the red cell r.'iliki
than in plasma. Using the data from Roupthton '14'. Table
VIII shcws the coputed differences in solubility as
-36-
lable VIII
INPUT DATA--MODEL G
FreeExpected Energy InputSpec ies Parameter Species
AIR OUT
02 -10.940000 1.000 02
C02 -7 690000 1.000 CO2
N2 -11.520000 1.000 N2
H20 -36.600000 i.000 H+ 1.000 OH-
PLASPA02 0. 1.000 02(02 0. %.000 CL,2
0: C. t.000 NZH, 0. 1.000 H* -0. -0. 1.000 ePLASM
OH- 0. 1.000 of,- -0. -0. -1.000 *PLASM
CL- 0. 1.00C CL- -0. -0. -1.000 ePLASM
0oi 0. 1,00o NAA -0. -3. 1.000 *PLASM
K0 0. 1.000 Y# -0. -0. 1.000 oPLASN
CA++ 3. .."00 CA+t -0. -0. 2.000 oPL•SM
MC 0. i.000 MG*+ -0. -0. 2.000 oPLASP4
S04 0. 1.000 504- -0. 0. -2.000 aPLASM
HP04- 0. 1.000 HP04- -0. -0. -2.000 *PLASM
UREA 0. '.000 UREA
GkUCOS 0, 1.003 GLUCOS
LACTIC 0. 1.000 LACTIC -0. -0- -1.00U *PLASNNH4* +. 1.000 NH4# -r, -0. 1.000 *PLASMmcol- -21.350000 L.000 CO2 1.000 OH- -0. -1.00o *P.ASM
H2C03 -32.040000 1.000 C02 1.000 H+ 1.000 O0-
C03- 6.260000 1.000 CO? -t.000 Ho 1.000 0m- -2.000 *PLASM
H20 -39.390000 1.000 H* 1.000 014-
AIMPR 0. 1.000 NISCPL -0. -0. -10.000 *PLASM
RED CELLS02 -0.490000 1.000 G?C02 0. 1.000 C02Ný -0.50000U 1.000 N2H* 0. 1.000 H*Ox- U. 1.300 01-Cl- 0. 1.000 ZL-NA+ 2.193 339 1.000 NK+ -2.941575 1.000 K+
CA+* 2.?%1790 1.060 CA"#"MG*4 -0.151703 1.OCO MG-*S.4. -2.000000 1.000 S04"
SP04" -2.000000 1.0C" HP04AUREA 0. 1.000 UREACLUCOS 0. 1.000 c.UCOSLACTIC %.. 1.000 LACTICNH4# 0. 1.000 NN4*HC03- -?1.490000 1.000 CC? 1.000 O0-
H2C03 -32.840000 1.000 C02 1.000 H# 1.000 OH-
COV 6.120000 1.000 C02 -1.000 $44 1.000 nH-
H20 -34.390000 1.000 M# k.000 O"-
MISCPR 3. 1.000 MISCOEH@44 0. |.Oi0 M84
HRIO0 -pl..'30000 1.000 02 1.010 H64
-7 -
Tabl~e TX
COMPLETED DISTRIBUTION- -MODEL G
PRELIAINARY OUnon MOOfL %TO S!ATE NO PRnFltIN REACTIONS OflAND
£10 cut PLASMA RED CELLS
X-SAR 9.99121F 02.086144E 01 1.79S96t 01
PH -0. ?.39264t 00 ?.19424( 00
0? MOLES L)149OF 02 b.?1954E-05 6.842SSE-0S
C02 POLES S.26022F 01 6.953001E-04 4.1/*TOE-04MFRAC S.26d4aF-02 ?.40001F-05 2.4OAOIE-O'S
N2 ES I.S4000E 02 2.16367f-0'4 2.21662f-04S?.S4'6S)E-01 7.44139F-0 1.23S4SE-05
H20 MOLES 6.10?6'iE 01 2.e7139F 01 W.S4S98 0'MERAC 6.10791F-02 4~.94440E-01 9.94440f-.
H* MOLES -0. 2.10661F-06 -,.0690SE-ClAFRAC -0. ?.29516f-10 I.1S?06E-Gj9
OH- POLES -0. 3.Ol714E-0? 1,2121IE-07MFRAC -0. 1.06i?3E-OR 6.74909E-09
CL- MOLES -0. ¶S.74290E-02 2.26210E-02MFRAC -0. 1.90892E-03 1.?595SE-03
NA# M4OLES -0. 7.644SE-02 8.3144SE-03MEFRAC -0. 2.64752ZE-03 4.6629SE-04
K+ mnLfE% -0. 2.2q475E-03 4.2?S25E-02"MERAC -0. 7.95704E-05 2t3@OASE-03
CA+* MOLES -0. 1.1798SE-03 2.25S44E-04NERAC -0. 4.778?$E-O% 1.2SlblE-05
MG?4, MOLES -0. 4.02?31E-04 1.14to?6E-03
504.. MOLVS -0. 1.e4.6S3F-O4 3.4.O347E-04MFRAC -0. 6.19i02F-06 1.495O?E-OS
HPO4- POLES -0. 5.029591-04. 9.2704IF-04."WiRAC -0. 1.74.18SP-Ot S.161I6-O5
URFA POLE~S -0. I.q311)E-03 1.104SIE-0)PFRAC -. 6.7Oal9E-04 6.?0S79f-05
GLLJCOS MOLES -'.?.26019F-01 1.4bSS1E-03%tctAC -n. 7.A276461-0S 7.82764E-05
LACTIC MOLFS -0. I.4.5S4OE-01 5.74497E-04.PFRAC -'. .O%1zQE-O% ).14"RA(-os
NH'.. POL tS -. 4. 30i4Of-04 4. IO4VE-0'.MFPAC -0. I.4S830-05~.?%E0
HC03- MOLE S --0. 1.)R678S-15? . 2J If -O0)MFRAC -0. 4.80280F-04 1.49R459f-04
H2COý MOLES -0). 9.863I1F-07 6.15040~-OfMFRAC -0. 3.412459E-08 1.4%>.ý)t-08
Col. MOLES -0. 1.910118F-05 S.' AeeE-06MFRAC -n. 6.72284.E-07 k.tiulit(0?
MISCPR MOLE S -0. 4.706u00-04. I.?S05j0t-MFRAC -0. 3.01S12l-0% .. OS0Es
HS4 MM F S -0. -0. ).35)[email protected] -0. -0. 1.366HE-04
H6402 NOL 15 -0. -0. 8.1546)t-0)MFRAC -0. -0. 484AE0
-38-
modified, free energy parameters. The result is the
solution of slightly more 02 and N2 in the red cell milieu
than in plasma. Reference 2 computes these data.
Finally, this preliminary model incorporates an
illustrative reaction of hemoglobin with oxygen. Hemo-
globin oxygen~ation has an extensive literature (summarized
in Ref. 14) and is, of course, still the subject- of much
research. This preliminary. illustrat*ve model incorporates
one of the earlier theoretical structures of the oxygena-
tion reactions, Studying a simpler theory and its short-
comings clarifies some of the details of, and much of the
motivation for, the following two chapters.
In 1910, A. V. Hill [15] proposed that reaction of a
solution of hemoglobin with oxygen could be written
Hb + nO Hb(O2) , K (7)2 z n
where both K ard ki were to be determined from empiriccal
data from the laboratory. Using the Hill equation, as in
Ref. 14, the percent saturation of hemoglobin with oxygen,
y, can be computed from
100 y n (8)1 + Kpn
<$9-
where p is the oxygen pressaure in millimeters. The
saturation can be measured in the laboratory at various
p0 2 by a variety of means. Comparing the laboratory
experimental curve to that given by Eq. (8), one can
attempt to "fit" the laboratory curve by adjusting K and
n [14].
The discussion in the literature centers around the
fact that unique K and n cannot be found to fit the ex-
perimental curve over the entire range of oxygen pressure
from zero to, say, 150 millimeters. A deterrent to fit-
ting the curve under general conditions (i.e., physiologic)
is the difficulty of determining a satisfactory laboratory
experimental curve, particularly at the extreme values of
oxygen pressure [14].
But, for purpcses of this comparison, the Dill curves
[167 are satisfactory. They have been the standard for
many years [7], and were obtained under conditioi.3 reason-
ably well-controlled for the time.
Table VIII has, in the red cell compartment, a reac-
tion of oxygen with hemoglobin (cf. Table V). The re-
action constant, K, is determined in the model to give a
normal saturation (97 percent) at 100 mm 02' FigUre 3.
shows two cases (computed by varying the r0 2): two curves
-40-
(N
_ - _ _ _ - _ _ _ _ _9
_ _ _ 0
cE
0
rol C____ : jad) -io~o a uo__ 0 E
-41-
for the case n = 1 (as in Table VIII) and a curve for the
case n = 2.5 as Hill originally determined. The normal
Dill curve also appears. The Dill curve, of course, was
determined undet general physiologic conditions.
Hemoglobin has four binding sites for oxygen. So
for the case n = 1, in order to bind one oxygen per
molecule of hemoglobin, the hemoglobin was conceptually
broken into four parts, each with a heme, as in Eq. (7).
Thus, for the case n = 1, 9.09 rather than 2.2725 moles
of hemoglobin combine with oxygen. The reaction constant
for this case is K = 4.98x0-3 , or n K = K' = -5.30.
Reducing this constant immediately decreases the maximum
saturation without materially shifting the curve toward-3
the Dill curve, as for the curve with K = 6.74xi0 ,
K' = 5.0. The pH is 7.39 in plasma, 7.19 in red cells.
The Hill curve with n = 2.5 uses the proper average
number of moles of hemoglobin, 2.2725, per liter of blood,
but the fixed average number of oxygen molecules, 2.5,
attach to each molecule of hemoglobin. Thus, each hemo-
globin, Hb4 , has either 2.5 molecules of 02 attached or
none at all. There are no intermediate stages. An equi-
librium constant of K = 1.51x0 g, KI 2 -11.1 is necessary
to give 97 percent saturation at 100 mm 02 and the same
pH as before.
-42-
Some of the data for Fig. 1 (the n - 2.5 and n - 1,
K - 5.3 curves) are verified in Ref. 14 where similar
curves were computed by desk calculator (the values of
K are not given). This reasonabl:, validates the present
model under varying conditions. But the discrepency
between the empirical Dill curve and the Hill curves
necessitates better hypothesis and in improved model.
Subsequent biochemical hypotheses for the oxygenation
reactions are increasingly sophisticated [2] and, also,
the presence of CO2 and Ii+ are important. Hemoglobin
binds CO2 in the carbamino reactions as well as buffering
the H+ ion; both of these reactions effect oxygenation.
Reference 2 discusses these problems.
-43-
Chapter II
CONCEPTUAL COMPARTMENTS
1. INTRODUCTION
Chemically speaking, by far the most interesting
constituents of the blood are the proteins. These species
along with other complex constituents--the organic phos-
phates, the lipids, the carbohydrates--can combine with
other blood constituents potentially to create innumerable
biochemical reaction products. However, such a model, with
lists of possibly thousands of reactions, would be ex-
tremely unwieldy; and the thousands of necessary micro-
scopic reaction constants probably could not be determined.
The problem then is to make a model which approximates
the more complex real situation but, at the same time, has
exactly the same characteristics as the large model wherever
possible.
The folInwing elementary example illustrates this re-
mark. Consider a polypeptide having two identical but in-
depen'etr ionizing groups--i.e., the ionization of one does
not affect the ionization of the other. Thus, a representa-
tive molecule HRH has three ionization states and four
reactions
-44-
HRH H" R-- + H+, k 1
HR- -R- + H k3
-R. -R- + H+, k4
each with the intrinsic microscopic dissociation constant
ki o.: intrinsic association constant i/k But if the paths
of ionization are immaterial, one ran instead write [81
HRH -(HR--+ -RFH) + H KI
(2)
(--RH + HR--) (--R--,) + 'H, K 2
with constants K1 and K respectively, where KI and K2
are the apparent titration constants obtained from a
Laboratory titration curve, and HR- and -RH are indis-
tingui-hable, i.e., a single species. The constants ki
and K. can be related by the following reasoning: Con-
sidering the several mass action equations for Eqs. (1)
and (2), we may write, as in Ref. 8,
-45-
K - (-RH) + (HR-)] (ii+) . (-RH) (H+ H- +
1 (HRH) (HRH) (HRH)
(3)
1 1 + k2
-nd
KC-R--)+-Rl) + (R..)__2 (--RH) + (HR-)• +
(--R-- (H+)
(4)
- -I
k k3 k 4
The two chemical systems, Eq. (1i and Eq. (2), are thus
completely convertible, and equivalent in the sense that
the computed ratios of concentrations for (-S-)/GIIRH) at
equilibrium are identical, as are the concentrations of
(H+) and (-HR + HR-). The first case, however, requires
four reactions to represent the system; the second, only
two.
Tiis and otler types of reductions to be discussed
have a cormo,, 'perty: They '-ed,:ce the size of the
problem but not the simulation accuracy. Such reductions
.46-
are important in the computer model., since both the com-
puting time and the size of the computer req,iired for
a sclution increase nonlinearly with the problem size.
In biological systems, a problem could easily be too
large--i.e., have too many chemical reactions in its
representation--for even a reasonably large computer.
in the blood, serum albumin has 190 sites for active
hydrogen ionization, each of which falls into one of four
:lasses. Thcse classes are independent but not mutually
exclusive; i e.. ionization in one class does not prohibit
ionization ir. another. Because ionization can occur
simultaneously in all four classes, they monst be con-
sidered as simtultaneous events thoug;h not with equal
probability. SincL each site has two poss;ible states ar.d
there are 99 sites in the first class, 16 in the second,
99 Irt,57 in the third, and 13 "n the fourth, then 2 , .257 18 0602 2 1 10 states are pusSible--each of whilch must
be represented in the model. Obvioti31y. some reduction
is necessary.
While an equivalent representation can red(.ce the
size of complex systems, as in Eqs. (1) and (2). con-
ceptual coro)artmentalization is generallv an even -oore
economical device. As will be shown, the c.tnceptual
-47-
compartments are artificial compartments created in the
chemical milieu of the model to isolate classes of re-
actions on complex molecules when these reactions can occur
simultaneously. As in the example of serum albumin, the
reactions of independent classes may occur simultaneously--
thus, the same molecule must have multiple representation
to show all possible configurations. The procedure sug-
gested here is to identify and characterize the myriad
chemic, . reactions according to class (type of ligand)
and interdependence (mutually exclusive, dependent, or
independent), in _rder to represent each class of reaction
4s an isolated event rather than representing all possible
"Ltd-class species. Since the classes of reactions are
much fewer than the total possible number of species, the
reduction can be considerable.
-48-
2. CONCEPTUAL COMPARTM .TS
PROTEIN IONIZATION-- EXAMPLE
The blood macromolecules of principal concern are
serum albumin, the various globulins, and hemoglobin.
Because serum albumin is perhaps the simplest, most
stable, and best known, it will be used extensively as
an example. This protein is the most abundant of the
plasma proteins, about ten times more numerous than the
next most abundant. The pertinent characteristic of
serum albumin is its ability to buffer hydrogen ion in
the physiological range of pH, but its activity is by no
means limited to proton binding. Serum albumina also binds
small ions of either sign and even small neutral molecules.
To simplify the description of serum albumin for the model,
we will deal exclusively with thw hydrogen binding activity.
Tanford, et al. [17' and Tanford C187 (sources for
much of the factual data ork albumin) list among the
hydrogen-binding sites of serum albumir, 99 carboxyl
radicals with average pK w 4.02 at 250 C. The carboxyl
groups alone have 99 configurations of the ifflecule in
which it has lost just one hydrogen, 99098/2 configura-
tions in which it has lost two iydrogens. and so on, in
-49-
all 299 species in the milieu. But several ways exist
to reduce this variety to an approximate or an equivalent
system.
Assume first that the 99 sites are equivalent and
independent, that the ionization of one site will not
affect the probability of ionization of any other. An
equtivalence can then be drawn between the L-nization of
this polybasic acid and 99 monobasic acids, each with the
same pK and the same concentration as the protein. But
this equivalence is immediate if the protein is concep-
tually split into n parts. If fi is the number of moles
of hydrogen released per mole of protein per site then,
as for a noprobasic acid,
RH R- + H + k (5)
or
k -1 h (H+) (6)(RH) 1-;
where k is the microscopic dissociation constant for that
particular site. If this were the only site for ioniza-
tion, then cleariy th. ideal titration ciistant decermined
in the laboratory would be k. For n identical, independent
-50-
sites on each molecule of R, n times as meny protons will
be released at the same pH, but of course thn ionization
pK remains the same, i.e., -log k. Also, A nh is the
moles of protons for all sites per mole of R. Therefore,
A
n~l-Fi) (RH)
or
k--- (H),n-D
where, again, the empirically observed titration constant
in vitro would also be simply k.
For the similar case of n moles of rmonovalent acid,
each with the same concentration and pK as for R, the
titration constant will equal that of the protein, i.e.,
either system will buffer maximally at the pK of the in-
trinsic ionization gro,.ips. Then, in titration or buffering
action, the two systems would be indistinguishable except
for electrostatic concentration effects.
-51-
We note for future reference the difference between
the titration constants and the dissociation constants.
For the monovalent acids, these two constants are the
same and are given by the mass action equation, i.e., the
pK of the ionizing hydrogen. For the protein, the pK of
the icnizing class (Eq. (6)) gives the titration constant;
but the dissociation constant, given by mass action
equations where the protein is not conceptually split,
depend upon the number of hydrogens dissociated (and
perhaps the particular configuration of their dissocia-
tion). This difference is important because the apperent
mass action equation constant for a protein may be dif-
ficult to measure, whereas the titration constant can be
determined from the in vitro titration curve.
This discussion implies that, under the strong assump-
tion of equivalence 4nd independence of the groups of e
given class, for application to the computer model, the
simulation of the protein could be obtained by writing down
the chemical reaction, Eq. (5), and inserting this in the
model with constant k and in amount n times RH, i.e., n
monovalent acids. In fact, the simulation is not quite
so simple: the milieu would contain n times too much
protein and the osmolarity would be in error.
Ref. 8, Chap. 9.
-52-
However, in other respects--e.g., the buffering re-
action--this simulation would be perfectly valid. There-
fore, the model may reasonably contain an osmotically
isolated compartment for the monovalent acids. This
conceptual compartment will be impermeable to all species
except H+ ion and will not be counted in the osmolarity--
it will be a sink or source of H+ ion, a place to store
H+ ion effectiv, ly out of the milieu in proportion to the
pK of the monovalent acids which are availaule only with-
in the conceptual compartment.
The example of serum albumin in Sec. 1 permits more
soecificity. Table X gives five major classes of ionizing
sites on the protein, each with a distinct number of sites.
There are 17 imidazole groups, 57 '-amino groups, 19 phenolic
groups, and 22 guanidine groups, so that, as on p. 46,
approximately 1060 species are possible and an equal number
of intrinsic conscýnts could be required for a chemical
description,
The method of conceptual compartments replaces each
ionizing class with a single monobasic acid having the
"average" pK of that class, but with n times the mole
number of the protein, where n is the number of sites in
the class. Then, to not disturb the osmolarity (or the
-53-
Table Xa
TITRATABLE GROUPS PER 65,000 GRAMS SERUM ALBUMIN
Found by Amino AcidTitration Analysis
a-carboxyl (1) 1
0, y-carboxyl 99 2!89.9
Imidazole 16 16.8
*-amino (1) 1
c-amino 57 57.0
Phenolic 19 18.1
Guanidirne 22 22.0
Sulfhydryl (free) (0) 0.7
aAdapted from Tanford, et al. 179 .
total mole count), an artificial compartment is created
to contiin the nonobasic acid. This compartment has but
twu species, the ionized and un-ionized forms. The dis-
tribution between these two species is determined by the
pK of the acid and the H+ ion concentration of the liquid
compartment.
Thus, the serum albumin problem is reduced to a
representation containing just clever species: the protein
in the main compartment and two species in each ionization
-54-
class compartment. The remainder of this section shows
in what respect this problem with eleven species (plus
the remaining species in the liquid phase, such as Na+
Cl, OH, H 20, .) is equivalent to the e'trigin3l problem
of 1060 spezies, where "equivalent" means that the solu-
tions of these two problems will bc indistinguishable.
Shapiro r4' develops and proves in general an
equivalence theorem adaptable to the particular case of
serum albumin. Suppose, first, that the ionizing sites
on the protein are divisible into classes (e.g., carboxyl,
alpha-amino, etc.) and that within each class each ionizing
site is indistinguishable, the order of ionization im-
material, and the free energy associated with ionization
is strictly additive (i.e., the ionization of one site
does not in any way affect the 4onization of any other).
Second, suppose this entire protein-carboxyl system
is replaced in the mathematical model with a single mono-
basic acid having a pK identical to the equivalent hydrogens
of the protein but 99 times the number of moles of the
protein. Pre3erving the ,smolality requires a new phase
or conceptual com.•partment containi~ig only the monobasic
acfd in its tvo states, ionized and un-tonized the pro-
tein itself remains unreactive in, the original milieu
-55-
(again, to keep the osmolality correct). The protein and
the monobasic acid are related by a conservation of mass
equation.
Shapiro then shows that under these conditions the
total number of moles of ionized monobasic acid will
equal the number of ionized sites in the original large
problem, although the number of species has decreased
from 299 to just two. The remaining species other than
the protein and monobasic acid will be unaffected in
concentration or distribution. Further, the total thermo-
dynamic free energy of the system will be unaffected. We
forego the details or proof of the more general theorem.
This model then proceeds similarly for the other
classes of ionizing sites on the protein (Table X). The
pK of each class comes directly from Tanford, et al. [171,
who determine the pK from the titration curve with a
correction for the static charge on the molecu).l and for
the adsorbed chloride ion. Reference 19 discusses this
computer model in detail and shows th&t it reproduces
the titration curve for scruz, albwtin wl.thin two ionizing
sites over the range of pH from 3.5 to 10. The extreme
ends oi the curve require a more complex model.
-56-
3. BLOOD PROTEINS: PLASMA
The power and flexibility of conceptual compartmen-
talization permits further development of the elementary
and insufficiently accurate blood model of Chap. I. The
simulation will continue by incorporating further com-
plexities of the hemoglobin reactions and certain buf-
fering systena of the blood.
",m one view, the principal buffering system of the
bluod ,, the bicarbonate. With the loss of each CO2
rAIecuie at the lung surface, a free hydrogen ion is ab-
sorbed in solution by the OH ion from the decomposing
bicarbovate ion, thus making the blood mtore basic. At
the tissue level, the absorption by the blood of each CO2
molecule creates hydrogen ions, making the blood more
acid.
Thvse two sytems--twe production and consumption of
H ion by the bicorbon;tte react ons--need not be and pee-
hpps seldom are in exert balance, thus cakvising :espih.tory
acidosis or alk•noxi. However, the very low Co0, content
of the atmosphere retlative to the hblood permits CO, to he
ea ,cl e•rvei -. , the ltn•.s; the autonomic controls over
deptO -'t ract, -.:t -eirijation and blood klow rate tend to
balance prodiction ard absorption of H+ i:)n.
-57-
The necessity for this buffering may be eisily shown.
At a steady resting state, blood coursing through the
tissues picks up about 40 ml of CO2 per liter in changing
from arterial to venous. This is equivalent to the re-
lease of about 2 nmM of hydrogen ion, since CO, either
combines with H2 0 to form HCO0 and liberate a H ion,2 3
or forms a carbamino compound with hemoglobin and again
liberates almost one hydrogen per reaction. Since the
hydrogen ion concentration is normally only about 107
moles per liter of blood, this tremendous excess acidity
must be buffered. The net change in pH during the normal
transformation from arterial. to venous blood is only about
0.03 pH units.
Hemoglobin itself takes up approximately three-
quarters of the excess hydrogen in the oxylabile binding
of hydrogen on groups which become avoi.lable on the :wlecule
as the oxygen 1s removed at the tissue level. The oxygen-
ation of hemog-obin A'-.elds this hydrogen at the lung leval,
thus preventing tht blood from going basic from the loss
of CO2 .
The remaining excess. hydrogen at the tissue: level is
buffered princtp4.ly by the p*asmv proteins. the pwhosphate
ionizatioa systems, and bf the Pon-oxylabile sit- cn the
-53-
hemoglobin [20]. Plasma proteins, however, may account
for a major portion of the remaining excess, since the
imidazole groups of the serum albumin are at their maxi-
mum bufferine power in this important range of pH of the
viable system. Peferring again to Tanford [171, one
riillimole of serum albumin buffers approximately one
tillimole of hydrogen in a change of 0.1 pH point. Since
I:he pH changes about 0.03 points from arterial to venous
blood and since there is approximately 0.5 mM of serum
albumin per liter of blood, the serum albumin could buffer
up to 0.1.5 mM of H + durtng this change.
incorporating the plasma protein buffering system
into the rmathematical model requires modifying the gen-
eralized model of plasma protein described in Chap. I.
Sertm albumin accounts normally for 80 or 90 percent by
mole nuwnber of the total plasma protein, nd the various
globulins for most of the remainder. Fibrinogen is alsopresent, but I. nit thought to be an lmporrar buffer.
We assu•.•6c that the two principal proteins represent all
crf the important buffering power of the plasma fraction,
.ind have sim'lar ;itrf-tion curves, particularly in the
middle, viable range of pm. rhis latt•r assumption ig
not quite true 718,217. but considering the small
-59-
proportion of globulin and the reasonably small dif-
ference in the titration curves in the viable range, the
error will be insignificant--not more than 2 percent of
the miximum buffering power of the plasma proteins.
Finally, the titration curve itself is more sophis-
ticated than indicated in the previous section. fanford
[17] used a solution of serum albumin with salt added to
provide an ionic strength 0.15. Thus, the titration curve
incorporates the electrostatic charge on che molecule of
protein and the comn'. tive bindLng of chloride ion, The
pK of the hydrogen-bindinL; sites thereby varies appro-
priately through the raoige of pH. References 19 and 22
detail this problem. Of course, the behavior of the
protein in plasma may differ greacly from its behavior
in the 0.15 rmolar NaCI titrating solution. However, this
error can temporarily be discounted, since Chap. I1. will
demonstrate with Astrup dlagrams and ,iore stringgent tests
that the simulated biood doe• buffer properly.
Setrum albumin binds anions, cations, and even un-
charged particies by incompletely tinde str.od mechanisms -8'.
However, Cl probably binds to the posit',vely charged
amine Froups; this reaction is a function •f the pli and
the concentration of clitoride ioni. Scat~hard. et ai. "23.
-60-
for example, discuss the binding of Cl", which is iim-
portant in the respiratory function of blood; Loken,
et al. '241 treat the biniding of calcium. These reac-
tions change the net charge on the protein molecule and
thus affect the buffering power of the protein; but by
tying up permeable ions, they also affect the Gibbs-
Donran equilibrium. Neither of these systems are ex-
plicitly in effect in the model, where serum protein may
bind up to 50 percent of the calcium, and up to 10 ions
of Cl per molecule of protein, depending upon the pH.
The binding of these ions by other proteins has not
been studied as closely; in pz.rticdilar, hemoglobin's role
is net wel! u,•eratoo6. We presume that hemoglobin and
other complexes withir the ýell may also bind these small
lon'. Altbough v.e 'mty assume that none of the single-
valent alkaline cations are bound, both the double-valent
Mg and the double-valent anions are tied up in various
ways. Lic lizcrature regardi•rg these intracellular re-
actions is not as definite as for CI and Ca, a.d no
actempt has bten made to account for them.
-61-
4. BLOOD PROTEINS: RED CELL
Hemoglobin is only one important constituent of the
red cell. In fact, the water content is approximately
99.45 percent of the total moles; hemoglobin is only
about 1.5 percent of the remainder, or about 2.30 mM per
liter of blood. However, hemoglobin is more prominent in
terms of mass: each liter of blood contains about 160
grams of hemoglobin--all in the 450 cc occupied by red
cells. The hemoglobin protein, minor impermeable con-
stituents, and the membranes of the red cells (which are
very thin) increase the volume of the cells over that of
the pure water content by about 120 cc per liter of blood.
The physical-chemical structure of the red cell
interior way orient the hemoglobin molecules with respect
to each other, perhaps linked in a fluid crystalline
lattice. In any case, we assume an homogeneous fluid
throughout, with reasonably steady composition states
(within measurable error) when -.he cell environrrent is
constant. A change in the environment resultii in a change
in the cell interior--i.e. , in the rate of it3 mect -nisms--
ond consec.jently a new steady state.
Resides hemoglobin, inorganic electroiyteg, organic
permeable molecules and ioos, and water te red cell
-62-
contains a complex of interesting and active substances,
most of which are charged. The kinds of these particles
(organic phosphates, enzymes, chelates, etc.) are so
numerous and their specific contributions :so varied that
they are usually lumped together in the electrolyte balance
as simply X
The milliequivAlents of this total are more easily
calculated than the actual millimoles. By assumirng that
the red cell milieu of a given volume of arterial red
cells is neutrally charged, one can ccmpute the total
charge attributable to X -- about 10 milliequivalents of
negative charge per liter of blood. Then, since the
osmolarity of the milieu should equal that of plasma,
and the males of each other substance are report-J, one
can determine that there should be about 7.5 millimoles
of X, only an approximate calculation.
From this, however, the average charge per average
molecule is between one and two, about the average charge
of inorganic phosphate or of a hemoglobin molecule near
the viable pH. Accordingly, the mathematical model carries
7,5 mM of X per liter, but (as Chap. III proves) for the
present model the net charge need not be specified. This
-63-
is a temporary procedure for handling the mixture of
substan -,s denoted by X , whose de:ta:.Is may be added
gradually to the model as specific reactions data become
available or important. However,, the situation for
hemoglobin differs considerably, since its details are
important and relatively well-known.
Hemoglobin's chemical reacti.mns important to respira-
tion are the reversible combinations formed with:
1) Oxygen (via the four iron atoms in the hemegroups);
2) Carbon dioxide (via --Nl groups, probably those
in heme and lying near the iron atom);
3) Hydrogen ion (via the -COOH, --INH2 , and -- NH3
groups, a significant number of which are oxy-labile in their ionization; i.e., the oxygenattachment affects the bonding energy).
The Adair [251 equation for the oxygen dissociation
of hemoglobtn under standard conditions--viz.,
_• (alp + 2a8p2 + 3a 3p3 + 4a 4p4 )tOO 4(a+a~~a~ 2 + 3 p3 +a 4 p4 )v 100 4(a 0 + a 1P + 42P 2+ a +a I4
where
- 1- - - - . --- -__ - ---_
-64-
y percent saturation of hemoglobin,
a0 = 1,
(8)•j• -•K.
j i-1
p - oxygen pressure in millimeters,
and
[Hb 4 (0 2 ) 19 - Ki p[Hb 4 (0 2)i:I" (9)
-- is used in this modcl, as in Ref. 2, for the generation
of the empirical saturation curve.
Given an empirical curve, y vs. p, from the labora-
tory, thie analytic problem is to find the unique set of
four parameters, the ratios ao: a1 : a,: a 3 : a4 , or simply
the apparent Kj, that enable Eq. (7) to reproduce the
cur,;• within the model. Because of the oxylability of
the ancillary reactions of hemoglobin, however, these
apparent Kj may differ from the intrinsic Kj of Eq. (9).
In this circumstan~ce, Pauling '26' correctly wrote Ki
for the apparent c:onstants in the milieu and Kj for the
intrinsic constants of "pure" hemoglobin. The solution
of this iripasse lies in finding analyticel transformat. i.-ns,
-65-
based upon structural, physical, and chemical hypotheses,
which will carry one set of constants into the other,T
K -K1
The apparent macroscopic K,
,21'
KI
Hb4(O2)j.1 + 02 - Hb4 (0 2 ) j for each j , (10)
are not identical, and we have
-.Ly I00
Kip + 2KIK~p 2 + 3KiKKp 3 + 4K1K2'K3K4P (
41+ KIP + K;K~p 2 + K K;K;p3 + KIKK2K3
for the Adair Eq. (7).
Reference 2 discusses the use of the computer in the
derivation of the oxygenation constants. This study
utilizes the ratios derived independently on the computer
and by Roughton, et al. 727,287, namely. i : Ki : K; :
K' :: I : 0.4 : 0.24 : 9.4. The absolute values are ad-
justed to give the. proper saturation of hemoglobin at p50,
the oxygen pressure for half-saturation. Chapter III
-66-
discusses the quantitative goodness-of-fit for these
conebtants.
This simulation assumes that the transformation T
is affected only by the ancillary oxylabile carbamino
and hydrogen ionization reactions. In deriving T, we may
reasonably assume (141 two oxylabile hydrogen ionization
sites per heme group on hemoglobin, and 16 sites per heme
for non-oxylabile ionization. Identification of the
position and kind of these sites on the actual molecule
is not yet definite [141, but for convenient reference
this simulation classifies the oxylabile sites as a
carboxyl and an alpha-amino, the non-oxylabile as iden-
tical ligands on histidine groups. Further, we assume
that the pK's of the oxyyabile groups are sufficiently
far apart that only three species with respect to the
oxylabile heme groups appear in the milieu:
K KCOOH COo - COOR R R' (12)N~3 3
where R is a heme group.Evidently, the CO2 molecule is bound only by the
amrbaitati sn cie on the rh*.ht of Eq. i12), with the
-67-
amino in the NH2 condition. The apparent constants for
oxygenated and reduced hemoglobin differ, and therefore
the equations for the model are
Hb + CO2 ' Hb(CO 2 ) + H+ K - 6.4x06 (13)
and
HbO2 + CO2 " &bO2 (CO2 ) + H+, KOC - 2.5x10"6 (14)
where the symbol Hb indicates a 1/4 hemoglobin molecule,
and/or a reactive group on an amphanionic heme. Sim-
ilarly, the pKi for hydrogen ionization will have sub-
scripts referring to oxygenated or reduced hemoglobin,
and the apparent pKi may correspond to any number of H+
equivalents ionized. The equations relating to the first
stage of oxygenation are:
Hb4 + 0 - Hb402 K'
HbP Hb + H , -5.25.4 4
Hb4 KHb4 + , pKRIH 7. 9 3 (15)
HbO .- Hb - - 5.7542 42 pK2 457
-4 2
-68-
where KI in the first equation is the intrinsic reaction
constant for the attachment of 02 on any of the four hemes.
Ki is the apparent constant measured in t:he laboratory in
the presence of H+ or CO2 reactions. Equations (13), (14),
and (15) specify the interrelated ruactions with respect
to the first oxygenation state. Computing the ratio (in
moles) of oxygenated to reduced molecules yields (sup-
pressing the square concentration brackets)
b4 02 -K'ro IHb14 0 21
Hb 0+ Hb 0++Hb 0- + Hb 0 C74 2 4 2 4 2 4 2 (0)16
b +Hb + +. b- (16)4 4 4 ~4 2
Alternatively, using the equilibrium constants from the
individual mass action equations, Eqs. (9), (13), (14),
and (15), p0 in millimeters, and concentrations of each2
species (suppressing the square brackets). yields
+ K rn KH K) 01
Hb 0, lp + KIP K p + Kp±-K
Hb4 H + K COi KO0Xa H- H+ + KRC
-69-
or
H+ KO1 CO2
1-+ + - + K KKOz (H+) 2 OC 01
H +_ __ 4- 1+ CO° (17)'KRZ (H' ) 2%K 0
where numerator and denominator are divided by rHb4, and
multiplied by 760 mm. Thus, a complex ratio modifies the
intrinsic conatant K I to give K, the apparent conscant
for first-stage oxygenatLon in the presence of ancillary
reactions. This result, Eq. (17), resembles derfvations
elsewhere, e.g., Pauling r-26";.
Equation (11), corresponding to the Adair formu'ation
for the dissociation of hemoglobin--i.e.,
•p+ ,3 . 4ajp + 23ýp + a;P3 + 4a 4P
a + *P + )+ +
-- shows the constant a' " K" NOW, if
-70-
a 2 - K' K2'
a3 K' K' K;123
a' - K' K' K; K'
wheve the KI are the apparent j th-stage oxygenation con-
stants in the milieu of reactions, then
+ +KO + C12 KO0"-KOz 1 H+ C))2 O
a.' a ji- H12 1O (19)H + I CO 2
+ I + -+ KOTRH (H+)"
gives a for j-1,...,4, the intrinsic Adair constants, by
an argument stmilar to that yielding Eq. (1).
This mathemattcal model incorporates Eqs. (19). A
different model may result, howevzr, if the relations of
the several reactions of heuoglobin are eventually ,!o-
termined to be dependent in other ways. some of the H÷
ionization must be oxylabile, but certainly not all; and
-71-
H+the oxygen absorption will depend on H lability. Simi-
larly, remote areas of the protein and heme groups may
bind CO2 as non-oxylabile carbamino, although carbamino
formation always depends on the pHl. Also, some of these
reactions may be mutually exclusive--e.g., the CO and 0
reactions, and the Ca++ and H1+ reactions, not considered
here.
In the format of the mathematical model, each stoi-
chiometric chemical reaction requires a separate entry in
the matrix of reactions. Thus, if five ways exist for
02 to combine with Hb4 and four ways for H+ ionization,
and if these reactions are not mutually exclusive, then
20 entries would represent the possible combinations.
However, conceptual compartments convert this "muliipli-
cative" miodel ii,to an equivalent "additive" model of r-ine
entries. In addition to the principal red cell compart-
ment, the latter model computes carbamino reactions and
the H + ionization of reduced and oxygenated heme in
separate but dependent conceptual compartments.
Equations (19) relate to a model which computes the
carbamino reactions of oxygenated or reduced heme in the
corresponding ionization coryartments. This procedurc.-
is efficient and does not yield multiplicative combinations
-72-
because Hb +, Hb, Hb , and HbCO2 are assumed to be
mutually exclusive species, CO2 combining enly with the
amphanion form. Also, the present model regards the
product of Eqs. (13) and (14) as completely ionized, an
assumption which is probably not much in error at viable
pH.
Finally, the computer program incorporates Eqs. (9),
(13), (14), and (15) as lists of the expected chemical
reactions and species in appropriate compartments. Table
XI shows the red cell and subsidiary conceptual compart-
ments for the current mathematical model. In the red
cell, the Adair formulation has four stages of oxygena-
tion of hemoglobin. The coefficients of the terms re-
duced hetme, oxy heme, and histidine give the number of
equivilents assumed in nj moles of hemoglobin for sub-
sequent reactions in the conceptual compartments. Thus,
the coefficients in the red cell compartment count the
taoles of reduced and oxygenated hemes and the number of
oxystable H+ ionization sites, which are assume(' to be on
histidines. The two subsidiary ccipartments each contain
the two stages of ionization and a carbamino reaction.
The amounts of reduced hem., oxy heme, and histidine
-73-
U0 U 0 0 0l 0A.00
2 X - x x
000,~0n0000 " 0000 0
08080000 0 oo
o (000 0 00 00 0 00
0 0 0 0 0 0
1-4 I I t II I
x 0CN
c' 0 0 0 CO 0 00 Cz o 0
8 C: C I ' 0 0z C;
z
0 C
0-4
tI- -~ 4.h
A6- .~ . 44
* 44 5
-74-
srecies in the conceptual cotrvartments derive from the
red cell via conservation-of-mass equations called
"transfer rows." In all case3, the free energy parameters
e--e the natural logs of the several pK's in the mole
fraction scale which give the best fit to the hemoglobin
titration curve and to the amount of carbamino formed
in the oxygenated and reduced cases.
It now remains to show that the results of the con-
ceptual compartments and transfer rows in the model are
simply mass action equations similar to Eqs, (15) and
(17); i.e., that H+ ion and CO2 reactions with hemoglobin
have the hypothesized effect upon the oxygenation
reactions.
Write the mass action equations from the model
treating the transfer coefficients as a species (see
Table XI). Consider the first-stage oxygenation of Hb4 ,
the row "Hb402" in Table XI:
(Hb4 02 ) (RHeme) 3 (OHeMe) -15.7665
(Hbb4) (02) e (20)
-75-
where the symbol RHeme stands for a reduced hehne of which
Hb4 02 contains three, and OHeme represents an oxygenated
heune. The concentrations are in mole fractions and the
exponent is log base e of the equilibrium constant, i.e.,
the intrinsic constant K.' Equation (20) is, the.n, the
first-stage oxygenation with extra terms which transfirrl
the apparent oxygenation (as measured '•y the actual con-
centrations) into the intrinsic equilibrium. Equation
(20) shows this more clearly if written
(Hb4 02 ) 1 -15.7665
(!4) (0~ 3 21"(1I 02) (RHeme) (OHeme)
where the concentrations on the left are measured.
Investigating the nature of the transformation on
the right side of Eq. .'21) via the mass action equations
for the chemical equations in the conceptual compartments
(Table XI) yields, comparing Eqs. (15):
K+(RHeme)OH' -22.4397 (22)S(H 2 Hb )
where
-76-
(RHeme) - (HHb ) (the zwitterion)
S(b)(H -21.0582KRI (RHeme)
(.Hb') (H+)2 e-31.4723
KRC (RHeme)(CO 2 ) - 4
and a similar set from the oxyheme compartment. These
exponents are in natural logs and mole fraccion scale.
Consideriag that the mole tractions of all species
in a conceptunl compartment must sum to one,
%I%~C (GO 2)(RHeme) 1 +- +K - (23)
LKRZ (H+OHI (H+) 2
and, Abnilarly,
(+ +K + K '(CO 2)(OHem~e) + 1 -+ K,.3
(OZH) ' (H +)2
-77-
In addition, the condition that one hemoglobin yields
four conceptual hemes with no change in free energy gives
Hb : 4(RHeme) . (24)4
SubstituLing Eqs. (23) and (24) into Eq. '21) immediately
results in Eq. (17). A similar development yields Eq.
(19). The ancillary reactions in the tPodel thus affect
the oxygenation reactions in the hypothesized way, i.e.,
Eq. (19).
-78-
Chapter III
VALIDATION OF THE ODEL
1. INTRODUCTION
The addition of the serum albumin buffering and
hemoglobin reactions permits further testing of the mathe-
matical model begun in Chap. I. These tests will subject
the improved model to varied chemical stresses and note
in detail the qualitative and quantitative responses.
Generaily, these computer validatfon experiments will be
similar to laboratory experiments, so that results in the
two facilities may be compared. Emphasis is therefore
placed on reproducing laboratory conditions as nearly
as possible, so that the ex~periments are comparable.
The subsequent discussions of partictilar experiments will
further illustrate this point.
-79-
2. THE STANDARD STATE
The standard reference state for the blood simul2tion
continues to be the arterial blood of a resting, adult
male, as in Tables I and II of Chapter I, where various
stages of an incceasingly complex model approximated these
reference data. The incorporation of serum protein buf-
fering and heinoglobin chemistry--the final stages in this
series of experiments--requires another comparison of these
standard data witb those from the present model.
Tables XII and XIII list the input for this model,
called "Standard Model BFFR-l." In comparison to Model G,
Chap. I, it contains several important additions. The
species protein (PROTN) in the plasma compartment now has
subcategories associated with the conceptual compartments
Site 1, Site 2, Site 3, and Site 4, where the carboxyl,
imidazole, (-amino, and phenolic groups. respectively,
are ionized. The miscellaneous protein (sometimes called
"X ") in the red cells also is ionized at a& average pK
of 6.79. The hemoglobin is oxygenated in the four Adair
stages and undergoes ionization and carbamino reactions
according to the data developed in Chop. II. (Table X1II
also shows the subcoupartm-ents of the red cell.)
-80-
c 500 00,00 0 00(P1a
>0 00 Ok 0f
iA0 o . 4. A0 k1
0
C ~coooo
0-co~o -
41 2 -4.
fm P-4 0 c.c
0 II0z C.
4j00 W. A6 .--. ,- - -- A 1
4000
O 00 00
*P4 0 0 S 4, 4 0 0 - On 500 S0WOg i gi g g gg t l
0~ %A j 0 X X00O
4b .4 - -.
(A %W6 *z1a eo *o
Table XIII
EXPECTED OUTPUT SPECIES
FreeExpected Energy InputSpecies- Parameter Components
ve, Aso&00 .00
ca) o.ca.,660 0
551 -1.6040"i0~ "o 1 00- -0-0
a,.3 *.0emo N0 5 .6046 1.6 041110 t18
cm. 686Wo04 1.60 on" -ac.. 02W6.0 -0.66 ai~ 9
W.0::. 0.CO*6 1.090 0*0. -0.0414 #.41141111e oft*18
6a.0 0 0 6 .4 0 0 .c 1 0 . - 4 1. 40 4141 -- ). O ft F 01.61* 1 0
Pa. 0o 4.k. too* Ioo a"~e t:41.0 oft SetDA6..1 1.o o .0'90 .006 -4141414 6.111110 oft "'I
up ~ ~ Is o.. 1 "4 1.06f f."m ANN%* 066 1.6 611
OO,,C $ f6 K R-.000 Ua"-030 .e 9.041 69*t 11&
.86 0.40E 9.00" wo am
IU.0140 .0044 Lai -1.066 :11100 as- -).Of oft a S
Peat.0. 000o40 1.*44 4I9Lpt -".0* acct- -6.06 e1. -i.60oml
640 Elk k%
Cal .04041M 1.00w to.?NJ -c%e00" W.004
t.600 a-%-
444.-l044ts 1.400 &.a010%
ago- . 4 "Aft 1.500 E6* t.49We5414 a.''06 1.641443 W4 I.0043 4.0*1
40404 -I've vt 1.0
tM4tI U."44.006 q'i 0 w
-0440 .94Q -40,6W -Mi
1.'0* e43' -1.666 0;we -;." a
$.su"" 3 4.60f -4 -1.600 04- 6 -. " 19-
-*%.I* . ". Wofa.1.000 L4 -*.44 .s.6 W1 11-
6#44*-.44 -44.f6 4.00 64LI 10
49 1"4" .~0 .6 6* -604',0.
60610--6 44'6)46 4 16654 l9016 1.6016
412-r % 1 t "
60 It.e96 2n0 I*ew e.66 to 6
OR ý. -*.o.65 *046 4w. -4.&W t.w06 oft*60
pt. .46644 "W6 arms -4.*44 oft ok 16
ti etl: i
W4% %44,4008 116*4046 ~ ~ ~ ~ ~ ~ ~ ~ f Vb9 .0.f aftolp~ e 16646 .0 * 41
.5164M itS4
-82-
Table XII gives the total of species in the liquid
phase in moles-per-liter-of-bood. The species H+ and OH"
are combined in the model according to (H4.)(OH')/(H 2 0) - Kw
to give water. The total moles of miscellaneous macro-
molecular species ard protein for plasma and red cells
come from Documenta Geigy £291. Table XII also shows the
gas phases for equilibration.
The free energy parameter. of TIable XIII resemble those
of the previous models (Chap, I). Each additional chemical
reaction, however, also requires a parameter. For example,
consider the carboxyl ionization of setum protein given
in Site 1, Table XIII. The conceptual compartment Site 1
shows two reactions. The symbol RCOO represents a carboxyl
site on a serum protein Molecule. This site has two pos-
sible expected states related by
(R XG " K((RCOOH)
where 4n K - -12.40503 with the 'noncentratiors in molee
fractions, or -loglOX PIN - 3.64, a "best fit" average
value fromi the titration data for carboxyl ioinization sites.
Similarly. the remainng ioni-.atior, parameters are
derived on a m~le fraction scale from the best taboratory
estimates as disctussed in the previous chapter.
-83-
The oxygenation equilibrium constants have the
(Roughton) ratios derived in the previous chapter. Table
XIII expresses them in the mole fraction scale for all
constituents of the mass action equations. While the
ratios of the oxygenation constants are fixed, the
absolute values are adjusted empirically to giee the
correct oxygenation at p50, the oxygen pressure at half-
saturation. This empirical fitting is the best procedure
presently available.
Table XIV prints the computer output for the standard
arterial state. All species appear in moles, mole frac-
tions, and moles-per-liter of water in each compartment.
Similarly, Table XV is a coraputer solution for the
distribution of species in standard venous blood. The
differences between arterial and venous result entirely
from a change in gas pressures. For arterial blood, pO2
100 M•r., pCO = 40 mm; and for venous,DO - 40 . m, pCO22 2 40MPO2
46 mmw. Table XVI sumnarizes these data for some of the
.i.portant species, and compares them with similar data
from the reference literature.
Table XVI has few i&ortant discrepencies. The mathe-
matical model uses a slightly higher (5 percent) solubility
S~4
-84-
FLA19 %-A A I CILL ( l S 14 Kt
P-* .P90619 01 I.?O0900 01 9.19128* 01
PH .Illol1 00 Y.IIINSF 00 -0.
07 :0'012 0.,119-* 0.6" 10 " .11': :00./t.10 1.11sl0t09.0 1.01 -0 .149141.02
.It .;0 l.71mi01-04 1.ll61l8f-O0U l 4I00
(l V0IC$ ).I16117-0% 2 .lll,0-004 ?:S400)10 01
991 ?..133O .1: 14S Z0 I.400-O
9/L~l 4.11 403 ,.OlolO-0o .13 .01,0
"-? "t 01 10 l.a 1 1o4 0 1.? 09 10 0.0 )0 I01Arks C .9.00S1-0 q.0l-1I 6.1041S -0
N9it .7" l..I1o:000 S .-11900( a 01 ~ qO 1 .11000
-n90(3 '.0390ll-0S. l.6 99193-0 -.0.OPMAC .107 .0 a .30130 0.11.111t) 0.0191%0-00 .0-131-t 00 -0.
nN3- 01% .014- 1 .1d/ t0l10.0 0
C 1 0191 1.9 a Table XIVC1- 01111$ i0.60plo1-0 J.401 It-O1 J3
"It 90a1 I.0 014 -0 I~l.110-0 -0.
-0.."O STANDARD ARTERIAL BLOODL0 901I0h1 ?.%oIl-0 0 il00-01 -0.
A 91Ant0i 1. P00lif-0a1 4.4P,001l-at 0.9/101 Nos 0.00190 f1 1. :I01 -0.
0* 9013% l.1103'13..0 41 0l -0 .0.
CA. : 01I F01 I6.094 -% 1 11101:004, :.0.9/L"1* 4.01094/3 j - 1 0. 1103 -1 -0
%It0 MAI 90.o I.AlliI-o 694 04-0l -0.
.C.. :(It0I1 4.tl .O I ':"' ' 1 1 MINIM 4911130 140At ACI )91
10 . 131 Mid .,1101-&04 1.191J.0-01 -0).93001 ~ ~ ~ ~ ~ RO~P k.Il94 13009-%-.6*10990 )V0.0* OHM 0-V1.410001NP
S04. -O1tt% 14,030 i.9041 04 -0.031 .4190 -..09*1 4%l'9004 100I0.l 0 Is-RAN to 1.6aOO.F-04 k.mnls-u -0.6iuf
0/39*1 1.Il0110-U4 i.110140-Al -0. lo .410.. 0 U
PO.91010 1.41400f-04 I.1914i0-04 -3).00 mA911 4.0'4AfO 1013* .000613-00 -0.:10.93% I40I9fl-.-)
93901n .. is7111 -Ol 1.66$110-01 -0. 93001C 17240?70-01 0 -0.9/10)0OLF k.S1419 3-0 l.1103IS26 -0 .a103 MR. 3 -U. 1:40. F:0 1 0010 -0
00I ~ ~ ~ t SO39 ".1010 4100.It - 0. 1IA -0-n.07. 0
k.4. 0 I 4.*9404.-Ol 4.?014.91-04. -0. 96901- -0.tf 1 .3l4 3 *0.
939A(n 4.Spq3)9. P :.910 0 : -0.%'"'l-0NMI Utps 1..314q00.0 1:06%163-04 -0.A 9330 0. 1.130 .01 0.
1*010 003 .31. 03fq-O I.0 9 9 of -a "Ho9.- "(10% .0 .04 1-3 E 0-I'LMYU 0.1944401-0 1096800-0% -0. 3030. 431$-.-.I1l0.I
9/vf :1070 L4.301`11-01 A.14l01-03 -0 90* -0. ))1: 9406-O
o3001NAD k011s 3..331-0 -0?1O-1 o
0/01 1.0030-1 F?10 013 0l -0. t 0 m w l-
I10 01 .9*41I3-'01 4.000-?-00. fPD fjti93U9 01ROF 1.0141-o * " .4"0U'0 -3o3.GS11,
0/03 .90J-41.0000: -010)- 9010 .40 4. 4.1416 00-03 -0.1"'Z 0'
93001 P 0ti .19112F.307 I.6?0941-0 -01.391013) !.411199.04 1.411O)0-01 -0a-m,0139900(I0
93911Z .910-0, I:.n04S0-o, -0.
" "F0os 9333610 .0"4.13 f10om
.'L."' -0.7 1.441000f. 100 J .001 Stoup.0 0L11.1 mi -0.
-0 /01 0 -0. %.0100 -0. 99901f- 0a010 a0 0
99l"0 1 -0 1 I# . Wl ),41-as -0. 99901A i.116103-07 -. 0. 0 -0 . 09/1010OLI -0. 5 %3199-i01a -0. 090 911'.0
100-J 0 0
09000 ~ ~ ~ ~ RC0 000W0ULZl-31-.91* 0 l0700 -0. 0w3901 -0. 4610130 -0. 0*9.3 901 .0. -0.9913 -0j.-0
901312 -0. *.1.0140.10(1 -0.2 9303911-0. : -0. :.4900 0.:
No*4 9013 -t$:0. ""11 37-0 -0. 96300 900C091 -0.16 0" 1 0.0000 -0.
9/1010o -0. 1 U011ot-05 -0. RAN" KO10 -0.i - . 2 '1 01. *.onioo-0 ,
3440 90100 -0. A.,.?-0 -0. S 0 0 . .9 3093991 0 -0. 1.I94I10-04 -0. lot a0 3 -0. 0.: " J4 1 01 0 01 6 -0911010 -0. 0~~.l040I0 30 ~ 0 0 .000
-85-
I3 1.3 6 6 d0 7.1 olf03 00 -0.
30o? 0.4,1*0 .17106 11.6 Of0
311 IS .3113130-0 1 -1.00 917000 1 -1.46%1:1 019/Lol) I.11111313- 1 I.449163-011 41.6611/00 03
0-M611 A."Golf0*l1 4 ?sl.%1I01-06 1.46119( 07
670 "01.1' 2:.'S11.102 O 1:1181113 0 3.011*10 %l
. $ ,0 l ., I _.0 10 q..0I3,1 0 :-0 -0:. _
3111 .0190-?l.176011 o Table XV./I30A S.4I0133 -03 All 3109 -0.
CL- 3.71 f 011.l*1ot 13-.0-01 -M, ? 0.
FIAC 01 1.92b049-01/ :.?00: 0 STANDARD VENOUS BLOOD"It3"M I .91601"I-01 1601 -0 -0.
(/)ZI I 1. 1*06-IC 01 a.It 31430 -0.a
.10 31110 1.,406600-01 3.1tr 1 4 3I6 0 :
O99960 ?64j1SE63-01 2.6.1"t03-0 -0.vn/l6 I..Nl 000.0l .117,13 0
C* A "lt S I. u . .61jl -0 .1 l0~ 1 -O430560 11I111- 1 :1 9 1 ?1I16o0 -0.
?/O .4m06l-l I.&11~l-0419 -0.
&19C 1..8690F 1 .3 000-01 " -0.a/L-10 V.1,9l17 I/ 1.11-0.1 -0.no- tm thi N
$f,4- "M /U 11 . 31 3-0 3 36,l1 10-03 91RA A"1 '1 16 06 1 1N6116300f69
t "1136 1 " j14030.1 4 0.0111813-011 -0. "Me. /. : 01.111 :139 1 0 0l .41 0 1
. 903*0 S.17 4*6 1.0W014-1 -0.'-33- 131 z.,l* 6 -0-.
P/I'11U 4.610-V11W-041 . 6h Iss116-011 :0. as0 .701~l-.-I
629111- O0111I% h1.164911
04
`1-)1960"-04 -0. Im V3 91% 1 .6411 0I-04. -0. 00:303F : 11 .11l3I? 3-0 1. ! .9113-09 0. 3*36 :0:0A& .0%13113-%.? 0 -0 .
4M. 0 'is 1.419N4f0 4 116 I 2 .16.040 -0.:0
-I lc .:1 61 f. 1:04 1. "W".l-0 1 -0. 6036 U11 .1l130 011611w~~~~~~1C14 611..1 .11611l1 :** .73111..0.:
6/~li. 7.13l001 .63710-01 "F.Ar 31136 -0. 4. 00'63- -0.
930 .3116-011 .13l01.1 *a -0 10.10 -0l .l
3/111 .140" -3 .114090 -0. 06* 001..ll31610 0
A00 A. ' 14 -'.1 10.3 )I1640-0 -0. f30 9.ac -0. -0. 0 -0.1101*6/3 .Q(C1 .1:11.614 10.Cl 1.041606t-01 -0. 9 C -.-. 4360 -0
0(011-~OCP" 00111 I. -a*31*1l 1./3 0 03096C "Ill 96-0 I.1930 -0.F t 0 60,o,0
3/1.61a1 )'30.11l10/ /.1O0f140-0l -0
07 03 1103c Il 301001 ""I",163-0 -0.N ufs-0 0./I.10 .. 146. 0 :_116.03 '~o ..1 10-0 -0. "AC -.-. 69 -o
" "011 " "001 I.1114110-0 1.1611 0-14-0
32C01 l.1/1*01 ll0111-06 - 0
I301 M 71. .1 0)0%-0 6 1.106, )(-a& -0.3019 61.11 .9 103 4 -0-. .
""3/.61 '0" 1.1191-1 -0. 4 -01 1A 0"300 I0.1 .2.1170-0 1. AS
-0.
If-0. .1 s 0.
E3436 301. 011 0.I.1133130-06
-0. 3*36 9.39631001-'1. 0. .
PAWN *00f :0. 31 141-01 -0.
./t.7/U -0. 4.1.619y0q0t060 301 0 *4*11-1-0 0
6360)~~QOO .0.01 it769300 0 360-. 0. lOolIE-O -0. -0.
3*360l Acc1 -0. :00LO94-09-0.M69000 -0.0. 9.9910 0.3/1.670 -0. lml3011*Otis1 -0.
.6 360 S 0 2 OF -ti -0.6 00
3/ :1 01. 3.131001::.0 . 313401-.-. ~ .301010
-I0m60 -0..) OCIL30- -# S31 01w -0. -. 0 .4390
3/.1 0. of*"$*10-0 -0. I03% -0. -0. 1:0.61104.1:01110
-86-
r-4 t 4
414 C- .4 - SM ) in fn en ' 0 0 M '0 .
0 41 .4Go
0~~. 0: In e;
" ! "- c4IC~ S -4 V.. U e4
cc 0 ' 0 ~ '. 0
Ei
0 C 0 0a,0 .
M u4 M C4 00
00
C4 cc w- V. C
0 00-1 w In 41 co m-4 0 0 04
< . 1 x 00 fnO IT 0 % u 0. 4m1 0 T LA 10 00 c ' a,. 0 as
1~ 00 a.4
M~0 a-s a 4 C
0d 10 C-4
co 'oo o 0-
'A .4 1 t f 0 '0 It 4 u 0%
0-4 o* '0 4 4% 0 0
I In
o 8 -u
414- a N C4+ q o 8
0 u z x z hd >
-87-
constant for oxygen than did the calculations of the
standard reference. This slightly increased solubility
causes a small rise in the saturation of hemoglobin, but
the absolute level of the oxygenation constants for
hemoglobin are adjusted empiri.ally to fit the saturation
curve. The carbamino CO2 is also more abundant, in ac-
cordance with what are believed to be better laboratory
data. The hematocrit is computed using volume factors
(cc of water per cc of plasma cr red cells) of 94 percent
for plasma and 72 percent for red cells.
In addition to comparing the data of Table XVI with
the literature, many other criteria exist for evaluating
a model of blood chemistry. From Table XIV, one can-l
compute, for example, the ratio of Cl ion mole-fractions
between plasma and red cell,
S. .. . . 1.94 - 1.45 , (2)
(Cl ) RC
while Berrstein 30"' gives 1.44 for this ratio. Since
The prenent data are frorm F.,.W. Roughton. F.R.S.Cambridge Universitv. private correspondence. 196S.
-88-
Cl ions are assumed not actively pumped, the Cl" gradient
is a measure of the Gibbs-Donnan gradient and of the
specifIc ion potential across the cell membrane. In this
connection, as in Chap. I,
(C) (HCO) (N + ) (H+) (OHPL a)k -3L 4)RC RC PL (3)
RC 3RC 4PL H)PL (O-)RC
for the ratio for all single-valent ions having the iden-
tical apparent level of the active pump mechanism. This
study assigns a zero-level active pump to species satis-
fying Eq. (3), and names these ions the "free" ions in
contrast to ions associated with a non-zero-level active
pump. The simpler models of Chap. I could also show this
ratio equal to the square root of the sulfate ratio, but
it does not hold here, as will be demonstrated below. The
constant before the bicarbonate ratio is, here, 1.15. The
HCO3 ratit is not the same as the Cl, owing to the
increased solubility of red cell CO2 over plasma and a
slight difference in apparent pK for the red ce1l bicarho-
nate reactions. This Memorandum thus postulates a zero-
level active pump for the bicarbonate ion and accounts
-89-
for the anomalous bicarbonate ratio by differences in
intracellular ionic activity--although these two phenomena
may be difficult to distinguish.
One of the major discrepancies in this model of human
bloc l arises from an inconsistency in the variable bio-
logical data. Whercas the literature [29,301 gives the
Cl ion concentration ratio as 1.44, it usually gives the
H+ ion concentration rAtio at around 1.57. This H+ ion
ratio is computed from the difference in pH between plasma
(pH 7.39) and red cells (pH 7.19), a difference of 0.2 pH
[7,311. If neither of these iuns is actively pumped,
the cause of the difference in concentration ratios, if
it exists, is not clear. Of course, a change in the
activity of H+ ion or of Cl- ion is possible, e.g., by
interaction wlth the proteins.
In Table XIV, the model has the "correct" C1 ion
conccentration ratio '311. In Table XVII, the model has
the "correct" H +ion ratio 7'. In ,oach table, the other
ion of the two has an error in the quoted ratio. The
change !n gradient of the free ions was obtained, in
this case, by increasing or decreasing tht average nega-
tive charge on the serum proteins by 4 percent, the lower
average negative charge giving the higher chloride ratio.
-90-
Il1I'1 Paps. 1*IIVt UP- t 11" o"'_ 5l-0kISIO *5
P15%PA sit) 4.141% Alm nut1
3. *Is1F o CI .MV.'1' 01 9*99Il00 l 0
Y .I.1#41 oc 1.Ilol4 00 -0.
:' 01 4 ... 676l1-0% 6.SIl?tF-OS I .1114-lf010PISO 1.ls~llC5 1110411l1-O0 1.1thkIt-Ot
ynl.f of~Ill C -1.11614f 01 -1.790191 01
*PG&E .4101O .I.40194tsI0 _ j.6loss,- 0:
61.it . I.0 )l~ 03 .1.061021 kh 1.061141t 01
:Ott t1110 l l03I)-04 .111161 04 .1 400of 03
, ,A 1._l~..01 I.1l,41-IO 1:S46017101
lt"lll -Il ,1" of-10110 .1.1*11O1 01
111 IIIS 3. o,11 01 .100,011-01 10.30101 0I
l~lIC .'lS~l(-01 S%6401 as-01 .0400
t.&(1 .1901 C 1. 1 4.1 01O -. 0: 110
P101 1..III06I.C.:(10 -t0.
1~~) 1..tFt I t~l lI -. 00550 0 -0.
:0. Table XVIIP31.3.1-0.3911 00.4S~l.60 00 -0.
:1AfQ~f.)4-0.MODIFIED ARTERIAL BLOOD6 ~ ~ ~ o -o00 3%% lI0 .j091#6%f03 -0.
A. .1) ~.ll5* %' .. 901it 00 -0.
w5** 6 11 Iq. %-t~llI -01 .35041? -0 -0.
6111. .l~lIa.0 II10111 -0.11.0Of -XIt no6, 00 NCl1at 001-0.
6Ifl .'01111C% 6 ..l4Ill0% -0.
%J- at -I % 1.U9 wal .1.1010100 -0. 9-A .10561 61t1, 0.1661610111 4.6%6001.01II
I. .01104-t-06 I.9I5- 0
/I1l -. 15 01 II 1.1.0%ks LI -0. ,ol. :01t% ..003 05-.0
ll0* 6)1I 0160-U4 *.p18?21-0- -0.4 .. ,.to13" -s 06 .1`0S00100-U -0. -a.~l- Pll 10 .0ro031-100 v0
63)1, -. 1551 01-.014h11 JI -Jl. f War I .I141 P- r1 -0. :
"1lPU5- Not1%1 1.10-I s191 Of-L4 -0.N- O tg. -110 3u.qt53I.0 -0. -0
PiPAc I 11 10.l Ph 1S- v 6 1. it3- 0% 0: Kim c 1:4% ,6 ?1 -I W -0. -0.
601111- I 3160601 lu$)s111 01 0.-
RCAlA. :1 t1 10 . %P1 too Io% .0. -.
Put0 I .011.-C04 101c ..10 -0.f- -00
61131IIISIII-1.06S79f 0I -0. lipfa. :at1t. -0. I.110' :00
it15)6 ).51l1 l.ibib54106 -0.
q..?31 1.0661 10 .1 0 At 00 -0. PIPc .0. :4.qbt45- 0 -0.
Ac I 61 .04010 1 ... qtt -0 S.510111 -040 04fuD?- 61151 t II S. 1090 0if
.,..c N.0056o.lo- lI 1FpoiI0 -0. toIAC -0. 3.110001-I -0.
-ISI '.10f116 Iou -1.010701 01 -0.
0(o5* 0.5ooo : 6.0% -oI. 1.11111-00 -0
"0*60 % IIt* 000-0 I. qlos lOS -0. 6165. :W 0 . I.I.OI-I
U10 .'03-0 q.4%.r7f-01 -0. 6h 0
.,L." -1.9.l -0. -a-116O
Jt v1.l %06' I. `10*pf06- S.4116. .01 -0.
60111-.5006141 00 _%o~~ Il 0 ~ o o .155 l
.4(A01. r'1' I. ol3 1, , 4.01 .01l-Ol -0. %a.A~o" I~i"
a1(1P11 FI.I161-l 00 .AS14-0f -0.
"61 5 .5 5 l .0 . 14 0.c -0.P/1", 41 ..I~l 1 -. 001011l at, -0. 0.00I-1 1.66f(2 101a-l
5311130 .1:10l 1,0 10:.1l 0 06165C ~ ~ ~ ~ "a 1.OAI1 .3010 0
Pa511 (n I1% -0. f.to5-l 0 m011 POLIO 9. 0 1- 0. 0 -0.
4(00-~~~ ...1 ,1350050 -0. 0.
6*51)1 -0. .%b. ~ll0' :0.t1 -?u1 -UI. J-61651f0 01 -0. 50*663l *01*0S -06.101-l-.
IlIAC -Il. l.OIIII-0l -0. -1
..SP. C0* -0. of-at1.0 -0.
.6.01 IO fl -0. .1.041 ofvs i- l -0. pl -01.95 -I-.-.
c100 6110 -. .l%00 -0.*156 . :atI -0 . -0 l.*s0-: 06001I 0 -. "I ": -01 0 : -0.
*6*663-1 JoCI f -0. 0. 1064,111-A. 0 .
"11*05 n, -. 0. S.I519'S 00 4 -*0* "f1~ 00 . -0. -.005.0 0.
611.-0. I4.146t0I-00 .0.
6005311 -?. 104. -0.,l -I 0. off 1404 1 Va .a. C0 .0191
A.506 )1411% :0: 1.11604IM l 0 .40 0 0 o .1 6 0 -
614 n.1. .2 0 1-O -0. 1*.- 1010 -0 --0. 0.Olt10-Go
6010311 .0. - .1000%I.5 1*5 o -0. "f. :0. .991
*1
-91-
This small inconsistency has not been resolved. The model
with the correct chloride ion was chosen for further com-
putation, since it also has the "correct" hematocrit ' 7 1 .
Foi the free ions, Eq. (3) holds and permits the
following definition of specific free ion potential:
(Cl)PL ,RT Jn - = RT(0.3716) - z F E , (4)
(CC)' Cl
where:
-I -l
R - Gas constant - 1.987 cal deg mole
T - Absolute temperature - 3100,
z - Valence of ion - -1,Cl
* -l -lF - One laraday - 23,0b0 cal volt mole
E - Specific ion potential;
or
615 9 t (0. 3;,lt)( 1)(2'J.060)
- -9.9 millivolts (S)
for the rod cell free specific ion potential.
-92-
The specific ion potential as measured by Cl ion
gradient differs Irom that which could be computed simi-
larly from Na+ or K+ gradients, because of the active
cation pumps. The direction of the observed Na+ gradient,
and hence the wtde. gradient, is opposite to that given
by the Cl specific ion potential. Ti.it specific ion
potential would, in fact, concentrate the positive ions
in the interior of the cell (see Chap. I), just as the
negative Cl" ion is concentrated externaliv. Therefore,
the Na+ pump must overcome the (Donnan) electrostatic
gradient, as well as raise Na+ ions up to the concentra-
tion ritio observed. Analytically,
(Na+) PL (CI-)PL - 2.113+------ - - -) . 69 x 1.45 - 8.25 = e (6)
(Na+) RC (C1')RC
where the exponent on the right is exactly the free energy
pArameter, in Table XIII, for the Na+ pu-p. Taking the
log of both sides,
(Na)PL + (Cl') PL( . 2.11, 739 (7)
(.a RC " )R-C
-93-
Or, if we write
- (~x) 1 (8AF' =T t n Wx-2 (8)
then
(Na+) pLAF + - RT n--n - - 615.9(1.739) cal/mole
Na+ (Na+) RC
- 1071 cal/mole . (9)
The interpretation of Eq. (8) in this context is actually
not -s clear as Eq. (9) would suggest. Thus dF' replaces
the usual AF, because the ideal equation (9) gives only
the theoretical work required to move an infinitesimal
increment of Na+ ions (plus the concomitant negative ioas)
from one concentration to the other while maintaining the
system in the present state; hut it ignores any energy
lossas irn the membrane, the concenz..rakion tffects in each
phase, or any mechanism which maintains the gradient.
Particularly, Eq. (9) does not measure the work required
to obtain tý.e presenL 'Aistribution from classical equl-
11 br i Ur.
-94-
The K+ ion is concentrated in the cell interior, so
the Gibbs-Donnan gladient aids its active pump. There-
fore,
(K+) PL (CI') PL() - -3.0220 - -n - -3.3945 (10)
(K+)PC (C")
where -3.0220, from Table XIII, is the free energy param-
eter r'ap:esenting the work per mole to maintain the ob-
served potassium gradient.
Computing the "potential" of the K+ ion from Eq. (4)
gives
(K+)PLRT 'n +z F' E+
(K') RC K K
E K5-3945)K + (+1)23,060
S-9.07 mnilllvaits ;
and for Na +
-95-
E -615.9(1.798)
Na + (+1)23,060
- +4.64 millivolts
However, neither of these last two "potentials" are
significant for the activity of the cell, since the cation
mobilities are so low in relation to those of the anions.
For example, the "membrane pocential" ut? the cell, given
by Ref. 32,
E [T Pk(K+) RC + PN(Na +)RC + PCi(Cl) PLEm + - • R ~ • C, ~ .... PL(
F Pk(K+) pL + PNaNa+ ) PL + PC(CI-) RC
where the P. are permeability coefficients, reduces to
just Eq. (4) since PNa ` Pk " PCl"
Tables XIV and XV show a somewhat misleading distribu-- - C+•+ +4-
tion of the species SO4 and HIPO, as well as Ca and Mg4 4
These species represent the so-called acid-soluble phos-
phates. the non-protein sulphate, and the douhle-valent
cations obtained by precipitation. While the distribution3
of these species in the model correspond to results quoted
in the literature (cf. Table XVI), the "distrlbutior," as
- . .. ..-.- -i , J.--*1- A
-96-
such is not well-defined. These species are much more
likely to combine in various forms, as well as bind to
the protein, than are the singly-zharged species of either
sign. But these reactions have not yet been incorporated
into the present mathematical model, which imposes theo-
retical concentration gradients on these ions to create
in the two compartments the moles of each species observed
in chemical analysis.
While Tables XIV and XV give a detailed distribution
of species for arterial aid venous blood, Table XVIII
graphically presents the differences between arterial and
venous as percent change from the arterial standdrd for
both the plasma and red cells. These data present several
interesting facts requiring, for explanation, complex
intercompartmental relationlships. For example, in changing
to venous blood from arterial, the hydrogen ion concentra-
tion increases proportionately more in the plasma than in
the red cells, whereas Na+, K+ * + and Mg move f.
red cells to plasma, and C1 moves from plasma to red
cells. Since in the model the gas pressure change alone
initiates this pattern of movement, i.e., pO2 changing
from 100 mm to 40 and pCO2 ch&iging from 40 mm to 46, it
requires explanation.
9-97-
Table XVIII
VENOUS BLOOD ST I4DARD MODEL ftFFR- 1) HOLE FRACTION
CHANGE FROM ARTERIAL
COIWAS fts "1~ 00~1 sSCI0. sarhsf C"44.1 mm@ S144VAt
"-66 -.10 ~ -.* -toP .0 to. 0o
"1~ . i 5 . 101 ,44 -al 4 1 i~
'.9 04.1641 I.10-01 -1. 11106-% $,
441t1. .. 19Ej &%1:04-06 -0.'.
'11 p)4 If ul 1.4611 at -0.? all
ofu'l -11.I 0 .'?. %S 00 1 . 0tCE44(0)- 1.4 05 :34 ".04 au .1:0111
061 i*5f.01 of9lQ 3' IS 1.014t II '. "0':o' S.moa
(4.t 1.5A t4 is4t~ 4.. 0.
*6.)..3 0-0 .31 00 -."C04 a . I I- i.-41%#-0I I. .4.
mk,*. '.4111- JI %. )(got -0 -. Ct 5 50
44.CI( S "*.11' "0 .4?Ih-UI 0.1,, 0.Vt.00%%5
" "44(1 40154 47 144. 04. 0.4
'94.1 4..)l5 - 94 .''5F'0 -0.4.* '44'4141l15ISlg,1153*~44~.' I.C4 -0 l -50%1 1:19.V5134111111114148flS5
~...4 .44~#CO .4.l-Il to:. 0 45 1145#51195115*I5ISSI~lIsmII
4.4~'4 4.1411,7 4 04at44 -1.. as
.d4.0"0' A-'* ~*.41 * 544U~49 1
s:' i
-98-
Table XVIII lists concentrations of species computed
in millimoles per liter of H 20, although the plot is in
percent change in mole fractions. In either case, the
change in the water concentration appears as zero because,
even on the mole fraction scale, the molality of the solu-
tion (and consequently the mole fraction of H 20) is
practically constant and is the same in each compartment.
Water moves, of course, under the influence of various
stresses applied to the blood; when ions cross the
membrane, a proportionate amount of water goes along to
maintain the equal osmolality. The changing concentra-
ticn of fixed protein in both compartments shows tikat
water moves from plasma to red cells in the change from
arterial to venous; the concentration of plasma protein
increases and that of red cell protein decreases. Tables
XIV and XV, where the water movement is also evident,
demonstrate this more clearly.
Blood plasma is less buffered than the red cell
milieu (evident also from the Astrup diagram1 !;tidies Af
separated tractions -33'). When the pCO,, rises, the pit
Of Lhe plasmna shifts proportionately -xie than that of
red ceils. Thus, as tOe CO2 accumuah~tes in the blood and
-99-
(via carbonic anhydrase in the red cell) forms HCO3 and
H+ .the hydrogen ion is buffered--principally by protein--
while the bicarbonate ion redistributes according to the
Gibbs-Donnan relations and in indi-wect response to the
cation pumps.
In the system model, the bicarbonate ion is just
another permeable, single-valent, negative ion, though
subject to intraphase reactions. Hence, it finally ac-
quires the gradient of Eq. (3), i.e., before and after
the CO2 accumulation. In this view, the apparent Cl
shift (Hamburger shift) is merely a redistribution of the
total negative ion pool in response Lo an additional in-
crement of permeable negative ton whose corresponding
positive ion is buffered. Most of the additional HCO 3
ion, however, is found in the red cell. because the
increment in acidic pH1 shift is imaller than that of
plasma. By the Henderson-Hasselbach equation, the product
(H )(HCO 3 ) is directly proportional to pCO 2 ; but with a
positive increment in pCO2, if the (H ) does not increase,
then the (HCO3) must. Thus, as Table!s XIV and XV demon-
strate, a large proportion of the produced Ilk)," stays in
the red cell where the H+ io.n is highly buffe'red.
S . . .. . . .
-100-
Actually, these statements depend more complLtely
upon the intercompartmental mechanisms than has been
suggested. Fixed proteins in the red cell and plasma plus
the active cation pumps are the principal determinants of
electrolyte and water distribution, because of the inter-
play between the fixed charge on the protein and th3
active pumps. As the pCO2 changes, protein buffers the
H+ ion, altering the net fixed charge. Now a propor-
tionate change in the charge on the plasma versus the
red cell protein exists for which the Donnan gradient
would remain constant, but such an exact proportional
change would be coincidental. Generally, in the physio-
logical range, the hemoglobin buffers better than serum
albumin; the charge increment thus differs for the two
proteins, and ions must move. As ions move across the
membrane, taking along osmotic water, the cation gradients
also shift (the Na+ gradient increasing mind K+ decreasing)
since the apparent activity cf the cation puu,ps is a
function oL the ion reservoirs from and into which the)
transport ions.
In addition, tIhe hemoglobin oxygenat',on changes
slightly with pH and pCO 2 (the "Bohr effect"), causing
a sligh't shift o. all other species. Table XVIII also
-101-
shows the changes in the oxygenated species of hemoglobin
in venous b-)od relative to arterial blood, although these
changes exceed the graph boundaries. The fully oxygenated
species Hb4 08 decreases considerably while the remaining
species, insignificant at high saturation, increase greatly
at low pO..
THE MATRIX OF PARTIAL DERIVATIVES
Blood in a normal unchanging stc-,iy state is probably
very rare. Blood is a viable system, so that even when
held in a bottle it changes chemical ,omposition as it
metabolizes. In the body, it also has a constantly chang-
ing cher,,ical composition, at least cyclic with the cir-
culation time. Blood compositLon reflects even small
inputs or losses around the vas;cular tree, since every
species in the blood is--at least via somt series of mass
action equations--related to every other.
We may nevertheless assume instantaneous 3teady
states at various and particular points in the circula-
tion and compute the chemical co-position at these poitrts.
In addition, given a particular steady-state composition.
and :ivetn nn increment in any one of the inout ccciponents,
the expected response of each output species is comput-
able. If ........ are the input moles of species,
-102-
and xJ jml,....n are the output moles, then we compute
the partial derivative, bx /6bi, for each i and all j's.
Tables XIXA and B compile these partial derivatives.
Tables XIXA show the expected change in the output species
(left :olumn) per unit change in the input species (column
headings). Similarly, Table XIXB gives the expected
change in cotal moles in each compartment. Generally
speaking, small input increments cause additive responses,
i.e., the response to two small simultaneous inputs equals
the sum of the two responses when the inputs are added
separately to the same reference state. Consequently,
this matrix may be used to predict the systematic re-
sponse to small mixed inputs--e.g., the addition of NaCI
in water solution.
As an example of the matrix's use, first consider
the addition of 1 meq of Na+ ion to a liter of arterial
blood. Reading under the Na+ column, 9.13 x 10" meq ot
the Added Na+ will appear in plasma and the remainder,
8 b5 x 10 meq. will appear in red cells. The addition
also causes 5.99 - 10 mj of Cl ion end ,2.06 x 10 2 MM
of water to move from the red cells to the plas-nq. Every
other species will also change. H + ion decreases in both
compartments, but in the red cell compartment the OH" ion
-103-
Table XIXA
OUTPUT SPECIES I.NCREMENTS PER UNIT INCREASE
IN INPUT SPECIES
JAC'JlIAN. Cytif I
0Cal ?.1 " CL- 0
PLASRA -. 93 -0? -9.370630 .W 9930 3.0.30f-. 19302-.992 02 0".000 02 -2.614002 3.0A91-01 -1.769323-01 *.123-o I ?ol3- , -192930 -409030 -.9 114- -9.79)24.-09CO? -1.6705i22- .2.1323y -.1 1230 1.09210 -l.1 922-29 -. 03..-0...:00 9: -00 -4:90.011-0192 -9.tq99313-1 -S.13001-071. -2190903-u7 ?.929233-09 -2.3T11909-09 -3.936916-0049 2.90, -01 -23)-03
-5.8 1622-11 1.9192GE-20 -6.02118GE-12 4.3h69371-0? -4..0360 .2930 -99023. -2091-03-Dm-. 1.051133F-12 -3.RI137(-09 i.719721-2, -S. Y30933-06 S.793203-03 -3.S2304106-0 5.2982103-0 -7.013993-0?CL- _1.91311-9-.3-003 6.979313:-0 2.001103-02 -2.00192f-02 I.9994.*-02 3.999913-02-.1060NA. I2.9323 9 9. 12011-06 12.53".3-0 -1 .2119f-0 4.319221-02 9.21-0 .31-2 -29091-
-9.10312-07 2.69403- 3 - G.6.lI-1 -2.I0?&if-0Z 1.00l60 :.93901-0 0.90930 -!.20,,10-0CA.. -2 2030? 6.099.01 -07 -9.691219-08 -7.090671-01 2.1109236-001 ' .4416799-01 -2.1814,g-01 -6.204201-05
It.* ~ ~ ~ ~ ~ ~ ~ ~ 4 -293)30 2.9330 4902-1-.9930 9.39033-00 .1103 -5 21903-01 -2094973-
.49. *-199.60 . 990403-0 .796930 -. 109-0 9.76909 2.l-00 21 _O:-0 -0.12903-a9 26069323-03 -3.964013-0? 6.4S130 -0212-9329-09 -60.249:26,03 l.9026,111-09 1091-.
4zLA.12C 210222-01 -2033-32301-10010-0-.1203 -l739086-05 4.910,111-01 -. 1.9,911-0326394,90 o.f2230 -9090-01 .962993-06 1.0."F-0. '2.093-I103 -31091 411-02.3630 -1. 11.1. -01
1.(0 .200-06 _,.41, ?1-O *.220I 2.3 99 4:-2-291-04 -3.2160 S:2360 -2.903909-02L T C1.21 - 2 0 9 3 212 ?-13 .0 2 S ,)3 -0 6 * .22 26 -06 4 1 - .0 80 2 0 3 -0 1 _S. 9 9 2 1 1 1 -0 1 -2 .0 2 71-i 0 1 52 596 6 - 3.0 2 21 6 -1 02A20 17.3 1-09 f.299-0 I .1.31-09 2.2023 -2712-3-.92,0 1201-9-.9860Clki. 7.j790109 -2.006U993-0 61-i-01 2.26930 1.99903-0) -2.20L3063t-2 -9.2019)f-01 1.219 3-02 1.21633.-02.120S -. 620-2-.16070-0 -3 4.6l Io .112?01:e-1:163.6,993102-.691 026! -93303 01 2.0293 0t -w02 093079 -1. 146401-2 1.02104I-23 -1.'2232 -926210 1.9130 l311-6-.999-0-.000
w30Lcl) -7.141'291702 -2 14 '199-0 -299-0. -. 20293020. 299973-02 9I.S12376302 -2.039623 02 2.099302 0.11230: 29810 0 2 1613O S2201-92292-91002-9-.2416" -04 7.0199511.0co?. -2.)?003-09 .1.3600E:-01 .2.0193-08 -.1.929201-09 .02330 2.39I03-.9T1-S9~99
920 -).86113'3- -4s09.999293- -1.11"1001-01 -9.00730 9 .2 1 : 91, 30.3-4111300631!01-0 -2:0613:-0) 2.31796-03 1Or -9.002616t- I- 2. S 033A3I-Y0-01E123 -4.2 0000-0 Il-3.0I206,-0 -1.90-6 -19200 2.......
0t ..1- -9.S919313-2 -2.3212014-09 -7.1%94023-02 -9.50 79k022-k 5.09193-03 AT :1 .12016296 -21.979412609 ?.306196402L2 -1.491s201-01 9.01.389309 -3.0791192-0 -.2.0033-00- 1.003926-01 9.9910011-04 -31.993973-02 1.819319-0102* 2.9110 -962E-069 M 3.F06 t010. 9.7: 19,16-04, -9.9 I2912-02l -. 1:.19-01 6. 01:-001 4:t6$41-02
CA. .216301-.6099903-01 6 OAT :30, -4.90916-,04 -21 26 -03 s -. 79-03 `:%.91-0 .20260a0. 29032-0k1 -1047-9 1 2F.o39f32a-1 9.90973- _,. 192003-03 -9.1060:04 _.22190-0 I.0991-009 -.70900 .209-071. 1192120T1: -2.A:280441 : 2. .304,331-03 9.,0241:9063-0 2 - 7.91231-03 9.9207-04
999 0.1613-0 .13920-0 2.7930 2.19300 -2029-03 -2.122396-0 -0.799373-0 9.1334:-03K.3 l.012013-0l -2.9996of06-0 4.26401 ,-O -2I..270-09219930 0.9990 - 21-0 2.010 12604CA::I -2.09022:1-07 4O-O .118t 2.020933-09 -2.510213-01 -5.0710-003.0929-0 1.73902 1-03 -232.9- 2 042012-0$
UIA 7 091213-0 a 90191 2-061 -. 6191933-0 -2.29993-03 24. 096 1 31-03 39.2017 0-011 S2.31121-02 1.0 41`)11-0.1
,LC S - .:2.03 3-00 .19011(:0 7 -6. a '90 -0 -I396 3 0 1.9 6 3 0 3. 2093 0 129 32 3 0 3 9 0- ,2
F 03- -.~ 0 L 6 0 3.909329-00 1 '1.9 9 1-0 -2:.09 461 1 2 01 9 30' 1 . 9 I..29 61 3- 2-. 26111 9.39932;I -0292(0 -.9320301 .0223-01-1.9903-0 -. 2209931-09 -1.10.239303 1.09-0 -102 -09 1.,0.099-0
"COO. -7.09999-092 -1.4411036007 -062114(01-1 -0.99063-091 01.52224f3-0 1 $2.0900-0-21 175401-04 9.19116-0392 -. 931602 -266130 -. 3 3-02 -02.272051 0.202 02ISE9 2 2012 .9730
to' I -2.0821$8f9 IT-.2944090- 00 -2. 5f 60790 04 1.156691-04 -9.36916302 -9..932-0 1%.090-02 3.01303-03LACTIC--1.9022307 -2.901114-06 21.610190-07 -S.208201-01 9.2019916-02, 9. 992-031 2091-0 2 -1.1114I-2S01211-01.169A -2.922606-071 4.0114-1ito -6.%6151-08 1.199295(-03 -9.300931-03 -2.907903-041 21.93173s-09 9.90)13-03"6902O -9.933923-03 6.92923-O? -9.13903-0 2.)01-04 - 2.66[ .9Y3322-03 -9.6130993-09 t.61110111-02 k.6905"W-0296909 -2.206713-03. S.23971t-01) -1.909121-09 9Z.23933-00 -9.209961-03 -9.337933-09 2.073290-01 61.33923t-0493909 -2.122073-01 is.3030 _.19309 0.013993-06 -0.1)4456-09 -2.331213-09 27.902I93-09 9.0416623-OC60 .9980 -227930 -23990 -. 29-00 7.9213096-003 2.969O,-03 -1.67230 -. 993
6 00." I':l-.043943 0.00993309 0.111,1"-00 :,.,1,0-0 111230 -390430 .20-0 21906
"296.% -1.611933-04 9.193923-09 -1 ."974011-06 4.319624E-02 -2.9192931-02 -3.90)0932-03 1.0%299M-01 2.111099-03,_O,~ -129030 9.09230 of192-0* 414199001-0-2 .393993-03-0.9230332-01 91.0110631-02 83.29043-04
ICII -. 9176012.01730'9090976-0 6.52666-04~ -6.16006104 -1.814019-03 -1.6$6171-04 -2.271126-0
HSV1l 092 9.200973-09 -,401919:01 -4.9119-04, - '.7290-02) 1.1I3203ZI-02 ::3.9109:16-0 -:9.9021)20-0) -291 -0929604 -1.214131t-0 ,.2.133l-9 '"It900.' 0: 7649-02 -4.2940 -. 19-04 1.9700 0.9902I-.
.14602 2.993429-0. -1.959010-003 2.789329-09 - 0.4?03963-0 1.409901-02 1.66390(-03 -9.901313-03 -S.99406Z-00
OCAmrIA 2.1199-0 3.:220 2.929Is"6-09 -9.213391-02 9.212703-02 21.922013-02 -. 2.96390-02 -1.90402-03
01t3C 9-.3031 .0163-12: -1.32-2 2002-0-.09106-.94-0 7.234-0" .00302620 46.02 0163-09 9 91 81-09 -9 969693-09 7.1:5190 -02 -2.S21231-02 -3.6022 20-02 4. 6 001Z6 -02 2.99.909 -02
Hm.* 297 IaIM-0 6.20130 -. 104%292-01 9.19 ,92314, -222906-. 2 2-.223-01 -3.1022904- -4.0,16440-02 239030
* 3273 336U -. 02236-22 2.1091M93-2 -3.001932 -. 9010 2.91-0 1903-04 -. 0119-09 ....00(9-OS
ICORS- 2.363971-0 2021 WY-01 4.099096-0 -2 1:330 1.1909 -. 7330 .0310, .209193-09
o213 S30 -. 1" ',-2 1.9006 -1 .209941-01 -!:1.1309:23-0 2.09371-01 2.:6704 ITT -4 2.2239 4-O -1.0090 1-0
"B9991 -2.223123-09 21.362191o11116-09 -2217 -0 2.809030-02 -.2.04191-02 41.4921009-03 -9.9921`31-01 -9.644099-03DCA992 2.221223-09 -2.961SM-07 2.629291-09 -2.900091-02 02.90203-02-2.941091-02 2 .1141119"-02 9.6220264-0
a2163 IUNL HP 1231 1099 -20 2.63713-22 9.6906312-07-.392 -2.01226s-011 2.0622I-0s 11.0011016-0939.-11.51399-06 2.779313-01 -2.009993-04 3.219092-09 -2.211921-09 2S.90991-3,9 -S.160913-02 -1.9094-.09
639.1 2.320999-06 -2.761921-OS 9.3S41993-00 -3.20093f-09 3.233926-09 -2.9019731-09 -2.9294001-09 .3332
32334 SEAUN ".9911 229162 .7961 .13::1-.. 3330 -:44 .012030 0.:330 9. 119*-67
92 201926-092.00313-9 2.00.0 o0 1.923-929930 -2922-0 %:2.7304"-09 :-:.0229-00S1" 1% A.996C 9.3930 3.OS17-0 1::909' -;:.7913-2 -2:.S'936 00: -2o 00r0-.260
-104-
Table XIXA--continued
J1104l6A4, CYCLE I
CA.. MC... 1 04. "H11. 414 T ACTIC ,911 A,(l
641168A 1.411540 02 -9.190116E 01 -1 .80966F 02 -1.816511, 02 2.59061t 01 -4.4996l1 01 1. 441494it1 02 1. 4444 It 0002 6.14 110 3 04-4-.2156:-04 -. 4 ,01,621-04 h.17(1 -0S -. 0491, It1) _ .4 , 16 -'" 1. "o,9 -0610O .8 7 - :, -I.401-0, : 4. I *: Is aI O :4.,1'4'11-0 1 641-0 .0:1611-01 ".1811110, ".114, -ON092 2.121621-01 -4.3116-04. - .,_0 51 199S4101 -1. 4 1096-03 .. 4 ".01-04 -.,.1 9 IT1-04 1.1%94_-0 1. .1 S9.l-C-
H. -2.101011-0? -1.41SO11-07 -?299067-09 -1.46 9 -0 .1 I.284607- 3.723961f-07 I.%443l6-1q 1.441111-09ON: 9.626-06 I.4495RE-06 -1.64041:--09 -440,188(:06- .16226f-06 -130"0"34I,-l 3l6 -09:L4302242,6- 0 0 3.827421-03 :.,l1- 1 - 1 6001O 1:l010 1.40 1-1 -. 06t-1-..91-1CI, -7118-2-.:217-0 -.1 I7_1-, _489970 ...66261-0 -.l ) It -" 1.01 1e.1-0 2.o ' ,
-2.39191.:_02 -.1 410.1-0 I -4.21,7*6-01 I .41il L _0 2 _ 2.2211a 660.1 .1.1.1201-0 .1.4471-4k S.61441-6,CA.. R.S1006f-Ol -7_.2l11 10-1 l.012l87-Oj03 .11641-4 _ -4. 1617&-F03 S.487&9( -0 12.919""t-I6 e1.2971f-06MG.. -1.5121?F-012 .771`960 .16 9-0 1.9710 .411-1 0 . 73-0l .216 o ?.Il llj-1J6104. 1.09 o4I6-01 1.04441*-Ol 1.091 10- - 11.211- 0 1.01.11. -0g -4I 196 -1.066001 -06 -l.04031if-(.PO~4* . 6146mik-01 4 . 11109t- 01 -6. ",3II If- 01 1.67671-0I 4.106Z71-0I -62449421:1 -0 1. 4 k 641f-06- 1.41 4491 -061HZP04- 1.047790-01 I.)19*6I(-l -1.135 91' -0 1 I.019 741-03t 1.161170FI01 -2. 194 IOE-01 7.SI146E -01 7.9044-0k
f14 2l8I-0l -4.497661-01 -. ,)-l-.471obt-05 1.06442t-01 1.579191- 1 1.664101-01 7.66401-0?-914 2.0661-4 -. 7109106-1.1011-0 282011-1 .617991-0? -1.776431F-0% -I46110 -. 4668t1; 10
LAC ,IC I.4012-01 4.6410i1-0 I -.1a60st-f2 -. l 19.111-0 .21 1-01 .. 9102 If-0L -1. 10 144,(-04' -2.)0)4%1-0Ult A l.4680660 -111-01 -3.115200-0 -I1761-01 .613-1 -Il96-1 6.1606-IF-0l -l.01,61 -06141410$ I.2018l7-01 -6.4944SE-03 -1.4,169'f-02 -1.411519f-0 .9011l67-ol -1.628197f-03 -i.Ila171- 6 6.1606%f-CdNC0- 42661C l.10190f-Ol -1.71050'-01 -1.11138111E 1.064,62f-01 -L.51a1ZS-0t1I.61.1-81-004 9.612.191 -04"210 67470 -3.917221-06 -6.21t.1111 -6.%06171-06 8.6964,0f-07 -1.11'2Af:06 .00001-O9 1.O011014-06
cot. 1.0081 1-01 $.49314W-04 -1.617441- 4 .1531191,-05 2.6a,18-04* A.4970 tll78-Ll-.6 -0"20 1.10471 021 -91.629926 01 -1.809971 o7 1.111101f 0 '..116 03-X1060 .1210 .321CPROT% -l1 4 -J7 4. 10981t-06 9.21911-076 4 .10 ItIt-008? -4.310W2-08. 6..1169 11-06 - .006111-l21 6.012621-11
610 CIS -7.606601 02 1.543176 01 1.83164! 02 1.912si1 0z -2.3414499101 of 119 0 9.,? 196 - 11 9.79-0011 -1:0411111-03 41:116%111-04 6."110 7279-04 -6.9122191-0 1.800121-04 ;.6211,1-6:o 169, 1 1-0061
'62 -1469-1 7291- 4 1.9641.1-1 2.614-0 S -?.93070 1.1910 Il.9-' 1767
0 - .-402110_ 0 6 _1.91 401-0or7 l.,AI47It- 0& 1. S (2 SI 1 6 -9.164 01-07 1 .104241-0 A, .0 1911-04 " .0619011 .12F0 .41F0 .11f0 106O-7810 -0611100 -ta-9115 -09
17.- .. 1.0124110 0 -I.617142-0l 4.S9l13`1-01 14.681of001-0 -2.4181701-01 1.440IF030-1 .06911if-0t. 9.069611'-0$984 7.Solli61-0 I.6I7-l 4.1 8 0 4893-2 66284-0 -. 79-002 -2.121 , -0f -20 128-a4 1.l97981-l 4:1610 !01-01 '.2610 I19'f2 .216-02 -269 -0 _ -164 %-0 -'.82%91 06CA.. 1.4p$910.0 7. 11l4f- 0 1 -k.0 1:19(-01 -41.11616f-C' 4.),00F71- 01 -1.40116 1 - o - I. "600 D-61 -1.1' 9911-06
9$. .l12-01 7191-1 -1.03690:1-O - I. 19 171-q0I (,' .49SO I F l -9 .411011-0l - I.J1 2196 1.-06 -f2l40,I104. -3.0o9-410-0 -1.0446)f- 01 4.40I0ti1-O 4 .129-)IIV0 1 -1.0si "1:01- 4 .11%8-051 1.6 0610 .6 10NPI)4. -1.611Sa1-01 -4.645061-01 6.19O1LU-05 1.16fill1-01 .4.71S901-03 4.i004)F-01 -2.140961-06 -2I. 16 0 971t-06142804- -7.40009t-Ol -I.14166IF-01 1.191311-03 ).419641-01 -1.%1404f0 .419-01 1. 021911-06 i .021SIE-064.44. 1.054696-0) 4.456741-01 t .I 0 1 101 l.27I136-C) 4.6414%f-01 -2.1,12601-03 -1. 10061U-0 1 - I.1.6 IIf-071-04141 -6.66101101 4.69-0 7.? 960 l:1`3100:-04 l0 4 17 J,-02 3.644-06 -,1:4,146110 I4.1-J
0:1140 ..2."081-2 4441-001,46919 912481,602 11"',6317-0 1 3.028.91"001 I1 ,,I 't c06L 1. I 1)1,-01"-Cal- -2.069553-Ol -1.0.74if-01 1.0241-f-01 I.2141S -10"i -4.976119f-002 6.264671y-00? 1.17174E-04 I.Irbr4f104"-?COI -9.62p181-06 1.98610k-06 6.?6ss11-06 6.111111-06 -8.161661-01 1.60762f-06 1.9,I66%_11-0 l.9566st-08
a67 .l.- 10 .0060 -. 68609-01 -2. 11.191-02 41.21A. 1-00 -94 I7-0'I12."1914-0' 1.131.1-0o 2179 -. 4 l1 01 -9.00909303 6814- 1 2366-2-.11 611EO1-03 9.4614*26-01 -1.119101-001 -1. 119141-0
".R4 I .36096t-04 1.41194f-04 -1.080001of 166010 .6646-04 I.87C4-0 1.68710 .901-0""6402l .071- 41 4.160sof-04 -1. to244F -04 -1.074961-01 .9110 4914-4 .4010 .4010-1 014 2.10174 (-Ol 9 .146701-0U4 -6.9978671-04 -1:,1611.81-01 8.3b1)0t-04 -9:6 765:-04 )g? 17-06 1917-1B6406 1.00002F-04 I.216 n-4 -. 681-05 -1.2698-04 .12'421-04 -I1 1 111 -04 S.6 619-07 .6561Qt-07"-6403 -1.6049Ar-01 -1.59340-1-O 1.1?2677-05 4.110' It-0 1 -I.1714SIS-0 1-681901-01 -1.-160201 -06 -l. 6019t -0,
A610..191H 814 .q?1I10-0I 1.16909-0 2.817-03 9.?I422f-0I 3. 4.112 0-0 1 -1.910498-01 7.9ZI071-06 ?.9.?)nJ,-06H4214. ::142051-0:I 1.4071-0% -) .61 ""0 -6 7 06-01 1 .. 1460) -. 60304F-Ol ?:,110203-06 1. )%.?;, It-06".He.- 83264-04 1.4ZI6991-0 -12.6'",11":-0.4 011E606-4 9.,701016-4 -3.61101f04 6.81V, 41-0 .8941,.-0"6 l697-0 6.1210 -E_ 34109 -1 "09-04 6.310600f-09 -7. 35091F-01 f.2 761006 a.2 7-u61666. -1.124037-041 -6.401411-_01, A.8903,26-0', l.610SII-04 -1.841231-CS 6. 116It,' -0 S -I .of5S1 -0 1 -. so s I -_O
O0Y1M0 " S 41 -6.973 78*-03 -31. 690SX-03 2.662961-01 9.711119f-01 -1.411241-01 1.970481-0$1 -. 92 )(,0f -06-7.10-C.4460 .211-1 .l72- IT779110 - 0.68-1,E-002 9. 1049,101, -1019,0 .089- 1: -,: 89-0.
"1460- 114111-2 7747 1-0 -1.911 :-09 -1..00 06-0 ..0 00140 -6.77- 1 l.01, o 01 I.003621-"60?-- -:.944822f- -. 00-0 6.04101 .3110 .;69210 o.120,0 -I909- 9 "M.609-013CR80. -3.21618f012 -1.11026F-01 I.012,461-2 3476-0 l2I6-Ol 19.?42211011-02 -l.I4If,120-0 1.041121I
017176611 NO8 4.?17997-08 ).101211-06 -1.SSSD31-08 -6.04770-08 l.62170-09 -2.699$06-08 1.01160f-t0 S.94319t-11
80. ~129 f-01 4.760 1710? 9668-7-.24601 4310-2-.396 1.11031-04 1.110346-04:810* -3.3192l-0! -4.76017002 1.17601 l.94001-01 -4.141Z06-071 9.033901-02 -1.130171-04 -1.1301;F-04
S1I711 I i(Um4 -4.Z6671-06 -2.01171I-06 .1-96051-06 1.;9665(-06 -131 ?461F-06 2.00619f-06 4.3k1,4341-30o 1. I %??of-So6100N -1.91041-04 -91:13191-01 9.774611t-09 -4.916098-01 -6.174911-01 9.11914f-05 1.76161~b-07 l.76LSC.-01
it00 3. 411-04 910671F-01 -5.6164 71 -(I' $094410-01 6.136F01-09 -9.0120O66-01 - 1. 191741(- 07- .7 9 -0
9I17 SERUM -4.100691-07 -2.134401-01 1.47166E-01 I.S181l0-01 -1.600401-07 2.0931IS-07 3.1166J21-ll 2.9311:5J-11684.43. -9.06191s-02 -1.4s1121-02 9.02549*-01 -1.743311-03 -'.919911f-0)1 .64046F-0) 2711101 272lC064A 40 1.061800-02 1.4s993f01-0 -9.021510-03 ?1.4)I16-01 9.919?98-0I -3.440441-01 -2 . F12196-01 -I .7119F-OS
s1I78t3 SERUM18 2.130361-06 9.610171-07 -6.160231-07 -6.726446-07 7.710026-07 -3..0219-06 -2?. 1340L -1-0 -2?.7212321-68471 -0.43780 -I.6121306- 3.009171-04 -6.11.1WIF-0 -1.10919[-0. 3.601291k-04 ).014401-0lj 1.0431al-0
797 1.41610f-04 3.61393-04 -I.012231-04 6.661917-09 1.317011-Cl -1-611470-04 -3.0166966-07 -1.086181F-07
SI17 SERU 160 3.433961-07 I.. 6730607-8.8697 -8.7116i0-06 3.072131-07 -1.364781-07 - I.164996 -Il -6.2 F1770- I.4 -2:09.1!0-01 -'996741.,-0. 614117f-06 -1.4,90ilt -06 -6.820606-06 9.661194f-06 1.91*141*-06 1.900841-06
6:4.4 2. 411t-01 l.011600-011 _::, a , 1-06 91.194761-06 6.09)90f-6-.0910 l9i9-06 -3.91 74of -06
Alo (17t -3.616176 00 -3.160616 00 -1..2097 00 -1.910931 00 -1.i01690 00 -1.16201f 00 -1.)7147T1 430 -I.11741 00
42 I.3206-0 -104710, -9.0446 -04 -9.0811024f 90611-0 -2.041114-04 -2.I9119F-01 61.21S.,56.01
~.20C -1.616111p n0 -I . 1l0 1f 00 -I.776971 00 -1.620061 00 -1.460731 00 -129300 -1.374 OFII l . -1. 377341 00
-105-
Table XfXA--continued
4~1l,~ -.. l'141 '-1.O0466&t 0J a . I I- F it01
N) -4. 4M 14 i-0 1 *4..11?111-J It 0 ..(l0AIf-0 1
c- -N. I IM511f 01 *9.R9110I-01 4.16.?1)( 0041 S4~e901 0.1 - I. , 116 F-flI -:.106lll-o
4,4 I01 14(-us -A.)A11tN!-06 - I.I0.10?- 0,
C L - I - l ob. 1 1-t *9-'. BOA M -0 1 1 27.9%I l74 044-* e.111011i o*.0 .I'4%7&1-rit -. 66 zI t- )I
%'7 -. qN%6nt-Oo -&.97117'4-o. .oOC
44> -I'1 r991 7-0)I -ll1101?lO F .190..Ift-01
"Cll -. 170kb 001 61.',08 111)0I -1.15969F -00I.t0 F r. A t-01 -7.8161190-U- 1.8164 It-0?
Albf"t -. 101-01 -*. 180',Rq -0. -4.9 1101 -01.2P4,, -1Ioi ol A.1.049r17 01 1.09090 0)
0.) S.1101111-01 -. 101171-0'. -I.604771t-04-1 -9. 1 198af01 -1. N779S1-0 -0110-01
L.4.l -P. 4104OF-0i 11.01181-oi 5.696 ?of-020.4- A .dI . 0114 f-01 MM 170-0.11 190h If -at
CC.- tS .01110,) 9Z.408111-0 -1.101091 00'4 . -1.1000 0 0 I.0 ?.11r..01 S -06 6. 161 st- l l0
'1 - A.h a IF- 0I 1 --. lsl0 .1100-0 l 11711-0.
PC,.. -1.011901.)F0 *S.*.,11-OR -4.911001101
"AIDCi. -l.0000010 'P.011596( at -I. 199119If01¶0. lsl?017-0l *.7.11191-04 -I.84110I-0l
COI0 .I'1416-02 .0.9 F F9of-0 - - 4. nI aI- 0"4P1. 7.6380111-02 9.171-1 1l1 -01
'4* -1.98039F-06 1111970 .09'400-06
'4A -1.)014l%01 on 09010 -1.61z9'7.1-K.TI l.1010 .090991-01 -0.91110-01CA.. 1.'.?199*-01 i.811-l-.140iZI-0l
ýC1UCO I.ItOOFF-0I -1.61698(-04 -J.4411l9-01"COO-. 1.104.1OF01 0-1J.61110111-01101 10
'401 1.-116111-02 .1.0sym8-01 -1.40l9¶l-011
C 1 .100141o-01 -5.4190016-05 089010t.C)0 1. 110401 01 l..900l' -0? -1.bq619711 0
URI&' -0.41)18-0l1 .3801-011 -. 191060-01"110r)s -I.600'.l-0l I.%.105s-02 l.111-0244100' 20.0914It 01 -1.19177Sb-01 -).5301F0-01
.4Ao~l .Io01l0 ?.07106o1-06 -6.0209Si-06
Col471 .4 -I.000000-0) -2.90501-04 l4.810000I-0-11)l -1.97411- I'f0$ .101041 0? -1.1910200-""ft.- -I.07719f.-0l Z.170011-Ol 5-.70011I-01"IR11- -4.01749f001 7 1.ao- IM 11.0116911-01HC111. l6.100611-l at .17091t-04 I.LI111111-01
64al'40 -146'9l0-M-0 k.779110-01 W1.10111710M64W0* -4.091101-0? l.S10969-Ol 9-1491111E 00
'4N~fM11- 1.61154,.0-01 1.4001101-01 I.S0901S-01"MO*1. 0I.011100-lI -*.2070011-O -1.90740f-01
0I -RAI--41 -7.S00)l-01 1.171400-01 1.1104000102.a-0 -1.1.17is101 00 .SD9401-01 7.1107701 0*CARS 1.4610119F00 *I.IF01017-0I, 4.6.0900900
URY1711 04 1.60%441-01 -1.1190174-01 1.44400t900"EO"I.Ro 4%9il00-0l 31.609991-09 1.11119161f004(140- -1.k691,0t 01 9.76971-0. -1.16S1166f 0.
'.111 tlu 1.614000010 -1 .804101-02 -1.7411211I-01111*11.4. 6.710165 0-01 .091111-02-1.10075U-01
'.1711AL "A10 -1.70401 1 11F4l-07 1.1 ?F0 .6000 110119I4444l 9.116701t00 1.7111801-011 1.64-0411-00
laic)1 ?.7Z0911P-0 -I.1111001-011 -9.910111100
%lift0 qFU .9000070 O 1.6%I071-07 -9.i4S011-0k@(D. 1.91946f 01 1.0011811-09 I.0I010If-01ACO- I.7111111-0i -I.0*910 -1.01849t 01-
All ftOl. I11'010I 1.99241-0I Wi.110701 0RAI. 1.2111-01 ol -. 111911-0l -1.11401.9101 0
( ~ q1(01 1. 640101-0 4 1.101 1111-Ol 1.79100
'4 -. 0S177-01 -1.061101-04 -1.97,6611-01
90 -9.096101-0t1 *.1691511-02 1.6.41914f 00
-106-
Table XIXB
INCREM4ENTS IN TOTAL MOLES IN EACH COMPARTMENT
PER UNIT INCRL.ASE IN INPUT SPECIES
JACUI)AlA'. M~IT I
I'LA&-A a16.St0 -4%?010 -. lo4E -OC mo IalA - hI 19I twa o I )1 It /.(.,4110 O., -1.0l111)1 L?
R-40 CELLS O?411-i-.48-? ?44q-l-.II9 01. .1644411 Ul4 ?1/919 III -/.Us4al1 0/ ..Ok ,h
0)E4Vtl -1 *. 10 %4 it 0% *S.08.949-06 1."119,b& -08 -I . ?.ý I909*0 -o 1 1 IM8F -01 9 9Il. 1. - l .4 ?-0 j -1 IhI'l ,
nslaf -41. aftio0- I .84JI -l -I. 1?/49 -11d l.0 /?-1 p.V1.4I9-UA - l 1'..9. At ." I if-04 .'4 09os-W -J p
$ill? '.14U. -l.0194149-U I..4304-11, -1. 11 If -1r1 -k 1.4%/-A tn 1.I9%-01 1 .10, -)1l -J. 1 449t -0 1 6. OU* 0l-0.
$1114. %f9ij- I.S649991- -//4'1l .1foilf-I/ 1. oil141-04 - 1. )k1141 -0I -L.S110119-111 .. 11)%14-0 1 6.S1 qI- UI
AIR n191 1.06.610 00 3.04,4101: U0 .0.4619 no 1.211061 00 -t.110001 00 -1.16101*1 0 -L.01114[ 00 -1.446891 00
/009I144. CYCLE I
CA.. 96. SI. 004. 4"41. LACIIC UO& G 61C-1 S
PLASMA OZ1'.~0 -%.S90%11 ')I -1.9040qb 02 -1.46611[ 01? .Sqo811 01 .4.449ni1 1 l 3~1( 00 1.k49419 00
RFD CELLS -P.0.Iko, 0/ S.1141off 01 I.1?490 1.91/11 0.14 .445 01 4.7/119F 11A .191O 9.11 I949-C0
alhlMF 101 AMR .9111101-01 1. 14,90"t -0 1 -1.0174111-01 -.901111f-0 1 1.4 14,1)(-0 1 -1.9104491-Ut I.V1.10/I -00, ?.91 W"' -(,6
OmlI4ImF '(149 -*.911 189-0 -1. 749049-01 1.8,9/941-01 9.,19191-01 -1.411241-01 1.07044F-O1 1. 9: 001 -V6 1.2I.! -Cc.
01 44RI" 4.1797-osj-0 /..07/111-06 -l%911C9-.041'1t0A I l.4/?ofl f -061. 6491-WAlm I.021641-i0 %.961190 -lI
%fill SERUM -4.?44741.0l.o *.0%1191-04 l.)9t*%I-04 I.19490-06 _1."46410_0 /..006 191, -0@ 4. I 4 " - 10 1. 1 id at-I n
Slit] stau. ?.1101*9-06 9.65009 pf,Pf -6. Wilf -1- -6. V14,40-01 1.?%00ll,-01 -l0I OeI-09 -Oh / 1401 -I() -1.1tll/it-l0
41194 SERUM 1.411941-01 I .'67 10C-0 1 -S./*46191-C, -0. 1l1416 If 1 .0 Il -It -0 . 1o19-l*.4 ?t-)1 1. 61 #,1- 1A -6 .? 1, 1 ,I1
AlI OUT -1.464118 00 -i.490849 00 -l.??0199 00 -1.9109it 00 -10.0169t 00 -1.10011 k) I0 -. I N 00 -L. illil It'D!
JACORIA4. tcYCI. I
MlSC11h498 *1clI0FUM wok.
991S49 -1.11419t 01 -1.044666 0.1 1.119I19 0?
4*0 CELLS ?.kisibi 01 z.06sma1 0. *1.F11011 U?
AF09410 9948 -1.64%441-01 1.400271-07 I.W(Il01-01
0911490 n14 144490 -1.400111-0? 1.4.4009 00
CIVITAPLt H" -1.904414-01 l.I?801-01' 1.600009 01
%I11 itI oUp 9.90001101t -I.6%0818-08 144190
SIM %taum,9 1.f.10000101 -1.4064,11-01 -1.1I1190-08
S1111 Situ. 4.100009 0!8oi 990 1.4.16,11,1-06
S11114 Slav" 1.900006 01 t111~011-07 .9410
Ala 0out l.1189SI-01 -%.ap0l01-0l 9./lOl09 00
-107-
decreases considerably more. In response to Na+ alone,
then, the red cells would go very acidic and the plasma
alkaline. Also, partly in response to the pH change,
hemoglobin oxygenation decreases.
This experiment cannot be performed in the laboratory
since a negative ion must also be added. If I meq of Cl
is a- 'ed simultaneously, however, the total response is
the sum of the two responses shown in the columns headed
Na+ and Cl. Thus, there is still a net Na+ transfer to
plasma; and although alnwjst two-thirds of the added C1
remains in the red cell--including the Cl transferred
in the Na+ column (5.99 x 10- meq) and the di.ptribution
of the added Cl (6.44 x 10 meq in plasma)--there is a
net inc:ease of 9.57 x 10"I meq of Cl" in the plasma
compartment as well, as a net transfer ,;f water to plasma.
These predicted effects will be proved when dry
NaCl is added as an experimental stress. Other details
of the response to NaC1 can be predicted from the matrix
as well as the response of the svytem to other small
stresses. For large stresses, s5trh calculations must be
#In the 'octorai thesis of Eugene MaKnier, MD.,
Temple Un~versltv Medicai Cent _r, Department of Physiology,Philadelphia (unpublished).
-108-
validated, since the partial derivatives are valid for
only that system state for which they were calculated.t
OXYGEN DISSOCIATION CURVES
Any simulation of the respiratory chemistry of blood
shuuld reproduce the standard oxygen dissociation curves--
the oxygen-loading and -unloading functions of normal
blood--when the pressure of oxygen to which it is sub-
jected varies from, say, 1 to 100 mm. This test is
fuodamental.
Since 1942, the data of Dill l161 on oxygen dis-
sociation, from the Mayo Aero-Medical Unit have been the
best available for human blooo. However, improved tech-
rtiques (the pH and oxygen electrodes) and more carefI.,"
attention to detail (especially at very low and verN high
hemoglobin saturation) may prod _c an improved curve 34'A
The new data may modify the presently best available
saturation curve by as much as 2 percent of its value in
some regions of pO,2. This airount is negligible for
clinical applications where calculations are seldom within
See the discussion of Mouel F in Chap. 1, Setc. 3.p. 31.
F.J.W. Roughton. F.R.S., private communicatlon,l9b•.
-109-1
5 percent; but in biochemistry, a 2 percent saturation
errur in the curve-fitting procedure may modify the
oxygen association constants by 100 percent or more, and
lead to an entirely different hypothetical microchemistry,
particularly in the associated reactions of hemoglobin.
The present model, as Ref. 2, uses the Dill data for
validation. The solid lines of Fig. 2 are the Dill curves
for human blood; the plotted points are various experiments
with the model, The points marked by circles result from
a series of computer experiments (described in detail
below) wherein the pCO 2 was determined empirically to give
the stated pH of the Dill curves at 100 mm of 0 These
points reasonably fit the laboratory data--w.ith less than
I percent rrcr. This order of error may not evc.> be
detectable clinically, but may be crucial for a research
hypothesis.
The "normal" curve (rCO., - 40 mm, plasma pH - 7.4)
of Fig. 2 is, of course. 4eneraLed with all "normal"
vilues in the liter of simuiated blood. e.t;. , normal buffer
ba1st- hemogiohbin, and gas pressures--all variables with
standard valas '-'orr the literatare. The circle data
ooints gho4 the starmiard copLater results holding pH
constant over the range et- p1,. Ttie laboratory report 16"
ovoacc -
E C.
0 0
-0o
-o -)
E >
o 0
x s
-0C
o z 6
E >
(xU
Z)11u~lwn~ps0&1ua0-a
-Ill-
does not indicate whether the pH and pCO2 were monitored
throughout the range of pO2 in the wet experiments. The
maintenance of both these parameters is now known to be
critical in obtaining an accurate saturation curve.
Dill computed ' pCO, for given pH values and vary-
ing p02, assuming constant HCO3 content. In fact, the
HCO3 varies uniformly with pH which, in turn, varies with
p0 2.. Consequently, to maintain a constant pH over the
rnge of pO2 in the saturation curve experiment, either
the pCO2 or the non-carbonate base must gradually change
to hold pH and HCO3 constant. If pCO 2 is held constant,
either NaOH or HCl must be added to hold the pH.
Conversely, to obtain the extreme pH ranges of 7.2
and 7.6 indicated in Fig. 2 by varying pCO2 onl., pCO2
values different from those quoted by Dill are recessary.
Whereas Dil computed a pCO2 of 25.5 for pH 7.6 and 61.3
for a pH of 7.2, the corresponding pCO2 values are in
fact closer to 20 rtr and 85 rmm, respectively. The
Henderson-Hasaelhach equotion also verifies these latter
ftiures, when HCO 3 varies in the presence of a protein4
buffer having pK 7.2. The crossed pctnts lying on or
See Edsall and Wvman, Ref 8, p. 569.
-112-
near the corresponding Dill curves were obtained by
changing the pCO2 to these new values and recalculating
the saturation curves at pH 7.2 and 7.6.
For all three of the curves represented by the
czossed points, the computer results are satisfactory,
i.e., practically indistinguishable from the Dill curves.
This is important to several pertinent theoretical ques-
tions. For example, the interaction of CO2 and H+ with
oxygenation is evidently simulatable, if not in exact
detail than in overall effect. Again, note that the
oxygenation parameters were determined for hemoglobin in
solution and not for whole blood [27. Yet when the
parameters were incorporated in the whole blood, they
gave the dame curve; i.., probably no U4naccountable
change occurs in the oxygenation of hemoglobin incorporated
into whole blood compared to hemoglobin solution.
A further implication of the goodness of fit concerns
the effect of pH on the satiration curve. From pO2. 1.00 mm
to p0.7 - 1 mm, the pl along the soturation curve drifts
basic by a-bout 0.1 pI unit. (This amount of drift is less
than that predicted from hetoglobin solution experiments--
0.2 , pil ýnits 14'--because the blood is nore hivhly buf-
f re.) However, the reasonable fit to the Dill curve
-113-
indicates that the pH was not controlled in the wet
experiments, and must have drifted approximately the
same amount.
With constant non-carbonate base in the blood and
constant pCO2, the pH of normal blood goes basic when
the blood deoxygenates, and vice-versa [141. The exact
amount of this effect varies both with the absolute
saturation and the absolute pH. Beginning at 100 mm pO2
and pH 7.4, deoxygenation causes the non-ccnstant pH
curve to deviate from the constant p13 curve, but by an
amount different than for the 7.2 or 7.6 curves [357.
The oxylab~ie side reactions of the protein produce this
phenomenon, related to the Bohr eff ct. Certain ionizing
groups, as yet not precisely specified, change pK ac-
cording to the state of a nearby oxygen site. Generally
speaking, the pK of nearby sites decreases as oxygen
attaches, although for certain sites it may rise with
oxygenation :36. Also, some of these sites are evidently
carbamino sites, since tve absence of COI, reduces the
shift in pH effect :35].
To test the difference between a curve run at
constant pH and one in hhich the pH drifted. HCI or NaOH
was added as though a laboratory pHStat had been used in
-114-
the wet experiment. The circle points plotted in Fig. 2
indicate the restilts.
While the error from the Dill curves is biochemically
important at low pO2, the absolute amounts of these effectP
are difficult to verify in the literature, because of the
absence of definitely comparable experiments. In several
published experiments which study the pH effects on
oxygenation, principal results are on hemoglobin solution
r14,28,36,371. Since the whole blood is buffered more
heavily than hemoglobin solution, the quantitative values
for whole blood are not evident from these experiments.
Several other papers note that for whole bllod the de-
crease in pH shifts the saturation curve to the right,
i.e., decreased oxygen saturation at the s... ! p0O 1as
expected); but generally this shift in pH is accomplished
by changing pCO., _38,39". In short, a careful rest of
this computer experiment w(ild be interesting.
TU!E "'ASTRUP" EXPERIMENTS
Figure 3 reproduces an Astrup-type chart for clinicnl
determinations of the state of whole blood. The straight
Copyrigt,t RADIOM?7'.R. Copenhagen.
-115-
* 4/$ '• r
I 4. -
S\ ,,
' -' /
.4
Fig.3---Astrup- type experiments
-116-
line A (from the literature '_40') represents an experi-
ment with normal adult male blood; D is a comparable
experiment computed from the present model.
Generhlly speaking, the slope of the lines drawn on
the Astrup chart frow, blood samples is proportional to
the buffer pcwer of the blood in respiratory acidosis.
The steeper the line, the better the blood will buffer
or increase in pCO2 (i.e., H+ ion). Thus, the line A--
or the line D (since they differ insignificantly)--gives
the nor:.al buffer slope for blood with the standard
amounts of buffering constituents. In Fig. 3, blood at
pH 7.38 and at 40 mm pCO. contains the amount of "standard"
bicarbonat;- given by the Henderson-Hasselbach equation,
here 23.8 meq/L blood. With other conditions constant,
but c'ianging pCO,2 the buffered blood will normally follow
line A (or D) intersecting the hemoglobin content curve
At 16 (17) gra.%m percent.
The L-ffering ,:ons-itu-ents are, in decreasing order
of importance, the carbon dioxide system itself, .•o-
glohin, plasma proteins, organic and inorganic -?hosphatri,
and subsidiary systems. Togethe-. thest sever~l con-
stituents represent tie anions capable of binding hydrogen
ion in the viable pH1 range and. except -or the CO, system.
are called the "buffer base" of the blood. The carbonate
system is not part of the buffer base because it is
(indirectly) the CO.? which is to be buffered. "Buffer
base" is an imprecisely defined term, since the hydrogen-
binding capacity of the proteinate and phosnhate factors
varies so considerably with pH, their "buffer power" being
directly proportional to the slope of their titration
curves at a particular pH. Nevertheless, the term "buffer
base" ks useful for the narrow range of pH in clinical
applications. At normal concentrations of protein end
at viable pH aild pCO 2, the imidazole ionizing groups of
protein plus the carbamino reactions and phosphate com-
plexes normally buffer as though about 45 to 46 meq per
liter of ionizing sites existed.
Additions of acids or bases, as in metabolic acidosis
or alkalosis without respiratory compensation, shoult
change an available buffer base of the blood by just the
a,,-int of Acid or base added. Because the solvent water
has a low ionization constant. the odded excess of hydrogen
or hydroxyl ion can not stay in scluticn in ionized form;
these ions musit essentially either bind to a hydro#n
binding site or remove a bound hvdrogt-n--th-is red.Icing or
increasing, respeztively. the availahle btaffer base.
-118-
When the buffer base is non-normal, indicating a
metabolic alkalosis or acidosis, the buffering reactions
of the blood to increased pCO2 are approximately the same
as for the normal. However, the buffer line on the Astrul
diagram will shift to the new range of pH corre3ponding
Zo the new value of the buffer base. Lines A' and D' of
Fig. 3 are, respectively, the ideal responses of whole
blood in the Astrup experiment and the response of the
computer model to changes in pCO2 after a2ddition of 10
meq of HCI per lite-, The slope of the buffer lines has
steepened slightly because the lower pH1 is closer to the
middle :& the range of imidazole ionization, so the
buffering is more efficient. The lines D' did not move
a full ten units or the base exce3s curve, although 10
meq of acid was added.
This discrepancy is difficult to rationalize. Ihe
literature n-tes that the standard curves are based on
blood samples from 12 persons, and that the same titra-
tion results were obtained with hydrochloric, lactic, or
acetic acids [40]. Yet at constant pCO2 , the computer
blood buffers HCI better than real blood; i.e., for
constant pCO2 , 10 meq of HC1 changes the pH less in the
model than the same amount added to human blood.
-119-
The discrepancy is large--about 10 percent of the
buffering capacity in this central pH range; and its
direction indipates that the model is already buffering
too well--i.e., the model does not need an additional
buffer system. On the other hand, the blood used to con-
struct the chart was probably not viable blood; it was
probably damaged by centrifugation, changes in temperature
and gas pressures, and by the addition of a fluoride
stabilizing agent [401. Some dennturation of the protein
by physical forces, plus slight hemolosis, plus fluoride
agents' interference with cell glycolysis could easily
make the blood a 10-percent-less-efficient buffer.
Line B in Fig. 3 is the iso-bicarbonate line, i.e.,
the line along which pH + log pCO2 - K is constant, and
the HCO 3 concentration is consequently also constant.
Edsali and Wyman [81 give line E for the pCO2/pH relation-
ship in 0.02 M sodium bicarbonate solution; the HCO 3
concentration is nearly constant. They also zemark that,
for this simple solution ca•e, the total COI, T, carried
ts relotively invariant with pCO2 , the change being only
from 1.7.i to 22.0 mM per liter, with a change in pCO2
from 0.45 to 63.0 mm.
-120-
Blood, however, for efficient action must have a
different property. Relatively large amounts of CO2
must be picked up or lost with small ii-.,:rements of pCO2
between venous and arterial blood. This is accomplished
through better buffering of the blood than the simple
solution affords. If the pH is almost constant, then, by
the Henderson-Hasselbach equation, as pCO2 increases
HCO3 must increase proportionately. Thus, buffering the
hydrogen ion is essential to CO2 transport.
From an earlier mathematical model of human blood 2
not containing tVie imidazole reactions in plasma and red
cell protein, line C in Fig. 3 indicates the buffer power--
and hence CO2 - transporc capacity. Rotating this Line to
the present increased buffer capacity (line D) was a matter
of including the titration reactions of plasma protein and
of hemoglobin. Of these two groups of proteins, that of
the red cell is more important becRuse of the carbanmino
reactions of hemoglobin with CO2w oxylabile ionizing
groups. and the large grotip of non-oxylabile sites.
-121-
REFERENCES
1. Dantzig, G. B., J. C. DeHaven, I. Cooper, S. H. Johnson,E. C. DeLand, H. E. Kanter, and C. F. Sams, "A Mathe-matical Model if the Human External RespiratorySystem," Perspectives Biol. & Med., Vol. 4, No. 3,Spring 1961, pp. 324-376; also, The RAND Corporation,RM-2519-PR, September 28, 1959.
2. DeHaven, J. C., and E. C. DeLand, Reactions of Hemo-globin and Steady States in Lhe Human RespiratorySystem: An Investi,,Aation Usinj Mathematical Modelsand an Electronic Computer, The RAND Corporation,RM-3212-PR, December 1962.
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DOCUMENT CONTROL DATAIORIGINAT;NG ACTIVITY 2a REPORT SECURITY CLASSI FICATIO0N
THE RAND CORPORATION Zb. GROUJP NUSFE
3 RE'"OT TITLE
ViE CLASSICAL STRUCT11RE OF BLOOD BIOCHEMISTRY- -A KATHEMAIICAL MODEL
4. AUTHOR(S) (Lost name, fi~st 4smt. tuvtGil
DeLo'nd, E. C.
5 REPORT DATEGoYTLNOFPGS 6N.OFRS
-July 1966 1547 CONTRAC.T OR GRAN~T No. 6 RGNTRSRPR o
AF 49(638)41700 -92P
9a AVAILABILITY/ LIMITATION NOTICES 9bý SPONSORING AGENCY
DDC I United States Air ForceProjeCt RAMD
10. &8%rRACT It KEY WORDS
A mathematical simulation of human Biochemistryblood biochemistry tiA~t includes the re- Medicinecults of a detailed chemical analysis of Biologyhuman blood under a variety of chemical Physiologystresses. Mathematical simulations of in.- Modelscreasing degrees of complexity are devel- Bloodoped. A rudimentary blood model assumesthe convenitional rcles of the fixed pro-tein_., the neutral electrostatic chargecon~straints, and the active cation pumpas the major characterlistics of hemostaticblood. The microscopic proper~tles of theproteins, particularly their bufferingbehavior, are incerpor,&ted into the modelby a mathematical procedure that assumesthat the serum albumin and the variousglobulinus represent all of the importantbuffering pover of the plasma fraction. A~model of the respiratory biochiemistry ofthe blood, embodying the result..; of thepreviou5 biochemical structural detail, Ais
tested under vario.,,s conditions. Proper-ties of the mathenrAtical model. such ,Ps
Sgas exchange. bufferinog, and respo'nse to-hemical ntress In the iteady stritt, arepractically indistinstuishahle from thosepropeýrties of real blood vith-.n the limitrof th~p present validation program. .135 PP9