lecture 9' - introduction to modeling structural transitions

8/14/2019 Lecture 9' - Introduction to Modeling Structural Transitions

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Introduction to Modeling

Biopolymer Structural Transitions


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General Principles

Most measurements reflect average behavior of a

population of biopolymers.

e.g., a proteins native state is an average over many similar

conformations, all of which exhibit activity.

Statistical Thermodynamics allows a discussion of:

how a population is distributed over the accessible states;

the resulting mean values of the physical properties.

Two types of average behavior:

time-average of a single molecule;

instantaneous average over an ensemble of molecules.

Fundamental Assumption:

time-average and ensemble-average behaviors are

identical.


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Modeling Structural Transitions

Statistical Thermodynamics most useful for modelingphenomena involving multiple states: transitions between biopolymer states;

assembly of complexes from multiple subunits;

binding of multiple ligands to macromolecules.

Our focus: Modeling Structural Transitions Formalism originally developed to predict biopolymer melting;

also useful for predicting protein/nucleic acid structural

transitions. Outline:

Lecture 9: Intro. to Modeling Structural Transitions.

Lecture 10: Structural Transitions in Polypeptides/Proteins.

Lecture 11-12: Structural Transitions in Nucleic Acids.


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Lecture 9 Introduction to

Modeling Structural Transitions


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Modeling The Two-State Transition

In the simplest case:

A biopolymer is modeled as having two only states, A and

B.

if the two forms are at equilibrium, mass action gives:

Consider an ensemble of such biopolymers

a large number of identically prepared systems;

roughly 10

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copies. which is at equilibrium; (constant V, P, N, T)

What is the probability/member of occupying B?

From simple probability considerations and the law of mass

action, we know:

P = B / A + B = K / 1+K .


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The Statistical Weight

Let state A be defined as the reference state energies of all conformations then defined relative to that of

the ref. state. generally, this is defined as our state of zero free energy.

Then, let the statistical weight (i) of any state i: also be defined relative to the reference state,

in terms of relative equilibrium concentrations.

For our simple, 2-state system, we have 2 weights:

A = [A]/[A] = 1 B = [B]/[A] = Keq

Now, PB for can be rewritten in terms ofi:

PB = B/(A + B) = B/(1 + B)

Weight of the reference state generally denoted by o = 1.


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Generality of the Statistical Weight

i appears to be a simple redefinition of Keq.

however, i is more general:

Keq is a macroscopic measure of all products to all reactants.

often includes several pathways

In contrast, i relates occupancies of each state, i.

to that of the reference state, of weight o.

is essentially a micro-equilibrium constant;

very useful for systems with more than two states.

How do we estimate statistical weights,

for a physical system of interest?

by computing the Gibbs Factors.


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The Gibbs Factor

At equilibrium, the statistical weight, j of state j:

is related to the standard Gibbs free energy (Gjo) of

formation of state j

relative to the free energy (Goo) of the reference state:

Gjo

= Gjo

Goo,

In particular, j = exp[-Gjo/RT];

R = molar gas constant.

In this form, j is called the Gibbs Factor. For our 2-state system, PB can be rewritten:

in terms of the Gibbs factor of state B:

PB = exp[-GBo/RT]/(1 + exp[-GB

o/RT]).

the state with the lowest free-energy is most favorableo


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The Partition Function, Q

The partition function, Q of a system,

is the sum of the statistical weights of all system states:

Q = ii = i exp[-Gio/RT].

For our 2- state system:

Q = exp[-GAo/RT] + exp[-GB

o/RT]

= 1 + exp[-GBo/RT] .

Q facilitates computation of system probabilities:

the equilibrium probability of occupying state j:

Pj = j/Q = exp[-Gjo/RT]/Q.

Q also allows computation of ensemble average

quantities:

such an the mean free energy or entropy of a biopolymer.


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Computing Ensemble Averages

The ensemble average of observable quantity, X, at equilibrium

is computed from Q by simply taking a weighted average:

= i Xi exp[-Gio/RT]) / Q = i Xi Pi.

Here, Xi is the X value characteristic of state i.

For our 2-state system, we have:

= (XA exp[-GAo/RT] + XB exp[-GB

o/RT]) / Q.

= (XA + XB exp[-GBo

/RT])/Q It turns out that every intrinsic macroscopic quantity

will correspond to an ensemble average.

Q thus contains a complete specification of thesystem: if we know Q, we should be able to predict system behavior.


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Example 2 Flipped Coins

To illustrate probabilistic nature of our treatment:

consider the simple, analogous example of 2 flipped coins.

Each coin can land either heads (h) or tails (t). 4 outcomeshh, ht, th, tt (system microstates).

System (both coins) can assume 3 macrostates (j):

State HH: j = 2 headsunique (g2 = 1only hh).

State HT: j = 1 headdegenerate (g1 = 2ht and th).

State TT: j = 0 headsunique (g0 = 1only tt).

so the macrostate, j is the total number of heads.

Assume both coins are fair: So, the coin states (h, t) have equal weights of 1:

h and t = 1.

Weight of each system macrostate, jis given by:

j = gjhj

t2-j


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2 Flipped Coins: Average Quantities

Our Partition Function: given by the sum of the weights, taken over system states:

Q = j = (1)(1)2(1)0+(2)(1)1(1)1+(1)(1)0(1)2= 4.

Probabilities of State Occupancy:

for the 3 macroscopic states, HH, HT, and TT PHH = 2/Q = ,

PHT = 1/Q = 2/4 = ,

PTT = 0/Q = . (note: PHH + PHT + PTT = 1!)

Ensemble Average Quantity: Mean Number of Heads i.e., the mean number after many independent trials.

estimated by a weighted average:

= j j j/Q.

= [(2)(1) + (1)(2) + (0)(1)] / 4 = 1.

note: ensemble average quantities denoted by s. the expected result, from probability considerations.


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Modeling Structural Transitions in

Biopolymers

The 2-state system is conceptually simple. Adopting a 2-state model for a biopolymer:

Process described as an all-or-none transition; Then, weights of only two states are considered;

Useful for assessing the total Keq of formation.

However: yields no information about folding intermediates. e.g., partially folded states.

We need a more complete description much more complicated.

Simplification: Focus on 2o

structure limits the problem to transitions b/w well-defined states:

For polypeptides: between a random coil and an -helix.

alternatively, between an -helix and a -strand.

For polynucleotides:

between a pair of random coils and a double-strand (helix). alternatively, b/w a B-helix and an A- or Z-helix.


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Generalized Two-State Modeling

Each residue in a biopolymer may assume many states. e.g., ranging from fully helical to fully coil.

Simplification: residue-level, 2-conformation model. 2

ostructure formation all-or-none for each residue.

DNA: each base-pair either stacked or unstacked. or H-bonded or non-H-bonded, in our simple treatment.

Polypeptide: each residue either H-bonded or free.

Again, we consider the overall transition: from a reference or starting state (A) to a new state, (B).

But, each residue has either the properties of A or B. i.e., is in state a or b.

Limiting states: A = aaaaaa (all residues have the properties of A)

B = bbbbbb (all residues have the properties of B)

Many intermediate states: aaabbaa


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Ensemble Average Fraction of bs

Ensemble averages are also obtainable from Q by constructing a weighted average.

Example: Mean Fraction of bs Denoted by .

j and fj = j/N are the number and fraction of bs in state j.

is the weighted average of fj:

= j fj j/ Q = [(0)(1) + (1)(sN)]/(1+s

N) = s

N/(1+s

N).

Parameter s: related to mean free energy of residue conversion, Goab,by the Gibbs factor:

s = exp[- Goab / RT].

when G

o

ab < 0, is greater than . when Goab = 0, = , as expected.


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Model 2: Non-Cooperative

Transition from A to B, length N polymer. Reference state = state A.

again, residues have properties of A or B.

a = [a]/[a] = 1, b = [b]/[a] = s.

residue transitions non-cooperative ( = 1): each residue converts independently.

Result: Biopolymer has N+1 states

chains with equal j values are equivalent;

have the same weight, sj.

grouped into a single chain state, j.

states then enumerated by j:

N + 1 statesj = 0,, N.

Weight of state j:

j = gj s

j

= N!/[j!(N-j)!] * sj

.


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The Partition Function, Q

Degeneracy of state j:

Given by: gj = N! / [j!(N-j)!] = C(N,j).

the binomial coefficient;

number of ways of placing j balls in N boxes.

Partition Function:

equal to the sum over all states:

Q = j sjN!/[j!(N-j)!].

but this has the form of a binomial expansion, which

reduces to:

Q = (1 + s)N

e.g.; (N = 4): Q = 1 + 4s + 6s2

+ 4s3

+ s4

= (s+1)4

.


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The Propagation Parameter, s

In the non-cooperative model, the propagation

parameter:

expresses the probability that any residue converts from

a to b;

s = [b]/[a] = exp[-Go

ab /RT].

where Go

ab is the free energy change of conversion.

Intuitively, transition from A to B occurs a series of micro-

equilibria i.e., a series of 1-residue transitions;

Thus, s is the equilibrium constant for each micro-transition.


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Equilibrium Occupancy of State j

Probability of observing state j: i.e., one ofgj physically distinct, equivalent chains

with j bs;

denoted by Pj.

again, given by a ratio of statistical weights:

Pj =j/Q

= C(N,j) sj/(1 + s)N ; C(N,j) = N! / [j!(N-j)!]

For the special case in which s = 1, We note that,G

o

ab = 0.

Residue occupancies are randomsince a = b = 1; Each residue executes a random walk.

Then, the most likely state = most degenerate state. C(N,j) is maximum for state, j = N/2.

Example: For N = 4most likely state is j = 2. degeneracy, g

2

= C(4,2) = 6.


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Ensemble averages again obtainable from Q

By constructing a weighted average.

Example: Mean Fraction of bs:

Denoted by . fj = j/N = fraction of bs in state j.

= j fjj/Q = j (j/N) C(N,j) sj/(1+s)

N

= s/(1+s)

The mean Fraction of as is then: = 1 - = 1/(1+s).

so that

/ = s, as we expect.


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Utility of the Non-Cooperative

Model

Since each step is independent, this model

may be used to describe a random walk.

As well as non-cooperative binding of multiple ligands:

e.g., proteins with multiple, independent active sites.

Most Biopolymer 2o

structure transitions:

exhibit an intermediate level of cooperativity

neither the all-or-none, nor non-cooperative models suitable.

more sophisticated model required.


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Modeling Intermediate Cooperativity

ModelTransition from A to B, length N polymer.

a and b weighted: a = 1, b = s.

howevertransitions are cooperative (1 > > 0).

assigned to each ba interface, and each terminal b.

degree of cooperativity: relative size of s and .

= 1non-cooperative model.


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The Partition Function

Again, we have N+1 states each denoted by the value of j.

where j denotes the number of bs in the chain.

Each state has much smaller degeneracy:

Given by: gj = N-j+1; for N >= j > 0. Number of ways of placing a string of j bs in N positions.

Here, go = 1 (treated as an exception).

Partition Function: sum of the weights of all states.

Q = 1 + j>1j= 1 + j>1 (N - j + 1) s

j,


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Ensemble averages again obtainable from Q. by constructing a weighted average.

Estimating the Mean Fraction of bs (< Pb >):

j and fj are the number, fraction of bs in state j:

we have: fj = j/N.

Weighted average offjyields :

Note sums include a term sj, but not

polymer length contributes to propagation, but not initiation. s, thus experimentally resolved by studying vs. N.


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Degree of Cooperativity

Determined by the size of: sharpness of the transition increases with decreasing .

remember that s varies with T:

s = exp[-Go

ab/RT]

often called the cooperativity parameter.

Limiting cases correspond to our other models:

Case I - if


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Applicability of the Zipper Model

Applicable to a wide range of biopolymer transitions: formation of multi-meric complexes by subunit assembly.

e.g., viral coat assembly.

formation of regular structures along a biopolymer chain.

transition of a coil to an -helix (polypeptide folding);

transition of a B-helix to a pair of ssDNA coils (DNA melting);

The reverse process.

the B- to Z-helix transition (DNA).

Application requires assigning physical meanings to

and s. this allows the definition of each state:

in terms of an experimentally-meaningful statistical weight.

thus, allows estimation of the systems partition function (Q).


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The Nucleation Parameter,

Again, determines process cooperativity. non-cooperative processes:

should be modeled with ~ 1.

processes which exhibit a higher cooperativity: modeled with smaller values of.

intrinsic to each type of biopolymer: different values for different biopolymers

even if the transition appears to be similar.

Also specific to the type of transition: e.g., -helix/coil, -helix/-strand, etc

independent of Temperature and Length: Often also considered roughly independent of sequence.

Once determined for a given polymer and transition will be equally applicable to any instance.


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The Propagation Parameter, s

Dependent on the specific interactions: that stabilize or destabilize a and b.

Specifically, s = exp(-Go

ab/RT)

where Go

ab = Go

b- Go

a.

since Go

ab = Ho

ab - T So

ab, s consists of both: T-dependent component exp(-H

o

ab/RT);

T-independent componentexp(So

ab/R).

both of which can determined experimentally

or, if desired, can be estimated by simulation.


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Forward

In this lecture, we have described: the basic ideas behind statistical thermodynamic modeling:

as applied to state transitions of a polymer chain.

the Zipper model approximation: and its dependence on

as well as its two limiting cases: the all-or-none model (

lecture 9' - introduction to modeling structural transitions

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