introductory biophysics a. y. 2017-18 3. watermilotti/didattica/...edoardo milotti - introductory...

Introductory biophysicsA. Y. 2017-18

3. WaterEdoardo Milotti

Dipartimento di Fisica, Università di Trieste

Edoardo Milotti - Introductory biophysics - A.Y. 2017-18

Water is a very special small molecule: here is an incomplete list of its unusual properties

• it has a negative volume of melting• density max in the liquid range (4 °C)• several crystalline forms • high dielectric constant• high melting, boiling and critical temperatures• decreasing viscosity at increasing pressure• high mobility for H+ and OH- ions • both a donor and an acceptor of protons


The thermodynamics of water has a powerful role in the biosphere

biologically interesting region


Properties of some common solvents

highest thermal conductivity

highest heat capacity large thermal

diffusion coeff.

highest dielectric const.


large heat of vaporization


Subtle environmental effects related to the properties of water


Summer Winter

The unusual density vs. temperature curve stabilizes the temperature of deep waters in ponds and lakes.


The large surface tension – closely related to hydrogen bonds (see later) plays a role in the small world

γ = FL

γ = F

2

Work needed to increase area of one side of film

Fdx = γ dA = γ Ldx

Measurement of surface tension

Obviously this film has two sides, hence the factor 2


Surface tension of common liquids


Wetting, hydrofilic and hydrophobic interactions.

From Wikipedia


For water, a wettable surface may also be termed hydrophilic and a non-wettable surface hydrophobic.


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concentration of the various molecular players such as the case of the hydrolysis of ATP. Finally there are the cases where it is very hard to even define, never mind providing a concrete value. Examples of these subtle cases include the energy of a hydrogen bond, the free energies associated with the hydrophobic effect or the entropic cost of forming a complex of two molecules. While it is easy to clearly define and separate the length of a biological object from its width it is much harder to separate say the energy arising from a hydrogen bond from the other interactions such as those with the surrounding water. Together, the case studies presented in this chapter acknowledge the importance of energy in biological systems and attempt to give a feeling for energy transformations that are necessary for cell growth and survival.

Figure 1: Range of characteristic energies central to biological processes. Energies range from thermal

fluctuations to combustion of the potent glucose molecule. In glucose respiration we refer to the energy in

the hydrolysis of the 30 ATP that are formed during respiration of glucose.

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6

Energy of interactions in the cellular environment


196

persistence length. For DNA this length has been measured at roughly 50

nm (BNID 103112) and for the much stiffer actin filaments it is found to

be 15 μm (BNID 105505). The interplay between Coulomb interactions

and thermal effects for the case of charges in solution is governed by

another such scale called the Bjerrum length. It emerges as the length

scale for which the potential energy of electrical attraction is equated to

kBT and represents the distance over which electrostatic effects are able

to dominate over thermal motions. For two opposite charges in water, the

Bjerrum length is roughly 0.7 nm (BNID 106405).

The examples given above prepare us to think about the ubiquitous

phenomena of binding reactions in biology. When thinking about

equilibrium between a bound state and an unbound state, as in the

binding of oxygen to hemoglobin, a ligand to a receptor or an acid HA and

its conjugated base A-, there is an interplay between energies of binding

(enthalpic terms) and the multiplicity of states associated with the

unbound state (an entropic term). This balancing act is formally explored

by thinking about the free energy 'G. Thermodynamic potentials such as

the Gibbs free energy take into account the conflicting influences of

enthalpy and entropy. Though often the free energy is the most

convenient calculational tool, conceptually, it is important to remember

that the thermodynamics of the situation is best discussed with reference

to the entropy of the system of interest and the surrounding “reservoir”. Reactions occur when they tend to increase the overall entropy of the

Table 1: Length scales that emerge from the interplay of deterministic and thermal energies.

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energy and length scales


Bjerrum length:

distance at which the electrostatic interaction energy between elementary charges is of the order of kBT

Debye length:

decay length of electrostatic potential in ionic solution(see next slide)

Water has , therefore, at room temperature (T ≈ 300 K)

"r ⇡ 80

kBT =e2

4⇡"0"r`B

) `B =e2

4⇡"0"rkBT

`B ⇡ 0.7 nm


The Debye-Hückel theory brings us to colloids

A colloid is a mixture in which one substance of microscopically dispersed insoluble particles is suspended throughout another substance.


Distribution of ions close to a colloid (solid in liquid, Debye-Hückel theory)

The concentration of ions with charge q at distance x is proportional to the Boltzmann factor

and the total charge density is

In the proximity of the colloid, Poisson’s equation becomes 1D

n±(x) = n0 exp

✓⌥ q�

kBT

◆

r2� = �⇢

") d2�

dx2= �⇢

"

⇢(x) = qn+(x)� qn�(x) = qn0

exp

✓� q�

kBT

◆� exp

✓+

q�

kBT

◆�


Therefore, the Poisson equation becomes

and if we assume that the argument of the exponentials is small, we find

The solutions are exponentials with decay length (Debye length)

d2�

dx2= 2

q2n0

"kBT�(x)

d2�

dx2= �qn0

"

exp

✓� q�

kBT

◆� exp

✓+

q�

kBT

◆�

`D =

✓"kBT

2q2n0

◆1/2


The result holds also in the case of spherical symmetry

It is easy to prove that the solution of this equation is

This theory is used in many biophysically interesting cases, such as the electrostatics of viruses.

r2� =⇢

"

) 1

r2d

dr

✓r2

d�

dr

◆= �qn0

"

exp

✓� q�

kBT

◆� exp

✓+

q�

kBT

◆�

1

r2d

dr

✓r2

d�

dr

◆⇡ 2

q2n0

"kBT�(x)

�(r) =Q

4⇡"rexp(�r/`D)


The hydrogen bond

A hydrogen bond is the electrostatic attraction between polar molecules that occurs when a hydrogen atom bound to an electronegative atom experiences attraction to some other nearby electronegative atom.

The hydrogen is not a true bond but a particularly strong dipole-dipole attraction, and should not be confused with a covalent bond.


Biologically important H-bonds and functional groups


The strength of these bonds is intermediate between Van der Waals interactions (about 0.3 kcal/mol ≈ 1.3 kJ/mol), and covalent chemical bonds (about 100 kcal/mol ≈ 420 kJ/mol).

Note that the strength of the hydrogen bond in liquid water is

23.3 kJ/mol (≈ 5 kcal/mol) ≈ 0.24 eV/molecule

This must be compared with the thermal energy at room temperature

3/2 kT ≈ 0.04 eV/molecule


A quick estimate:

1 mole of water contains 2 moles of bonds ≈ 1.2·1024 bonds

and

the latent heat of vaporization (enthalpy of vaporization) @ 25°C is ≈ 44 kJ/mol

and this can be used for a rough estimate of the hydrogen bond strength in water:

3.7·10-20 J/bond ≈ 0.23 eV/bond


Average parameters for hydrogen bonds in liquid water with nonlinearity, distances and variances all increasing with temperature. There is considerable variation between different water molecules and between hydrogen bonds associated with the same water molecules.

(adapted from M. Chaplin, arXiv:cond-mat/0706.1355)


Bernal–Fowler ice rules (After the British physicists John Desmond Bernal and Ralph H. Fowler who first described them in 1933).

These rules state that:

• in ice each oxygen is covalently bonded to two hydrogen atoms

• that the oxygen atom in each water molecule forms two hydrogen bonds with other oxygens

so that there is precisely one hydrogen between each pair of oxygen atoms.


The structure of ice (hexagonal ice, the most frequent form of ice)

This figure shows thearrangement of the O atoms, as it was found early on, from X-ray crystallography.

Where are the H atoms?


... a possible arrangement of the H atoms in hexagonal ice ...


zoom in and observe the arrangement of the H atoms ...


The many possible arrangements of protons around oxygens in water lead to the

residual entropy of ice

i.e., a configurational entropic contribution that persists down to absolute zero.

Ice is the first substance where residual entropy was actually studied.


Linus Carl Pauling

Born: 28 February 1901, Portland, OR, USA

Died: 19 August 1994, Big Sur, CA, USA

Nobel Prize in Chemistry in 1954 "for his research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances”


Linus Carl Pauling

Born: 28 February 1901, Portland, OR, USA

Died: 19 August 1994, Big Sur, CA, USA

Nobel Prize in Chemistry in 1954 "for his research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances”

Nobel Peace Prize 1962, for arms control and disarmament, the only person who has won two undivided Nobel Prizes

See biography at this link http://www.nobelprize.org/nobel_prizes/peace/laureates/1962/pauling.html

http://www.nobelprize.org/nobel_prizes/peace/laureates/1962/pauling.html


J. Am. Chem. Soc. 57, 2680 (1935).

The basic idea in the paper is that each O atom is surrounded by 4 possible bonds and the H atoms can fill 2 of these bonds. Then there are

ways to fill these bonds.

42

⎛⎝⎜

⎞⎠⎟= 6


“checkerboard pattern” in 2D: black (white) sites are independent, because they are spatially disconnected; Pauling considered a similar configuration in 3D.


We can subdivide the whole ice lattice with N oxygen atoms in two sublattices, each with N/2 atoms. The atoms in each sublattice are independent, within a given sublattice.

Then, there are 6N/2 ways to fill the bonds for the O atoms in the first sublattice.

However, this is likely to produce a wrong configuration in the other sublattice. Since the arrangements in the first sublattice yield 24 = 16 ways to fill/not-fill the bonds for an atom in the adjacent sublattice, but only 6 are correct, the probability of randomly filling the bonds in the first sublattice and still find a correct configuration is 6/16 per atom in the adjacent sublattice.

Then there are approximately 6N/2 (6/16) N/2 configurations.


2D example


This means that the residual entropy of ice is

and for one mole of ice this corresponds to

Experiment:

S ≈ kB ln 6N 2 6

16⎛⎝⎜

⎞⎠⎟N 2⎡

⎣⎢

⎤

⎦⎥ =

Nk2ln 94= NkB ln

32≈ 0.405NkB

S ≈ 0.405NAkB≈ 0.405R≈ 3.37 J mol−1 K−1

≈ 0.806 cal mol−1 K−1

Sexp ≈ 0.82(5) cal mol−1 K−1


... this is in very good – although not perfect – agreement with experiment.

• The problem of the exact evaluation of residual entropy has led to the development of the so-called “ice models” in statistical mechanics.

• Exact solution of 2D “square ice” in 1967 (Lieb)

• There are no exact solutions, but only analytical approximations for 3D ice models

• Further estimates can be obtained from numerical solutions for 3D ice models


Surprisingly, ice is a “protonic semiconductor”, and can be fashioned into a semiconducting diode!

M. Eigen and L. De Maeyer, “Self-dissociation and protonic charge transport in water and ice”, Proc. R. Soc. Lond. A 247 (1958) 505


Stillinger& Rahm

an, J. Chem. Phys. 60 (1974) 1545


hydronium

hydroxide

Ionic species


The Grotthuss mechanism(C.J.T. de Grotthuss, Ann. Chim. 58 (1806) 54)


The shape of the potential changes as the distance between O nuclei changes, therefore the proper description of this potential requires two variables (position along the O-O axis + R00).

Low barrier, usually no tunneling

Strong, symmetric bond


(a very incomplete) reference list

• E. H. Lieb, “Exact solution of the problem of the entropy of two-dimensional ice”, PRL 18 (1967) 692

• E. H. Lieb, “Residual entropy of square ice”, Phys. Rev. 162 (1967) 162 • D. Marx, “Proton transfer 200 years after von Grotthuss: insights from ab initio

simulations”, ChemPhysChem 7 (2006) 1848• D. Marx, M. E. Tuckerman, J. Hutter, and M. Parrinello, “The nature of the

hydrated excess proton in water”, Science 397 (1999) 604

introductory biophysics a. y. 2017-18 3. watermilotti/didattica/...edoardo milotti - introductory...

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