Introductory biophysics A. Y. 2017-18 4. Basics of ...milotti/Didattica/...Edoardo Milotti - Introductory biophysics - A.Y. 2017-18 Multilevel statistical system Consider a macrostatedefined

Introductory biophysicsA. Y. 2017-18

4. Basics of statistical mechanics and chemical kinetics

in biophysical processes

Edoardo MilottiDipartimento di Fisica, Università di Trieste

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

Extremely short review of statistical mechanics1. Boltzmann factor

Heat exchange

Thermal reservoir

at temperature TPhysical system,

total energy E

exp − EkBT

⎛⎝⎜

⎞⎠⎟

Probability of finding system with energy E is proportional to

Extremely large number of degrees of freedom

Much smaller number of degrees of freedom


Multilevel statistical system

Consider a macrostate defined by

N1 particles at energy level E1 with degeneracy g1N2 particles at energy level E2 with degeneracy g2... Ni particles at energy level Ei with degeneracy gi...

the number of ways in which we can arrange the identical particles in the M levels is

and when we also include degeneracy, we find that the number of different ways to obtain the macrostate (thermodynamic probability) is

Ω = N!g1

N1

N1!g2N2

N2 !…giNi

Ni !…

N!N1!…NM !


We use Stirling’s approximation

and we find

Now the problem is finding the distribution {Ni} that maximizes the thermodynamic probability (this is the distribution that is observed with the highest probability)

lnΩ = N lnN − N( ) + Ni lngi − Ni lnNi + Ni( )i∑ = N lnN + Ni ln

giNii

∑

= N Ni

Nln giNi Ni

∑ N = Nii∑⎛

⎝⎜⎞⎠⎟

lnn!≈ n lnn − n


Maximization must be carried out constraining both the number of particles and the total energy

For constrained maximization we use the method of Lagrange multipliers and maximize the auxiliary function

lnΩ = N Ni

Nln giNi Ni

∑

N = Nii∑

U = EiNii∑

expression of thermodynamic probability

total number of particles is fixed

total energy is fixed

lnΩ + λN − βU = Ni lngi

Ni Ni∑ + λ Ni

i∑ −β EiNi

i∑


lnΩ + λN − βU = N lnN + Ni lngiNii

∑ + λ Nii∑ −β EiNi

i∑

∂∂Nl

lnΩ + λN − βU( ) = ln glNl

−1+ λ − βEl = 0

Nl = gleλ−1−βEl


Partition function (Zustandsumme)

Ni = gieλ−1−βEi

the partition function is used to determine many other thermodynamicalfunctions

N =X

i

gie��1��Ei = e��1

X

i

gie��Ei ) e��1 =

NPi gie

��Ei=

N

Z

Ni = gie��1��Ei =

N

Zgie

��Ei

Z =X

i

gie��Ei

U =X

i

EiNi =N

Z

X

i

Eigie��Ei = �N

Z

@

@�

X

i

gie��Ei = �N

Z

@Z

@�= �N

@ lnZ

@�


The entropy is a measure of the thermodynamic probability

and when we recall the thermodynamic relation

we find

1T= ∂S∂U

S = kB ln⌦ = kBNX

i

Ni

Nln

giNi/N

= kBNX

i

Ni

N(lnZ + �Ei)

= kBN lnZ + kB�U

T =1

kB�


A chemical thermodynamics refresher

1. Enthalpy

Recall that a change in internal energy is the sum of the heat absorbed and of the work done by the system

which is the first principle of thermodynamics, and that work can be further subdivided into work due to volume expansion (useless) and all the other work (non-PV work):

ΔU = ΔQ − ΔW

ΔW = PΔV + Δ ′W


Then

and we can formally restore the form of the first principle by defining the state function enthalpy

so that

ΔU = ΔQ − ΔW = ΔQ − PΔV − Δ ′W

H =U + PV

ΔH = ΔU + Δ PV( ) = ΔQ − Δ ′W

at constant pressure Δ(PV) = PΔV, as in most chemical reactions in the laboratory

if no non-PV work is done on the system, then the enthalpy change corresponds to the heat absorbed by the system


2. Helmholtz free energy

According to the second principle of thermodynamics

and therefore

Therefore, when we define we find

ΔQ ≤ TΔS

ΔU = ΔQ − ΔW ≤ TΔS − ΔW = Δ TS( )− SΔT − ΔW

F =U −TS

ΔF = Δ U −TS( ) ≤ −ΔW − SΔT


Therefore, in isothermal processes (where the system exchanges heat with a large heat bath)

and therefore the work done by the system is less or equal than the decrease of free energy

ΔF = Δ U −TS( ) ≤ −ΔW − SΔT = −ΔW

ΔW ≤ −ΔF

isothermal process


Therefore, in processes where no work is done or absorbed by the system

i.e.

and this is the condition for a spontaneous process with no work involved.

0 = ΔW ≤ −ΔF

ΔF ≤ 0


3. Gibbs free energy

The Gibbs free energy is like the Helmholtz free energy, for processes where the pressure is held constant:

and we find again that the condition for a spontaneous process, with no work involved, is

G = H −TS =U + PV −TS = F + PV

ΔG ≤ 0


Let’s summarize it again, just for clarity ...

then

and therefore for transformation at constant temperature and pressure and no non-PV work

ΔU = ΔQ − ΔW

= ΔQ − PΔV + Δ ′W[ ] = ΔQ − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦

≤ TΔS − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦

= Δ TS( )− SΔT⎡⎣ ⎤⎦ − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦

Δ U + PV −TS( ) ≤ −SΔT +VΔP − Δ ′W

ΔG ≤ 0H =U + PVF =U −TSG = F + PV = H −TS


ΔG = ΔH −TΔS ≤ 0 ΔH ≤ TΔS


Concentrations and Gibbs free energy

Entropy of mixing in binary solutions

n1 molecules of solventn2 molecules of solute

N = n1 + n2

Then the number of configurations is

Ω = N!n1!n2 !


Ω = N!n1!n2 !

lnΩ ≈ N lnN − N( )− n1 lnn1 − n1 + n2 lnn2 − n2( )= N lnN − n1 lnn1 − n2 lnn2

= n1 + n2( )ln n1 + n2( )− n1 lnn1 − n2 lnn2

= −n1 lnn1

n1 + n2− n2 ln

n2n1 + n2

= −N X1 lnX1 + X2 lnX2( )X1,2 are the volume fractions


Therefore the entropy change due to mixing is

and, assuming that there is no change in contact energy when the molecules of solvent and solute mix, the corresponding Gibbs free energy change is

ΔSm = kB lnΩ− ln1( ) = −kBN X1 lnX1 + X2 lnX2( )

ΔG = −TΔSm = kBNT X1 lnX1 + X2 lnX2( )= nRT X1 lnX1 + X2 lnX2( )= X1ΔG1 + X2ΔG2


We see that we can associate a free energy to each substance A in solution

and in particular, if we consider the free energy change with respect to standard conditions

and if we let [A]0=1M

volume fractionsconcentrations (mole/l)

1 mole/l

�GA = nART lnXA (nA = nXA)

�GA ��GA0 = nART lnXA

XA0

= nART ln[A]

[A]0

�GA ��GA0 = nART ln[A]


Chemical kinetics

The elementary reaction

can occur via a sequence of elementary reactions, with intermediates, e.g.,

A→ P

A→ I1→ I2 → P


Rate equations

The rate at which a reaction proceeds is proportional to the probability of bringing all the reactants in the same place at the same time, i.e., it is proportional to their concentrations, therefore the rate of the general elementary reaction

is aA +bB +…+ zZ→ P

rate = k A[ ]a B[ ]b… Z[ ]z n = a + b +…+ z

order of the reactionrate constant (notice that the rate constant has units adapted to the order of the reaction)


Example: first order reaction

A→ P ⇒

d A[ ]dt

= −k A[ ]A[ ]+ P[ ] = A[ ]0

A[ ]t=0 = A[ ]0P[ ]t=0 = 0

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⇒A[ ] = A[ ]0 exp −kt( )P[ ] = A[ ]0 1− exp −kt( )⎡⎣ ⎤⎦

The concentration of A decreases and it is exactly half the initial concentration when

A[ ] = A[ ]0 exp −kt1/2( ) = A[ ]0 2 ⇒ t1/2 =ln2k


Example: second order reaction

2A→ P ⇒

d A[ ]dt

= −k A[ ]2

A[ ]+ 12P[ ] = A[ ]0

A[ ]t=0 = A[ ]0P[ ]t=0 = 0

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⇒

1A[ ] =

1A[ ]0

+ kt

P[ ] = 2 A[ ]0 − A[ ]( )

The concentration of A decreases and it is exactly half the initial concentration when

2A[ ]0

= 1A[ ]0

+ kt1/2 ⇒ t1/2 =1

k A[ ]0



Equilibrium constants

We apply these concepts to the reversible chemical reaction

and we note that at equilibrium

i.e. the forward rate is equal to the backward rate, or also

aA + bB cC + dD

k f A[ ]a B[ ]b = kb C[ ]c D[ ]d

C[ ]c D[ ]dA[ ]a B[ ]b

=k fkb

= Keq


Then, the free energy change for the i-th species with respect to the standard state, per mole, is

and therefore, in a reaction, the Gibbs free energy change splits into parts that take into account the chemical bonds and the concentration changes

ΔG = ΔG0 + cΔGC + dΔGD − aΔGA − bΔGB

= ΔG0 + cRT ln C[ ]+ dRT ln D[ ]− aRT ln A[ ]− bRT ln B[ ]

= ΔG0 + RT lnC[ ]c D[ ]dA[ ]a B[ ]b

= ΔG0 + RT lnKeq

�Gi ��Gi0 = RT ln[i]


At equilibrium the free energy change vanishes

ΔG = ΔG0 + RT lnC[ ]c D[ ]dA[ ]a B[ ]b

= ΔG0 + RT lnKeq = 0

Keq = exp − ΔG0

RT⎛⎝⎜

⎞⎠⎟


Keq = exp − ΔG0

RT⎛⎝⎜

⎞⎠⎟

R ≈ 8.314 J K−1mol−1, RT ≈ 2.5 kJ mol−1@ 300 K( )

this is close to the binding energy of hydrogen bonds in water ≈ 5 kcal/mole ≈ 21 kJ/mole

Exponential dependence on DG0


Dissociation reactions

AB ! A+B ) [A][B]

[AB]= K

K is called the dissociation constant. K is large when the denominator is small with respect to the numerator (the substance is mostly dissociated).

K is measured in units of concentration. Notice also that when [B]=[AB] (half of B is bound and half is dissociated), then [A] = K.

Finally, it is common to define

pK = � log10 K


The dissociation constant of water is important

The concentration of water is omitted by convention.

KW = [H+][OH�]

log 1

0K

W

0.0028 0.0030 0.0032 0.0034 0.0036-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

1/T (K)Plots like this (where the indipendent variable is the inverse temperature) are called "Arrenius plots"


Redox reactions and the Nernst equation

Example of a Redox reaction (Oxydation – reduction: reduction = acceptance of electrons, oxydation = loss of electrons)

This can be divided in half-reactions (redox couples)


from

Voe

t & V

oet -

Bioc

hem

istry


Now consider the generic redox reaction

Just as in the case of the binary reaction

we find

aA + bB cC + dD �G = �G� +RT ln[C]c[D]d

[A]a[B]b

�G = �G� +RT ln[Ared][Bn+

ox ]

[An+ox ][Bred]


This is a non-equilibrium situation, where there is transfer of electrons (as in a standard electric battery), and we must use the equation for non-PV work

= Faraday constant

= E.M.F.

(Nernst equation)

�G = �W = �nF�E

FE

�E = �E� � RT

nF ln[Ared][Bn+

ox ]

[An+ox ][Bred]


Nernst equation is important in calculations of the membrane potentials.

For instance in the case of potassium ions, the following (equilibrium) version of Nernst equation holds

(o = outside the cell, i = inside the cell)

�E� =RT

nF ln[K+]o[K+]i


ATP (Adenosine Triphosphate) is a basic element in the energy budget, it is a temporary energy store, and a sort of molecular energy currency

3 phosphate groups

ribose

adenine ring

ATP


Typically, ATP is unstable, and its reduction to ADP or AMP produces heat.

In the cell environment, the energy released by the removal of one or two phosphate groups, is used to power other reactions (like protein synthesis) that are endothermal (and could not proceed without a source of energy)


ATP

ADP

AMP+H2O

+Pi

+PPi

ΔG = −30.5 kJ/mol(−7.3 kcal/mol)

ΔG = −45.6 kJ/mol(−10.9 kcal/mol)

Under typical cellular conditions, ΔG is larger, and is approximately −57 kJ/mol (−14 kcal/mol).


N.B. we shall meet AMP again, as a building block of nucleic acids ...





Halobacterium salinarum

Halobacteria are a class of the Euryarchaeota (Archaea) found in water saturated or nearly saturated with salt

Energy production in halobacteria


aerial view of the Great Prismatic Spring - Yellowstone


Charge differences can be generated by charge transport across membranes


Bacteriorhodopsinis a ~26 kDa transmembrane protein that acts as a light-driven proton pump in Halobacterium salinarum, converting light energy into a proton gradient.

bR is the only protein constituent of the purple membrane (PM), a two-dimensional crystal lattice naturally present as part of the membrane of the bacterium.

view from above

sideview(green lines define the cell membrane)


Top view of the purple membrane patch. The hexagonal unit cell is displayed in the middle of the patch, surrounded by white line defining the unit-cell dimensions.

(from http://www.ks.uiuc.edu/Research/newbr/)

http://www.ks.uiuc.edu/Research/newbr/


transmembrane view


Bacteriorhodopsinhas different conformational states that are spectrally distinguishable

absorption of light quantum

proton pumped into the outer environment

proton from cytoplasm


retinal (retinaldehyde): is one of the many forms of vitamin A (the number of which varies from species to species). Retinal is the chemical basis of animal vision.

It is the core functional element of bacteriorhodopsine (and many other light-sensitive molecules).



It has been speculated that charge, i.e., proton, transport is mediated by the formation of Grotthuss water wires inside the bRmolecule in the intermediate states.

Recently this has been experimentally confirmed.

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ATP Synthase, the engine of ATP production

“ATP synthase is one of the wonders of the molecular world. ATP synthase is an enzyme, a molecular motor, an ion pump, and another molecular motor all wrapped together in one amazing nanoscale machine. It plays an indispensable role in our cells, building most of the ATP that powers our cellular processes. The mechanism by which it performs this task is a real surprise.”

(RCSB – Molecule of the month, Dec. 2005)


• Two motors, F0 and F1

• F0, powered by flow of protons

• F1, powered by ATP

• Motors are connected, and one can force the other into a generator.





Materials 2013, 6 5833

thickness and protein-protein/protein-lipid ratio, this finding can be further extended to the engineering multiple-polymersome level life process with a higher architectural complexity.

Figure 6. (a) Schematic representation of ATP-producing polymersome (BR-ATP synthase-polymersome) (adapted with permission from [43]. Copyright 2007 IEEE.). (b) (i) Intravesicular pH change as a measure of proton pumping activity of BR-polymersomes and (ii) photosynthetic ATP production in the BR-ATP synthase-polymersomes. (adapted with permission from [85]. Copyright 2005 American Chemical Society.)

4.2. Reverse Osmosis Water Purification Membrane

4.2.1. Background

Aquaporins (Aqps) are membrane water channels, playing important roles in regulating water transport in cells and thus contributing to the water homeostasis of organisms [95]. From a practical application point of view, E. coli aquaporin-Z (AqpZ) with a histidine-tag has advantages due to large-scale protein production and single-step purification (Ni-NTA or Talon resin) [48]. AqpZ forms a tetramer (70–80 kDa, see Figure 7a) which can transport water across the membrane in the presence of an osmotic gradient. It is noted that osmotic water permeability coefficient (Pf) increases with increasing protein content, and decreases with a protein-to-lipid weight ratio >0.02, possibly through protein-to-protein interaction discussed in Section 3 (Figure 7b). AqpZ has been reported to selectively transport only pure water molecules across cellular membranes with a high water permeability (P ≥ 10−13 cm3·monomer−1·s−1) and a low Arrhenius activation energy (Ea = 3.7 kcal·mol−1) [48]. This corresponds to a water transport rate of about 3 × 109 water molecules/monomer/s. This exceptionally high water transport capability and sharp water selectivity of AqpZ make AqpZ-embedded polymer

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Biosynthesis within a bubble architecture

(a) (b)

(c) (d)

Figure 1. Encapsulation of polymer vesicles in a bubble aqueous channel. (a) Schematic representation of a proteo-polymeric vesiclereconstituted with BR and F0F1-ATP synthase. (b) The procedure for embedding BR–ATP synthase–polymer vesicles in the bubble waterchannel. (c) TEM micrograph of BR–ATP synthase–polymer vesicles. (d) Schematic diagram of a dry foam structure of bubbles withBR–ATP synthase–polymer vesicle incorporation. When spherical bubbles come together, dry aqueous foam formation takes place, formingpolyhedra. Plateau borders contain most of the aqueous solution.

solution. Bubbles were blown using a pipette with a smallaperture by expelling air. During bubble formation, detergentmolecules self-assemble to form monolayers on the innerand outer surfaces of a water channel containing BR–ATPsynthase–polymer vesicles. To test for the incorporation andfunction of polymer vesicles in the aqueous channels, weprepared foams by filling a UV cuvette with bubbles. Asfreshly formed foam showed free drainage during the initialstages, the cuvette was kept inverted in the dark to remove thepolymer vesicle solution not incorporated within the bubbles.

The self-assembly of detergent molecules in the bubblearchitecture was confirmed by measuring the pH of theaqueous layer before and after blowing bubbles by trappingthe fluorescent pH probe pyranine [15, 20]. Initial pHreadings of 7.1 in the detergent solution increased to about7.4 after forming the bubbles (data not shown). Since thepH of the water channel is related to the concentration ofdetergent molecules, we attribute this pH increase to detergentmolecules’ self-assembly into monolayers of the bubbles.Before taking any measurements, we confirmed the formationof dry foam, where bubbles take the form of polyhedra withnanoscale liquid films and Plateau borders (figure 1(d)) [21].To minimize gravity-induced vertical drainage from the waterchannels, foam samples were sealed and continually rotated ata speed of 20 rpm.

It is noteworthy that the pH values of the foam’s aqueouschannels decreased gradually (figure 2). Over the courseof 60 min, a pH change of 0.045 units was observed.We attribute this acidification to the detergent moleculesentering the water channel during the coalescence process.When bubbles coalesce, detergent molecules composing thecommonly shared water channel of neighbouring bubblesare supposed to self-assemble into monolayers of the newlymerged larger bubble. However, some detergent may bereleased into the water channel. This suggests that theinstability of the bubbles in this study induced a gradualacidification by increasing the detergent concentration withinthe water channels over time.

As the first step, we monitored BR’s proton pumpingactivity in the polymer vesicles. The generation of a photo-induced electrochemical proton gradient was measured bytrapping pyranine inside the polymer vesicles. Fluorescencewas first measured after incubation in the dark and any photo-induced intensity change was measured after illumination witha 5.0 W green LED (λ = 570 nm). Intravesicular pHmeasurements were performed in buffer solution using BR–polymer vesicles and BR–ATP synthase–polymer vesicles.Both systems in buffer solution showed an increase in theinternal pH with illumination (figure 3). That is, the generationof a photo-induced proton gradient resulted in alkalinization

2199

Biosynthesis within a bubble architecture

(a) (b)

(c) (d)

Figure 1. Encapsulation of polymer vesicles in a bubble aqueous channel. (a) Schematic representation of a proteo-polymeric vesiclereconstituted with BR and F0F1-ATP synthase. (b) The procedure for embedding BR–ATP synthase–polymer vesicles in the bubble waterchannel. (c) TEM micrograph of BR–ATP synthase–polymer vesicles. (d) Schematic diagram of a dry foam structure of bubbles withBR–ATP synthase–polymer vesicle incorporation. When spherical bubbles come together, dry aqueous foam formation takes place, formingpolyhedra. Plateau borders contain most of the aqueous solution.

solution. Bubbles were blown using a pipette with a smallaperture by expelling air. During bubble formation, detergentmolecules self-assemble to form monolayers on the innerand outer surfaces of a water channel containing BR–ATPsynthase–polymer vesicles. To test for the incorporation andfunction of polymer vesicles in the aqueous channels, weprepared foams by filling a UV cuvette with bubbles. Asfreshly formed foam showed free drainage during the initialstages, the cuvette was kept inverted in the dark to remove thepolymer vesicle solution not incorporated within the bubbles.

The self-assembly of detergent molecules in the bubblearchitecture was confirmed by measuring the pH of theaqueous layer before and after blowing bubbles by trappingthe fluorescent pH probe pyranine [15, 20]. Initial pHreadings of 7.1 in the detergent solution increased to about7.4 after forming the bubbles (data not shown). Since thepH of the water channel is related to the concentration ofdetergent molecules, we attribute this pH increase to detergentmolecules’ self-assembly into monolayers of the bubbles.Before taking any measurements, we confirmed the formationof dry foam, where bubbles take the form of polyhedra withnanoscale liquid films and Plateau borders (figure 1(d)) [21].To minimize gravity-induced vertical drainage from the waterchannels, foam samples were sealed and continually rotated ata speed of 20 rpm.

It is noteworthy that the pH values of the foam’s aqueouschannels decreased gradually (figure 2). Over the courseof 60 min, a pH change of 0.045 units was observed.We attribute this acidification to the detergent moleculesentering the water channel during the coalescence process.When bubbles coalesce, detergent molecules composing thecommonly shared water channel of neighbouring bubblesare supposed to self-assemble into monolayers of the newlymerged larger bubble. However, some detergent may bereleased into the water channel. This suggests that theinstability of the bubbles in this study induced a gradualacidification by increasing the detergent concentration withinthe water channels over time.

As the first step, we monitored BR’s proton pumpingactivity in the polymer vesicles. The generation of a photo-induced electrochemical proton gradient was measured bytrapping pyranine inside the polymer vesicles. Fluorescencewas first measured after incubation in the dark and any photo-induced intensity change was measured after illumination witha 5.0 W green LED (λ = 570 nm). Intravesicular pHmeasurements were performed in buffer solution using BR–polymer vesicles and BR–ATP synthase–polymer vesicles.Both systems in buffer solution showed an increase in theinternal pH with illumination (figure 3). That is, the generationof a photo-induced proton gradient resulted in alkalinization

2199

Scheme of a liposome with BR and F0/F1 “Bubbles” seen with electron microscopy from

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Artificial photosynthesis and ATP production in biomimetic materials


reference list

• J. K. Lanyi, “Bacteriorhodopsin”, Ann. Rev. Physiol. 66 (2004) 665• E. Freier, S. Wolf, and K. Gerwert, “Proton transfer via a transient linear water-

molecule chain in a membrane protein”, PNAS 108 (2011) 11345• H. Wang & G. Oster, “Energy transduction in the F1 motor of ATP synthase”,

Nature 396 (1998) 279• A. L. Moore, D. Gust and T. A. Moore, “Bio-inspired constructs for sustainable

energy production and use”, l’actualité chimique, mai-june 2009, n. 308-309, p. 50

• D. Gust, T. A. Moore and A. L. Moore: “Mimicking bacterial photosynthesis”, Pure & Appl. Chem. 70 (1998) 2189

Documents

Introductory biophysics A. Y. 2017-18 4. Basics of ...milotti/Didattica/...Edoardo Milotti - Introductory biophysics - A.Y. 2017-18 Multilevel statistical system Consider a macrostatedefined