4. Basics of statistical mechanics and chemical kinetics in biophysical processes

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1 Introductory biophysics A. Y Basics of statistical mechanics and chemical kinetics in biophysical processes Edoardo Milotti Dipartimento di Fisica, Università di Trieste

2 Extremely short review of statistical mechanics 1. Boltzmann factor Extremely large number of degrees of freedom Thermal reservoir at temperature T Heat exchange Much smaller number of degrees of freedom Physical system, total energy E Probability of finding system with energy E is proportional to exp E k B T

3 Multilevel statistical system Consider a macrostate defined by N 1 particles at energy level E 1 with degeneracy g 1 N 2 particles at energy level E 2 with degeneracy g 2... N i particles at energy level E i with degeneracy g i... the number of ways in which we can arrange the identical particles in the M levels is N! N 1! N M! and when we also include degeneracy, we find that the number of different ways to obtain the macrostate (thermodynamic probability) is Ω = N! g N 1 N 1 g 2 2 N 1! N 2! g N i i N i!

4 We use Stirling s approximation and we find ( ) + N i ln g i N i ln N i + N i lnω = N ln N N i N i i lnn! nlnn n ( ) = N ln N + N i ln g i N = N i N ln g i N N = N i i i N i Now the problem is finding the distribution {N i } that maximizes the thermodynamic probability (this is the distribution that is observed with the highest probability)

5 Maximization must be carried out constraining both the number of particles and the total energy lnω = N N = U = i i N i i E i N i N i N ln N i g i N expression of thermodynamic probability total number of particles is fixed total energy is fixed For constrained maximization we use the method of Lagrange multipliers and maximize the auxiliary function lnω + λn βu = N i ln g i N + λ N β E i i N i i N i i i

6 lnω + λn βu = N ln N + N i ln g i + λ N i i N i i β E i N i i N l ( lnω + λn βu ) = ln g l 1+ λ βe l = 0 N l N l = g l e λ 1 βe l

7 N i = g i e λ 1 βe i N = X i g i e 1 E i = e 1 X i g i e E i ) e 1 = N P i g E ie i = N Z N i = g i e 1 E i = N Z g ie E i Z = X i g i e E i Partition function (Zustandsumme) U = X i E i N i = N Z X E i g i e i E i = X g i e i E i = = ln the partition function is used to determine many other thermodynamical functions

8 The entropy is a measure of the thermodynamic probability S = k B ln = k B N X i N i N ln g i N i /N = k BN X i N i N (ln Z + E i) = k B N ln Z + k B U and when we recall the thermodynamic relation we find 1 T = S U T = 1 k B

9 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 ΔU = ΔQ ΔW 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): ΔW = PΔV + Δ W

10 Then ΔU = ΔQ ΔW = ΔQ PΔV Δ W and we can formally restore the form of the first principle by defining the state function enthalpy H = U + PV so that ΔH = ΔU + Δ( PV ) = ΔQ Δ W if no non-pv work is done on the system, then the enthalpy change corresponds to the heat absorbed by the system at constant pressure Δ(PV) = PΔV, as in most chemical reactions in the laboratory

11 2. Helmholtz free energy According to the second principle of thermodynamics and therefore ΔQ T ΔS ΔU = ΔQ ΔW T ΔS ΔW = Δ( TS) SΔT ΔW Therefore, when we define F = U TS we find ΔF = Δ( U TS) ΔW SΔT

12 Therefore, in isothermal processes (where the system exchanges heat with a large heat bath) ΔF = Δ( U TS) ΔW SΔT = ΔW isothermal process and therefore the work done by the system is less or equal than the decrease of free energy ΔW ΔF

13 Therefore, in processes where no work is done or absorbed by the system i.e. 0 = ΔW ΔF ΔF 0 and this is the condition for a spontaneous process with no work involved.

14 3. Gibbs free energy The Gibbs free energy is like the Helmholtz free energy, for processes where the pressure is held constant: G = H TS = U + PV TS = F + PV and we find again that the condition for a spontaneous process, with no work involved, is ΔG 0

15 Let s summarize it again, just for clarity... ΔU = ΔQ ΔW [ ] = ΔQ Δ( PV ) VΔP + Δ W = ΔQ PΔV + Δ W T ΔS Δ PV ( ) VΔP + Δ W then = Δ( TS) SΔT Δ PV ( ) VΔP + Δ W Δ( U + PV TS) SΔT + VΔP Δ W and therefore for transformation at constant temperature and pressure and no non-pv work H = U + PV F = U TS ΔG 0 G = F + PV = H TS

16 ΔG = ΔH T ΔS 0 ΔH T ΔS

17 Concentrations and Gibbs free energy Entropy of mixing in binary solutions n 1 molecules of solvent n 2 molecules of solute N = n 1 + n 2 Then the number of configurations is Ω = N! n 1!n 2! Edoardo Milotti - Introductory biophysics - A.Y

18 Ω = N! n 1!n 2! lnω ( N ln N N ) ( n 1 lnn 1 n 1 + n 2 lnn 2 n ) 2 = N ln N n 1 lnn 1 n 2 lnn 2 = ( n 1 + n 2 )ln( n 1 + n ) 2 n 1 lnn 1 n 2 lnn 2 = n 1 ln n 1 n 1 + n 2 n 2 ln n 2 n 1 + n 2 = N ( X 1 ln X 1 + X 2 ln X ) 2 X 1,2 are the volume fractions Edoardo Milotti - Introductory biophysics - A.Y

19 Therefore the entropy change due to mixing is ΔS m = k B ( lnω ln1) = k B N ( X 1 ln X 1 + X 2 ln X ) 2 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 ΔG = T ΔS m = k B NT ( X 1 ln X 1 + X 2 ln X ) 2 = nrt ( X 1 ln X 1 + X 2 ln X ) 2 = X 1 ΔG 1 + X 2 ΔG 2 Edoardo Milotti - Introductory biophysics - A.Y

20 We see that we can associate a free energy to each substance A in solution G A = n A RT ln X A (n A = nx A ) and in particular, if we consider the free energy change with respect to standard conditions volume fractions concentrations (mole/l) G A G A0 = n A RT ln X A X A0 = n A RT ln [A] [A] 0 and if we let [A] 0 =1M 1 mole/l G A G A0 = n A RT ln[a] Edoardo Milotti - Introductory biophysics - A.Y

21 Chemical kinetics The elementary reaction A P can occur via a sequence of elementary reactions, with intermediates, e.g., A I 1 I 2 P

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

23 Example: first order reaction A P d[ A] dt [ A]+ P = k[ A] [ ] = [ A] 0 [ A] t=0 = [ A] 0 [ P] 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 kt 1/2 ( ) = A [ ] 0 2 t 1/2 = ln2 k

24 Example: second order reaction 2A P d[ A] dt = k[ A] 2 [ A]+ 1 [ 2 P ] = [ A] 0 [ A] t=0 = [ A] 0 [ P] t=0 = 0 1 A [ ] = 1 + kt [ A] 0 [ P] = 2 [ A] 0 [ A] ( ) The concentration of A decreases and it is exactly half the initial concentration when 2 A [ ] 0 = 1 A [ ] 0 + kt 1/2 t 1/2 = 1 k A [ ] 0

26 Equilibrium constants We apply these concepts to the reversible chemical reaction aa + bb cc + dd and we note that at equilibrium k f [ A] a [ B] b = k b [ C] c [ D] d i.e. the forward rate is equal to the backward rate, or also [ C] c D A [ ] d [ ] a [ B] = k f b k b = K eq

27 Then, the free energy change for the i-th species with respect to the standard state, per mole, is G i G i0 = RT ln[i] 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 = ΔG 0 + cδg C + dδg D aδg A bδg B = ΔG 0 + crt ln[ C]+ drt ln[ D] art ln[ A] brt ln[ B] = ΔG 0 + RT ln C [ ] c [ D] d [ A] a [ B] = ΔG + RT ln K b 0 eq

28 At equilibrium the free energy change vanishes ΔG = ΔG 0 + RT ln C [ ] c [ D] d [ A] a [ B] = ΔG + RT ln K = 0 b 0 eq K eq = exp ΔG 0 RT

K eq = exp ΔG 0 RT Exponential dependence on DG 0 ( R 8.314 J K 1 mol 1, RT 2.

29 K eq = exp ΔG 0 RT Exponential dependence on DG 0 ( R J K 1 mol 1, RT 2.5 kj mol 300 K) this is close to the binding energy of hydrogen bonds in water 5 kcal/mole 21 kj/mole

30 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 = log 10 K

The dissociation constant of water is important K W =[H + ][OH ] The concentration of water is omitted by convention. -12.5-13.0 log 10 KW -13.5-14.0-14.

31 The dissociation constant of water is important K W =[H + ][OH ] The concentration of water is omitted by convention log 10 KW Plots like this (where the indipendent variable is the inverse temperature) are called "Arrenius plots" /T (K)

32 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)

33 from Voet & Voet - Biochemistry

34 Now consider the generic redox reaction Just as in the case of the binary reaction aa + bb cc + dd G = G + RT ln [C]c [D] d [A] a [B] b we find G = G + RT ln [A red][b n+ [A n+ ox ] ox ][B red ]

35 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 G = W = nf E F = Faraday constant E = E.M.F. E = E RT nf ln [A red][b n+ ox ] [A n+ ox ][B red ] (Nernst equation)

36 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 E = RT nf ln [K+ ] o [K + ] i (o = outside the cell, i = inside the cell)

37 ATP ATP (Adenosine Triphosphate) is a basic element in the energy budget, it is a temporary energy store, and a sort of molecular energy currency adenine ring 3 phosphate groups ribose

38 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)

39 ADP ATP + P i ΔG = 30.5 kj/mol ( 7.3 kcal/mol) + H 2 O AMP Under typical cellular conditions, ΔG is larger, and is approximately 57 kj/mol ( 14 kcal/mol). + PP i ΔG = 45.6 kj/mol ( 10.9 kcal/mol)

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

44 Energy production in halobacteria Halobacterium salinarum Halobacteria are a class of the Euryarchaeota (Archaea) found in water saturated or nearly saturated with salt

45 aerial view of the Great Prismatic Spring - Yellowstone

46 Charge differences can be generated by charge transport across membranes

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

47 Bacteriorhodopsin is 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 twodimensional crystal lattice naturally present as part of the membrane of the bacterium. view from above sideview (green lines define the cell membrane)

48 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 unitcell dimensions. (from /Research/newbr/)

49 transmembrane view

50 Bacteriorhodopsin has different conformational states that are spectrally distinguishable absorption of light quantum proton from cytoplasm proton pumped into the outer environment

51 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 br molecule in the intermediate states.

53 It has been speculated that charge, i.e., proton, transport is mediated by the formation of Grotthuss water wires inside the br molecule in the intermediate states. Recently this has been experimentally confirmed. E. Freier, S. Wolf, and K. Gerwert, Proton transfer via a transient linear water-molecule chain in a membrane protein, PNAS 108 (2011) 11345

54 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)

55 Two motors, F 0 and F 1 F 0, powered by flow of protons F 1, powered by ATP Motors are connected, and one can force the other into a generator.

59 from Choi and Montemagno: Recent Progress in Advanced Nanobiological Materials for Energy and Environmental Applications, Materials 6 (2013) 5821

60 Artificial photosynthesis and ATP production in biomimetic materials (a) Scheme of a liposome with BR and F0/F1 (c) Bubbles seen with electron microscopy from Choi and Montemagno: Biosynthesis within a bubble architecture, Nanotechnology 17 (2006) 2198

61 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 watermolecule chain in a membrane protein, PNAS 108 (2011) H. Wang & G. Oster, Energy transduction in the F 1 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 , p. 50 D. Gust, T. A. Moore and A. L. Moore: Mimicking bacterial photosynthesis, Pure & Appl. Chem. 70 (1998) 2189

Lecture 20. Chemical Potential

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