Thermodynamics of the Cell Environment Gibbs Free Energy

Home , Boltzmann equation, Semipermeable membrane

MAE 545: Lecture 11 (10/22)

Thermodynamics of the cell environment Gibbs free energy

G = E + pV TS = N µ i i i X energy pressure, temperature, chemical volume entropy potential of component i Derivatives of system energy Derivatives of Gibbs free energy dE = TdS pdV + µ dN dG = SdT + Vdp+ µ dN i i i i i i X X @E @E @G @G T = p = S = V = @S @V @T p,N @p T,N ✓ ◆V,Ni ✓ ◆S,Ni ✓ ◆ i ✓ ◆ i @E @G µi = µi = @Ni @N ✓ ◆S,V ✓ i ◆T,p In thermodynamic equilibrium system minimizes Gibbs free energy, when temperature T

2 and pressure p are ﬁxed! Charge dissociation in solution example NaCl salt + + binding energy interaction energy E 0 b ⇡ entropy entropy some kB ln v0 characteristic kB ln(V/N) volume

Free energy change for charge dissociation volume of the number of whole system dissociated pairs G = E T S = E k T ln(V/Nv ) b B 0 In thermodynamic equilibrium G =0

N 1 E /k T c = = e b B V v0 Entropy is the reason why many molecules dissociate concentration of dissociated ions and ionize in solution! 3 6.2.2 Free Energy of Dilute Solutions

The Chemicalwater free Potential solute of amixing Dilute Solution Is a Simple Logarithmic (B) Functionenergy of the Concentrationenergy entropy

To make progress on the question of how to write the chemical potentials for dilute solutions like those described above, we appeal to Figure from R. Phillips et al., lattice models. Lattice models have been used to great advantage Physical Biology of the Cell as a discretization trick that allows for the performance of combina- 4 toric arguments in a way that is analytically tractable. For our present purposes, we imagine our system (water solute) as being built up as + aseriesoflatticesitesasshowninFigure6.21.Theselatticemodels were introduced in Section 6.1.1 (p. 241) and illustrated in Figure 6.1 (p. 238). By restricting the set of allowed positions for the molecules of interest to the sites of a lattice, we have a countable set of distinct (C) states that can be enumerated explicitly.

We write the number of water molecules as NH2O,whilethenum- ber of solute molecules is given by Ns. Our objective is to write the total free energy of this system and then to obtain the solute chemical potential through the relation

∂G µ tot . (6.78) solute = ∂N ! s "T ,p Intuitively, the chemical potential really tells us the free energy cost associated with changing the number of solute molecules in solution by 1 as

Figure 6.21: Idealization of a dilute µsolute Gtot(Ns 1) Gtot(Ns). (6.79) solution as a system on a lattice. = + − (A) Cartoon of the actual solution in which the species of interest is dilute The reader is invited to explore this more deeply as well as the con- and wanders around freely in solution. nection to the partition function in the problems at the end of the (B) Three-dimensional and (C) chapter. We argue that the free energy is given by two-dimensional lattice idealization of the situation in (A) in which the 0 Gtot NH Oµ Nsεs TSmix . (6.80) molecules of interest are restricted to = 2 H2O + − visit sites on a lattice. water free energy solute energy mixing entropy

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One of the central ideas about entropy in this book is that there are afewbasicimplementationsofBoltzmann’sassertionthatS k ln W = B that arise over and over in carrying out statistical mechanical reason- ing. The ideal gas illustrates the way in which kinetic energy may be shared among a bunch of different molecules. The other crowning example in which we can simply calculate the entropy is embodied in the formula !! W(N, !) , (6.77) = N!(! N)! − (A) solvent which instructs us about the configurational entropy associated with (water) solute rearranging N objects amongst ! boxes. How can this simple formula help us think about the free energy of solutions? In Section 5.5.2 (p. 225), we showed that two subsystems are in chemical equilibrium when their chemical potentials are equal. If we are to consider an aqueous solution with some dilute population of molecules, or Mixing entropyalternatively, of dilute if wesolutions are to think about the interactions of gene regu- latory proteins and DNA in solution, how are we to write the chemical Let’s divide volume in smallpotentials of theHow various many species different of interest? boxes each containing one water configurations of water and molecule or one solute molecule.The Chemicalsolute Potentialmolecules of aare Dilute possible? Solution Is a Simple Logarithmic (B) Function of the Concentration NH O + Ns (NH O + Ns)! To make⌦ = progress2 on the question= 2 of how to write the chemical poten- N N !N ! tials for dilute✓ solutionss ◆ like thoseH2O describeds above, we appeal to lattice models. Lattice models have been used to great advantage as a discretizationSmix = kB trickln ⌦ that allows for the performance of combina- toric arguments inStirling a way thatapproximation is analytically tractable. For our present purposes, we imagine our system (water solute) as being built up as ln N! N ln N + aseriesoflatticesitesasshowninFigure6.21.Theselatticemodels⇡ N + N N + N S werek N introducedln H2O in Sections + N 6.1.1ln (p.H2O 241)s and illustrated in Figure 6.1 mix ⇡ B H2O N s N (p. 238). By✓ restrictingH2O ◆ the set✓ of alloweds ◆ positions for the molecules of interest to the sites of a lattice, we have a countable set of distinct Small number of (C) states that can be enumerated explicitly.Ns NH O solute particles ⌧ 2 We write the number of water molecules as NH O,whilethenum- N 2 ber of solute molecules is givens by Ns. Our objective is to write the Figure from R. Phillips et al., Smix kB Ns Ns ln total free⇡ energy of this systemN and then to obtain the solute chemical Physical Biology of the Cell  ✓ H2O ◆ potential5 through the relation

∂G µ tot . (6.78) solute = ∂N ! s "T ,p Intuitively, the chemical potential really tells us the free energy cost associated with changing the number of solute molecules in solution by 1 as

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One of the central ideas about entropy in this book is that there are afewbasicimplementationsofBoltzmann’sassertionthatS k ln W = B that arise over and over in carrying out statistical mechanical reason- ing. The ideal gas illustrates the way in which kinetic energy may be shared among a bunch of diﬀerent molecules. The other crowning example in which we can simply calculate the entropy is embodied in Chemical potentialsthe formula in dilute solution !! G = N µ0 + N ✏ WTS(N, !) , (6.77) H2O H2O s s mix = N!(! N)! N − (A) G N µ0 + N ✏ + k T N ln s N H2OwhichH2O instructss s B us abouts the conﬁgurationals entropy associated with solvent ⇡ NH2O (water) solute rearranging N objects amongst✓ ◆! boxes. How can this simple formula helpChemical us think potential about the of solute free energy of solutions? In Section 5.5.2 (p. 225), we showed that two subsystems are in chemical equilib- @G Ns riumµs when= their= ✏s chemical+ kBT ln potentials are equal. If we are to consider @Ns NH O an aqueous solution with✓ some2 ◆ dilute population of molecules, or alternatively, if we are to think about the interactions of gene regu- µs(T,p,cs)=✏s(T,p)+kBT ln(csv) latory proteins and DNA in solution, how are we to write the chemical potentials of the various species of interest? solute concentration cs = Ns/V volume occupied by v = V/NH2O Theone Chemical water molecule Potential of a Dilute Solution Is a Simple Logarithmic (B) FunctionChemical of the potential Concentration of water @G 0 Ns µH2O = = µH2O kBT To make progress@NH2O on the questionNH2O of how to write the chemical potentials for dilute solutions like those described above, we appeal to Figure from R. Phillips et al., µ (T,p,c )=µ0 (T,p) k Tc v latticeH2O models.s LatticeH2O modelsB haves been used to great advantage Physical Biology of the Cell as6 a discretization trick that allows for the performance of combina- toric arguments in a way that is analytically tractable. For our present purposes, we imagine our system (water solute) as being built up as + aseriesoflatticesitesasshowninFigure6.21.Theselatticemodels were introduced in Section 6.1.1 (p. 241) and illustrated in Figure 6.1 (p. 238). By restricting the set of allowed positions for the molecules of interest to the sites of a lattice, we have a countable set of distinct (C) states that can be enumerated explicitly.

∂G µ tot . (6.78) solute = ∂N ! s "T ,p Intuitively, the chemical potential really tells us the free energy cost associated with changing the number of solute molecules in solution by 1 as

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Figure from R. Phillips this section, we will explore a simple mathematical formulation of the H2O solution of idea of osmotic pressure. molecules macromolecules et al., Physical in H2O An immediate and important biological application of our analysis Biology of the Cell of dilute solutions is to the problemSmall of water osmotic molecules pressure. Thecan basic idea behind the emergence ofpass osmotic through pressure a is semipermeable that the presence of asemipermeablemembranecanresultintherearrangementofthose molecules to which the membranemembrane, is permeable which and blocks the setting large up of an attendant pressure. To seesolute this, consider macromolecules. the model geometry shown in Figure 6.22, which shows a container with an internal membrane that is permeable to water but not to the solute molecules of interest. As a result of the permeability of the membrane to water, the semipermeable membrane equilibrium state is characterized by equality of chemical potentials for the water molecules on both sides of the container. (Revisit Sec- Figure 6.22: Schematic of a tion 5.5.2 (p. 225) to see how entropy maximization implies equality G = N1µH O(T,p1, 0) + containerN2µH withO( aT,p semipermeable2,cs)+Nsµs(T,p2,cs) of chemical potentials.) 2 membrane2 with molecules in solution The chemical potential of the water molecules on the side of the in one side of the container and pure membrane with the dilute concentrationIn thermodynamic of solute molecules canequilibrium be solvent in thethe other. Gibbs free energy G is derived by appealing to Equation 6.85 and in particular by evaluating µ (T , p) ∂Gtot/∂N .Notethatherewemakeexplicitthedepen-minimized, which means that chemical potentials of water are H2O = H2O dence of the chemical potential onthe the pressuresame andon the both temperature. sides of the semipermeable membrane! The resulting expression is

0 Ns µH2O(T , p) µH O(T , p) kBT . (6.89) = 2 − NH O 2 µH2O(T,p1, 0) = µH2O(T,p2,cs) The equilibrium between the two sides may now be expressed via the equation

0 Ns µ 0 (T , p ) µ (T , p ) k T . (6.90) H2O 1 H2O 2 B 7 = − NH2O solute-free side side with solutes

Note that we have already asserted that there will be a pressure difference between the two sides by introducing the notation p1 for the pressure on the pure water side of the container and p2 for the pressure on the side containing the solute molecules. We now expand the chemical potential on the right-hand side around p1 as ∂µ0 0 0 H2O µH O(T , p2) µH O(T , p1) (p2 p1). (6.91) 2 ≈ 2 + ! ∂p " − As a result of the thermodynamic relation ∂µ/∂p v,wherev is the = volume per molecule, this equation can be transformed into N p p s k T , (6.92) 2 − 1 = V B where V N v is, to a very good approximation, equal to the total = H2O volume on the solute side of the semipermeable membrane. This relation is known as the van’t Hoﬀ formula and it gives the osmotic pressure as a function of the concentration of the solute.

Viruses, Membrane-Bound Organelles, and Cells Are Subject to Osmotic Pressure

Osmotic pressure arises in many diﬀerent contexts. From a biological point of view, the existence of osmotic pressure can give rise to mechanical insults that cells and viruses must ﬁnd ways to endure. In particular, cell membranes and viral capsids are permeable to

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Figure from R. Phillips this section, we will explore a simple mathematical formulation of the H2O solution of idea of osmotic pressure. molecules macromolecules et al., Physical in H2O An immediate and importantWater biological ﬂows from application region of of our low analysis Biology of the Cell of dilute solutions is toconcentration the problem of of osmotic macromolecules pressure. The basic to idea behind the emergence of osmotic pressure is that the presence of asemipermeablemembranecanresultintherearrangementofthoseregion of large concentrations. This molecules to which theadditional membrane water is permeable increases and the pressure setting up of an attendant pressure.and To the see this,water consider stops the ﬂowing model once geometry shown in Figure 6.22, which shows a container with an internal membrane that is permeablethe to water osmotic but not pressure to the solute is reached. molecules of interest. As a result of the permeability of the membrane to water, the semipermeable membrane equilibrium state is characterized by equality of chemical potentials for the water molecules on both sides of the container. (Revisit Sec- Figure 6.22: Schematic of a µH2O(T,p1, 0) = µH2O(T,p2,cs) tion 5.5.2 (p. 225) to see how entropy maximization implies equality container with a semipermeable v of chemical potentials.) membrane with molecules in solution The chemical potential of the water molecules on the side of the 0 in one side of the container and pure µH2O(T,p2,cs)=µH2O(T,p2) kBTcsv membrane with the dilute concentration of solute molecules can be solvent in the other. 0 derived by appealing to Equation 6.85 and in particular by evaluating @µ µ (T,p ,c ) µ0 (T,p )+ H2O (p p ) k Tc v µ (T , p) ∂Gtot/∂N .Notethatherewemakeexplicitthedepen-H2O 2 s H2O 1 2 1 B s H2O = H2O ⇡ @p dence of the chemical potential on the pressure and the temperature. ✓ ◆ The resulting expression is ⇧ = p p = k T c N 2 1 B s µ (T , p) µ0 (T , p) s k T . (6.89) H2O H2O B = − NH2O The equilibrium between the two sidesOsmotic may now pressure be expressed viadepends the only on temperature and equation concentration difference across the membrane! 0 Ns µ 0 (T , p ) µ (T , p ) k T . (6.90) H2O 1 H2O 2 B 8 = − NH2O solute-free side side with solutes

Viruses, Membrane-Bound Organelles, and Cells Are Subject to Osmotic Pressure

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If extracellular solution has different concentration of ions from the interior of cells, then the resulting ﬂow of water can cause the cell to shrink or swell and even burst. hypotonic isotonic hypertonic solution solution solution

c c cs,out cs,in cs,out cs,in s,out ⌧ s,in ⇠

Cells use ion channels and ion pumps to regulate concentration of ions and therefore also the cell volume.

(Note: cell membrane is impermeable for charged particles) 9 Osmotic pressure in bacteria

Bacteria have strong cell wall that can support large osmotic pressure (Turgor pressure).

⇧ 105Pa 1bar ⇠ ⇠

Antibiotics cause damage to cell wall and as a result cells rupture due to large Turgor pressure.

10 Energetics of ATP hydrolysis

How much energy is released during ATP hydrolysis?

ATP ADP + Pi

G = µ + µ µ ADP P ATP

[ADP][P ] G = µ0 + µ0 µ0 + k T ln i ADP P ATP B [ATP]c ✓ 0 ◆ }

12.5k T B Under physiological conditions: G 20kBT ([ATP], [ADP], [P ] 1mM) ⇠ i ⇠ Chemical potentials are typically deﬁned relative to concentration c0 ~ 1 M.

11 µs(cs)=µs(c0)+kBT ln(cs/c0) Law of mass action

kon example + Na +Cl NaCl + ! A + B AB H +Cl HCl + ! Na +OH NaOH + ! k H +OH H2O o↵ ! In thermodynamic equilibrium G = µ µ µ =0 AB A B [A][B] 0 0 0 k (µ +µ µ )/kB T o↵ = c0e A B AB = Keq(T,p)= [AB] kon

For general chemical reaction a R + a R + b P + b P + 1 1 2 2 ··· ! 1 1 2 2 ··· a i 0 0 [Ri] ( ai bj ) aiµ bj µ /kB T i i j i Ri j Pj = c0 e = Keq(T,p) [P ]bj P P ⇣ ⌘ Qj j P P Q Chemical potentials are typically deﬁned relative to concentration c0 ~ 1 M. 12 µs(cs)=µs(c0)+kBT ln(cs/c0) pH value of solutions

+ [H ][OH] [H2O]Keq(T,p) 14 = 10 c2 c2 ⇡ 0 0 at room c0 = 1M temperature

pH = log [H+]/c 10 0 pOH = log [OH]/c 14 pH 10 0 ⇡ How much free energy is changed when H+ goes to environment with different pH?

pH1 pH2 H+

µ µ = k T ln [H+] /[H+] 2 1 B 2 1 µ2 µ1 2.3026 kBT E = (pH2 pH 1) e0 ⇡ e0 Nernst electric potential E 13 Charge environment of the cell

cloud of mobile ions in water some salts completely dissociate + cations anions + + + + + + + + + Na Cl + + + K Cl + + + 2+ + + Ca 2Cl + + 2+ Mg 2Cl + + + proteins besides salt ions + + + there are also other + + mobile ions (H , electrons, + + phosphates, …) + DNA + Mobile ions screen heads of lipid molecules are electrostatic interactions macroions negatively membrane between macroions! charged 14 Electrostatic energy

+ r +z+e0 z e0 | | Coulomb’s law

2 2 q1q2 z+z e0 q1q2 z+z e0 Ec = = Ec = = ✏r ✏r 4⇡✏0✏r 4⇡✏0✏r Gaussian units SI units water dielectric constant vacuum permittivity 12 ✏ 81 ✏0 =8.85 10 As/Vm ⇡ ⇥ Electrostatic interaction is small for large separation Bjerrum length Bjerrum length Ec z+z `B e2 in water at room T 0 = `B = kBT r k T ✏ `B 0.7nm B ⇡

15 Poisson-Boltzmann equation Let’s assume some mean-ﬁeld electric potential ( ~r ) throughout the cell. Local density of mobile ions carrying charge z ↵ e 0 . 3 z↵e0(~r )/kB T z↵e0(~r )/kB T d ~r n e = N n↵(~r )=n↵e ↵ ↵ Z Charge density of mobile ions

z↵e0(~r )/kB T ⇢mobile ions(~r )= z↵e0n↵e ↵ X Poisson equation 4⇡ 2(~r )= ⇢(~r ) r ✏ Poisson-Boltzmann equation

2 4⇡ z e (~r )/k T (~r )= ⇢ (~r )+ z e n e ↵ 0 B r ✏ macroions ↵ 0 ↵ " ↵ # X For a given distribution of macroions Poisson-Boltzmann equation must be solved self-consistently for the electric potential ( ~r ) .

16 Debye-Hückel approximation Let’s assume that electrostatic energy due to the mean ﬁeld electric potential is small compared to kBT. Local density of mobile ions carrying charge z ↵ e 0 .

z↵e0(~r )/kB T 3 z↵e0(~r )/kB T n↵(~r )=n↵e d ~r n↵e = N↵ z e (~r ) Z n (~r ) n 1 ↵ 0 ↵ ⇡ ↵ k T n↵ N↵/V ✓ B ◆ ⇡ Charge neutrality

z↵n↵ =0 ↵ X Charge density of mobile ions

z↵e0(~r )/kB T ⇢mobile ions(~r )= z↵e0n↵e ↵ X e2(~r ) ⇢ (~r ) 0 z2 n = ` ✏(~r ) z2 n mobile ions ⇡ k T ↵ ↵ B ↵ ↵ B ↵ ↵ X X

17 Debye-Hückel approximation

Charge density of mobile ions e2(~r ) ⇢ (~r ) 0 z2 n = ` ✏(~r ) z2 n mobile ions ⇡ k T ↵ ↵ B ↵ ↵ B ↵ ↵ X X Poisson equation 4⇡ 2(~r )= [⇢ (~r )+⇢ (~r )] Debye screening length r ✏ macroions mobile ions

4⇡ (~r ) 2 2 2 =4⇡` z n (~r )= ⇢macroions(~r )+ 2 D B ↵ ↵ r ✏ D ↵ X

Electric potential for a Electrostatic interaction between macroions point charge ⇢ (~r )= z e (~r ~r ) macroions m 0 m ⇢macroions(~r )=ze0(~r ) m X zme0 ~r ~r / (~r)= e| m| D ze ✏ ~r ~r 0 r/D m m (~r)= e X | | ✏r Einteractions zmzn ~r ~r / = ` e| m n| D k T B ~r ~r B m