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Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Statistical mechanics of biological processes
1
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes • Describing biological processes requires models.
• If “reaction” occurs on timescales much faster than that of connected processes quasi-equilibrium: laws of thermodynamics can be used.
• Biological systems/processes involve large number of interacting molecules.
• Deterministic description impossible, resort to probabilistic description with evaluation of average properties.
• Statistical mechanics theoretical framework appropriate to quantitatively describe thermodynamics of processes at molecular level.
• Thermodynamic state functions interpreted through concepts of microstates of systems compatible with a given macrostate.
2
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes • Microstate particular realization of microscopic
arrangement of constituents of system/process of interest.
• Macrostate identified by a particular set of macroscopic independent parameters (e.g. E, N, V for an isolated
thermodynamic system) which affect dynamics of constituents.
Microstates compatible with a given macrostate are different possible ways system can achieve that macrostate.
• Statistical mechanics allows to calculate probability of each microstate under a set of constraints acting on the system.
• Boltzmann distribution probability determined by energy of microstate.
3
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes Boltzmann’s distribution: key equations
4
p Ei( ) = 1Ze−Ei kBT
Z = e−Ei kBTi∑
F = − 1βlnZ
E =1Z
Eie−Ei kBT = −
∂∂βlnZ
i∑
...
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Microstates in biology Lattice model very useful to describe statistical mechanics of molecular
recognition events (concentration arises naturally as key variable).
5
• Example with ligand-receptor
binding.
• Macrostates: bound vs. unbound.
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Microstates in biology DNA (or any other polymer) in solution
6
Shape of polymer (microstate) can be characterized in different ways:
• By using function r(s) to characterize positions of points of molecule
(s distance of point along molecule).
• By reporting coordinates of all atoms of DNA.
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Microstates in biology
7
DNA (or any other polymer) in solution
Shape of polymer (microstate) can be characterized in different ways:
• By using function r(s) to characterize positions of points of molecule
(s distance of point along molecule).
• By reporting coordinates of all atoms of DNA.
Definition of microstates not absolute, but depends on problem of interest!
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes Boltzmann’s distribution: examples of two-states systems
8
Transport through membrane channel: electrophysiology experiments
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes Boltzmann’s distribution: examples of two-states systems
9
Binding of ligands to a rigid receptor (no internal d.o.f.)
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding • Lattice model to describe thermodynamics of molecular recognition useful
because concentration arises naturally as a key parameter. Consider:
• L ligands, Ω boxes each with volume Vbox.
• Only two classes of states:
1. No ligand bound to receptor, all compatible microstates have same energy εsol.
2. One ligand bound to receptor, all compatible microstates have energy εb.
• pbound given by:
10
pbound =1bound
states∑
1boundstates∑ + Lunbound
states∑
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding • Numerator statistical weight of having 1 ligand bound and L-1 in solution
• Denominator configurational partition function for L ligands swimming in solution with no binding to receptor:
11
1boundstates∑ =Wmicrostates
1bound = e−βεb × L −1( )unboundstates∑
L −1( )unbound = e−β L−1( )εsol
states∑ =
Ω!L −1( )! Ω− L −1( )%& '(!states
∑ e−β L−1( )εsol
Lunbound = e−βLεsolstates∑ =
Ω!L! Ω− L( )!states
∑ e−βLεsol
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding • If Ω L as often happens, approximation holds:
• Introducing energy difference, concentration c and concentration standard c0 (L = Ω ):
• Probability can be written after rearrangement of equation as:
Weights to have zero or one ligands bound are 1 and c/c0 e-βΔε respectively.
12
Ω!Ω− L( )!
≈ ΩL
pbound =c c0( )e−βΔε
1+ c c0( )e−βΔεLangmuir adsorption isotherm
or Hill function with coefficient 1
Δε = εb −εsol c = L ΩVbox c0 =1 Vbox
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding
13
Estimate of parameters by choosing
Vbox = 1 nm3 c0 = 1 molecule/nm3 =
= 1024 molecules/l =
= 1024/NA M ~ 1.66 M
Langmuir adsorption isotherm
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding Langmuir adsorption isotherm
14
Value of dissociation constant Kd corresponds to concentration of ligands for which pbound = 1/2
Perfect balance between entropic and energetic terms of free energy of binding.
Estimate of parameters by choosing
Vbox = 1 nm3 c0 = 1 molecule/nm3 =
= 1024 molecules/l =
= 1024/NA M ~ 1.66 M
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Modeling biological processes Boltzmann’s distribution: examples of two-states systems
15
Expression of specific protein (transcription)
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression…
Cells express different genes in different amounts
at different times.
Regulation of gene expression is complex and requires several control mechanisms as well as degradation of
both mRNA and proteins.
Amount of proteins present at any time depends on all processes involved.
16
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression… Keep it simple: focus on first step, reduce complexity of gene expression by:
1. Considering only amount of mRNA produced by RNA polymerase.
2. Considering only binding of transcription factors (namely activators) to promoter region: if TF bound, then polymerase starts transcription.
Reduce problem to calculation of probability of polymerase binding to specific promoter region of DNA.
17
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression…
18
Further assumption (corroborated by experiments): all RNA polymerases are bound to DNA.
Probability of expression of gene
calculated considering competition between binding to specific
promoter site and non specific binding to all remaining ones along
1D lattice.
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression…
19
• Problem similar to that seen for ligand binding receptor.
• pb given by calculating relative weight of 1 polymerase among P binding to promoter with energy vs. binding to NNS non specific sites with energy .
pSb =1Sb
states∑1Sb
states∑ + PNS
bstates∑
εpdS
εpdNS
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression… • Numerator statistical weight of having 1 polymerase bound specifically.
• Denominator total partition function, need to calculate weight of P polymerases binding nonspecifically:
20
1Sbstates∑ = e−βε
Sb × P −1( )NSb
states∑
P −1( )NS = e−β P−1( )εNSb
states∑ =
NNS !P −1( )! NNS − P −1( )$% &'!states
∑ e−β P−1( )εNSb
PbNS = e−βPε
NSb
states∑ =
NNS !P! NNS −P( )!states
∑ e−βPεNSb
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression… • If NNS P, approximate:
• Introducing energy difference:
• Probability can be written after rearrangement of equation as:
• Weights to have zero or one specific bindings 1 and P/NNSe-βΔε respectively.
21
NNS !NNS −P( )!
≈ NNS( )P
Δε = ε Sb −εNSb
pbound =PNNS
e−βΔε
1+ P NNSe−βΔε
=1
1+ NNSP e
βΔε
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
StatMech of gene expression… Difference between strong and weak promoter can be attributed to
22
Δε
Δε = −2.9kBT
Δε = −8.1kBT
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Chemical potential of dilute solution • Chemical potential Gibbs free energy per unit mass change.
• µ of solute can be calculate from total Gibbs free energy Gtot:
• Total free energy can be written as sum of solvent free energy, solute energy and entropy of mixing:
• Dilute solution No interaction among solute molecules.
Energy of solute molecules given by sum of per-molecule contribution (εs) multiplied by Ns.
23
µsolute =∂Gtot
∂Ns
"
#$
%
&'T ,p
Gtot = NH2Oµ 0H2O + Nsεs −TSmix
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Chemical potential of dilute solution Lattice model allows to derive equation for chemical potential of dilute solutions
24
Ns NH2O
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Chemical potential of dilute solution Entropy of mixing Smix given by number of ways of
arranging Ns molecules within Ns + NH2O lattice points:
Exploiting Stirling approximation and low concentration c = Ns/NH2O gives:
25
Smix = kB lnW NH2O+ Ns,Ns( ) = kB ln
NH2O+ NS( )!
NH2O!Ns!
Smix ≈ −kB Ns lnNs
NH2O
− Ns
#
$%%
&
'((
Gtot T, p,NH2O,Ns( ) = NH2O
µN2O0 T, p( )+ Nsεs T, p( )+ kBT Ns ln
Ns
NH2O
− Ns
"
#$$
%
&''
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Chemical potential of dilute solution Deriving with respect to Ns after introduction of solute and reference
concentrations c = Ns/Vbox and c0 = NH2O/Vbox gives chemical potential:
which can be generalized to any dilute solution by writing chemical potential with respect to a standard reference state indicated by suffix 0:
26
µs =∂Gtot
∂Ns
= εs + kBT lncc0
µi = µi0 + kBT lncici0
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure as entropic effect Considering a binary spontaneous reaction between two species 1 and 2,
second law of thermodynamics states that ΔG must be non positive:
Δµ = µ1 – µ2 driving force for mass transport: if µ1 ≥ µ2 dN1 ≤ 0 and viceversa.
For dilute solutions:
Force arising in part from different concentrations of two species.
Identical reasoning applies to same species in two compartments 1 and 2. If concentration is higher in 1 and particles are allowed to
cross boundary between 1 and 2, they will flow from 1 to 2. 27
dG = µ1dN1 +µ2dN2 = µ1 −µ2( )dN1 ≤ 0
Δµ = Δµ0 + kBT lnc1c2
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure as entropic effect Consider a cell in water solution:
• Interior of cell very crowded concentration of proteins, DNA and many other molecules much higher than outside, while concentration of water much lower.
• Force will tend to move components outside and water inside cell.
• Mechanical force acting on membrane induces osmotic pressure.
How to cope with constant stress induced by osmotic pressure?
• Bacteria endowed with rigid cell wall outside plasma membrane.
• Pumps actively drying cells’ interior by expelling water and modulating concentration of ions and sugars, etc…
28
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure as entropic effect Osmotic pressure is a purely entropic effect.
• Rationalized using thermodynamics for dilute solutions.
• Consider two compartments, one with water and another with dilute solution, separated by semipermeable membrane (only water can cross).
29
• At equilibrium chemical potentials of water identical in two compartments.
• Chemical potential of water on side of dilute solution can be derived as:
µH2O=∂Gtot
∂NH2O
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure as entropic effect
At equilibrium chemical potentials in compartments 1 and 2 must coincide:
30
µH2O= ∂Gtot ∂NH2O
Gtot T, p,NH2O,Ns( ) = NH2O
µH2O0 T, p( )+ Nsεs T, p( )+ kBT Ns ln
Ns
NH2O
− Ns
"
#$$
%
&''
µH2OT, p( ) = µH2O
0 T, p( )− kBTNs
NH2O
µH2O0 T, p1( ) = µH2O
0 T, p2( )− kBTNs
NH2O
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure as entropic effect Difference in pressure in two compartments needed to maintain equilibrium.
If Δp p1 expand chemical potential on right-hand side of equation:
Since molecular volume is given by derivative of µ with respect to p:
Equilibrium condition becomes:
31
µH2O0 T, p2( ) ≈ µH2O
0 T, p1( )+∂µH2O
0
∂pp1
p2 − p1( )
∂µH2O0
∂pp1
= vH2Omol =V1 NH2O≈V2 NH2O
≡V NH2O
p2 − p1( ) = kBTNs
V→Δp = kBTcs
van’t Hoff formula
gives osmotic pressure as function of concentration of solute
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Osmotic pressure in E.coli Estimate of osmotic pressure in E. coli obtained by considering following assumptions:
• Concentration of inorganic ions ~ 100 mM, which means, since V ~ 1fL, a number of ions within bacterium of about:
Concentration as number of solute molecules Ns/V can be calculated considering that 1 fL = 109 nm3:
Since kBT ~ 4.1 pN ⋅ nm, Δp amounts to:
32
0.1M = 0.1NA 1L ≈ 6 1022 1015 fL→ 6 107molecules
c ≈ 6 107 109nm3 = 0.06molecules nm3
Δp ≈ 0.064.1molecules nm3 pN nm ≈ 0.25 pN nm2 ≈ 2.5atm
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Law of Mass Action
Chemical reactions at equilibrium obey Law of Mass Action, which introduces equilibrium constants setting ratio between concentrations of reactants and products.
• Can be derived from statistical mechanics, which also gives microscopic description of equilibrium in terms of stoichiometric coefficients and concentrations of species involved in reaction.
• Entropy maximization is a way to obtain equilibrium constants.
33
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Law of Mass Action Consider simple reaction with two reagents and one product:
A + B AB
• Stoichiometric coefficients νi must be introduced to count changes in number of molecules during reaction.
• In case above if A and B decrease by one unit, AB must increase by same quantity νA = νB = -νAB.
• At equilibrium differential of Gibbs free energy must be zero since G is minimum:
34
dG NA, NC, NC( ) = 0↓
∂G ∂NA( )B,AB dNA + ∂G ∂NB( )A,AB dNB + ∂G ∂NAB( )A,B dNAB = 0
↓
µAdNA +µBdNB +µABdNAB = 0
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Law of Mass Action In a general case, expressing all dNi through stoichiometric coefficients:
equilibrium condition becomes:
• For dilute solutions this condition can be written:
35
dG = 0→ µidNii=1
N
∑ = µiν idN = 0i=1
N
∑ → µiν ii=1
N
∑ = 0
dNi =ν idN
µiν ii=1
N
∑ = µi0 + kBT lncici0
"
#$
%
&'ν i
i=1
N
∑ = 0→ µi0ν ii=1
N
∑ = −kBT ln cici0
"
#$
%
&'
νi
i=1
N
∑
↓
−1kBT
µi0ν ii=1
N
∑ = ln cici0
"
#$
%
&'
νi
i=1
N
∏
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Law of Mass Action Exponentiation of formula and reordering of all constants on right-hand side gives:
Constant on right hand side defined as equilibrium constant of reaction Keq(T):
Inverse of dissociation constant Kd:
36
ciνi
i=1
N
∏ = ci0νi
i=1
N
∏ e−1kBT
µi0νii=1
N
∑
Keq T( ) ≡ ci0νi
i=1
N
∏ e−1kBT
µi0νii=1
N
∑
Law of Mass Action
Explains and predicts dynamic equilibrium by relating concentrations of reactants and products at a given
temperature and pressure.
Kd =1Keq
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Law of Mass Action • Equilibrium constant can be measured experimentally, but previous formula allows to set microscopic interpretation of chemical equilibrium.
• For instance, considering again simple reaction A + B AB one gets:
37
ciνi
i=1
N
∏ = cA−1cB
−1cAB1 =
cABcAcB
= Keq T( ) = 1Kd
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding revisited Example to highlight link between two ways of describing of molecular processes:
• Reconcile views by expressing pbound as function of Kd:
• Kd cast in terms of microscopic parameters of lattice model as:
38
Kd =L[ ] R[ ]LR[ ]
pbound =c c0( )e−βΔε
1+ c c0( )e−βΔε
Chemical " dissociation constant Kd: Statistical " binding energy Δε:
pbound =LR[ ]
R[ ]+ LR[ ]=
L[ ] Kd
1+ L[ ] Kd
Former result: dissociation constant Kd corresponds to concentration of
ligands for which pbound = 1/2
Kd =1Vbox
eβΔεLink between languages
of chemistry and of statistical physics
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Ligand-receptor binding revisited Note about experimental settings
• pbound depends on [L], that is concentration of free ligands and not total one.
• Important subtlety since in typical experiment we pipette in total concentration, while free concentration is determined by molecular properties of L/R interaction.
• Often to determine Kd convenient to work at concentrations where ligands are significantly more than receptors (excess of ligands).
• Free ligand concentration nearly equal to total ligand concentration.
39
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Thermodynamics of ATP hydrolysis • Energy released upon reaction ATP ADP + Pi depends on concentration of reactants and products.
• Considering change ΔG upon hydrolysis of one molecule:
• For dilute solutions:
• Considering change with respect to reference concentration (ΔGref):
40
dG = µidNii=1
N
∑ = µiν idNi=1
N
∑ dN=1" →"" ΔG = µiν ii=1
N
∑
ΔG = µi0 + kBT lncici0
"
#$
%
&'ν i
i=1
N
∑ = µi0i=1
N
∑ + kBT lncici0
"
#$
%
&'
νi
i=1
N
∏
ΔG = ΔGref + kBT ln cνiii=1
N
∏ cνii,refi=1
N
∏#
$%
&
'(
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Thermodynamics of ATP hydrolysis Choosing equilibrium state as reference implies:
Thus previous equation referred to equilibrium state becomes:
Since standard state free energy (at c = 1M) can be expressed as:
by adding and subtracting ΔG0 one obtains (expressing Keq in molar units!):
41
cνii,refi=1
N
∏ = KeqΔGref = 0
ΔG = kBT ln cνiii=1
N
∏ Keq
#
$%
&
'(
ΔG = ΔG0 + kBT lnci1M"
#$
%
&'νi
i=1
N
∏
ΔG0 = −kBT lnKeq
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Thermodynamics of ATP hydrolysis Considering hydrolysis of ATP and expressing concentrations in molar units:
Considering typical experimental values:
• ΔG0 -12.5 kBT
• All concentrations around mM range:
– [ATP] ~ 5⋅10-3
– [ADP] = 0.5⋅10-3
– [Pi] = 10⋅10-3
• Gives result introduced earlier:
42
ΔG = ΔG0 + kBT ln ADP[ ] Pi[ ] ATP[ ]
ΔGhATP ≈ −12.5kBT + ln
5 ⋅10−410−2
5 ⋅10−3≈ −12.5kBT − 6.9kBT ≈ −20kBT
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Cooperativity and Hill function • Many cellular processes are based on two allowed states only.
• Molecule or cell needs to be either on or off as a function of concentration of signal received turn analog signal into digital output.
• “Response” curves such as Langmuir adsorption isotherm not appropriate to describe these processes.
• Binding curve must be switch-like, step or sigmoidal function.
43
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Cooperativity and Hill function • Behavior seen in many biological processes, arises from cooperativity.
• Can be understood considering a simple process with two ligands.
• For sake of simplicity we consider ideal cooperative behavior: no intermediate with one ligand only bound to receptor:
L + L + R L2R
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Kd2 =
L[ ]2 R[ ]L2R[ ]
pbound =L2R[ ]
R[ ]+ L2R[ ]
pbound =L[ ] Kd( )
2
1+ L[ ] Kd( )2
Hill function with coefficient n=2
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
Cooperativity and Hill function More generally:
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pbound =L[ ] Kd( )
n
1+ L[ ] Kd( )n
Hill function with Hill coefficient n
• Increasing n increases cooperativity and thus sharpness of transition between states.
• Usually n is found by fitting binding data to Hill curves directly without reference to underlying origins of
a given Hill coefficient.
• In derivation assumed negligible [LR] binding is either all or nothing. Not strictly true measured value of Hill coefficient may not be an integer.
Biophysics Course held at Physics Department, University of Cagliari, Italy. Academic Year: 2017/2018. Dr. Attilio Vittorio Vargiu PLEASE NOTE! This material is meant just as a guide, it does not substitute the books suggested for the Course.
References • Books and other sources
• Physical Biology of the Cell (2nd ed.), Phillips et al., Chap. 6
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