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BPC-2018-Full.Pdf Background and Reference Material Probability and Statistics Probability Distribution P(X) is a probability distribution for observing a value X in a data set of multiple observations. It can describe either a discrete (i = 1 to N) data set or continuous function. For the continuous distribution, P(X) is known as a probability density, and P(X) dX is the probability of observing a value between X and X + dX within a large sample. All probability distributions are normalized such that Discrete Continuous N P 1 PXdX1 i i1 The mean or average value of X for this distribution is 1 NN XXPXXPXdX iii N ii11 The root mean square value of the distribution is the square root of the mean square value of x: 1/2 XX 2 rms The mean is the first moment of the distribution, the mean square value is the second moment, and the nth moment of P(x) is 1 NN XXPXXPXdXnnn n iii N ii11 The width of the distribution function about its mean is quantified through the standard deviation σ or variance σ2: 22XX2 Gaussian Distribution The Gaussian normal distribution is a continuous distribution function commonly used to describe a large number of independent samples from a random distribution. 1() XX 2 PXexp 2 2 2 2 The distribution function is completely described by its mean and variance. All higher moments of this distribution are zero. The most probable value, or peak, of the distribution is the mean, and the width is given by the standard deviation. The probability density with ±1σ of the mean is 68.2% Andrei Tokmakoff 06/09/2017 and within ±2σ is 95.4%. The full-width of P(X) at half the maximum height (FWHM) is 8ln 2 2 2.355 . Binomial and Poisson Distributions A binomial distribution is a discrete probability distribution that relates to the number of times of observing N successful events from a series of M independent trials on a binary variable (yes/no; up/down). For instance the number of times of observing N copies of the amino acid tyrosine within a protein sequence of length M. Alternatively we can say that it gives the probability that N random events happen during a given period of time, M. Like the number of barrier crossings (jumps) during a time window in a two-state trajectory. Consider the number of ways of putting N indistinguishable objects into M equivalent boxes: M ! M NM!( N )! N (Note: In the limit M ≫ N, Ω ≈ MN/N!). The choices made here are binary, since any given box can either be empty or occupied. Ω is also known as the binomial coefficient. However, unlike a coin toss, a binary variable need not have equal probability for the two outcomes. We can say that we know the probability that a box is occupied is po. Then the probability that a box is empty is pe = 1‒po. The probability that in a given realization with M boxes that N are occupied and M‒N are NMN empty is ppoe . Then multiplying the probability for a given realization and the number of possible realizations, the probability of observing N of M boxes occupied is M NMN M ! N MN PNM(, ) poe p po (1) p o N NM!( N )! This is the binomial distribution. The average number of occupied boxes is N = N po. A Gaussian distribution emerges from the binomial distribution in the limit that the number of trials becomes large (M→∞). The mean value of the distribution is Npo and standard deviation is Npo(1‒po). A Poisson distribution emerges in the limit that the number or trials (M) becomes large and the probability of observing an occupied box is small, po ≪ 1. NeNN PN() N! 2 The mean of the Poisson distribution is N and the standard deviation σ = N 1/2. Fluctuations in N scale as σ, and the fluctuations relative to the mean as / NN 1/2 . Then we see that fluctuations are most apparent to the observer for small N with the biggest contrast for N = 1.1 1. M. B. Jackson, Molecular and Cellular Biophysics. (Cambridge University Press, Cambridge, 2006), Ch. 12. 3 Constant and Units Practical Units and Identities for Biophysical Purposes 1 m 109 nm 1010 Å 1012 pm 1 nm 109 m 1 pN 1012 N 1 aJ 1018 JzJ 1021 J e2 230 pN nm2 230 zJ nm 4 0 At 300 K kBT 4.1 pN nm 2.5 kJ/mol 0.60 kcal/mol 1 e2 56 nm 40 kBT k T B 25 mV e 4 Thermodynamics First Law –– dU dq dw w: Work performed by the surroundings on the system. Mechanical work: linear displacement, stretching surface against surface tension, and volume expansion against an external pressure. dw fext dr da pext dV Electrical work: dw q E dr Magnetic work: dw M dB is field is external to the system or dw B dM if the field is internal to the system. q: Heat added to the system. – C: Heat capacity links heat and temperature. At constant pressure: dq pp CdT Second Law dq– dS rev T State Functions Internal Energy, U: Uq w dU TdS pdV dN Natural variables: N, S, V Enthalpy, H: HUpV dH TdS Vdp dN Natural variables: N, S, p T2 HT() HT () C dT 21T p 1 Helmholtz free energy, A or F: A UTS dA pdV SdT dN Natural variables: N, V, T more generally: dA S dT wrev where wrev p dV ii dN f dR da dq i Gibbs free energy, G: GHTSApV dG SdT Vdp dN Natural variables: N, p, T more generally: dG SdT wrev 5 where wrev Vdp idN i f dR da dq i For a reversible process at constant T and p, ΔG is equal to the non-pV work done on the system by the surroundings. Generally: (dG)T ,p wnon-pV . GA Chemical potential: i NNii Tp,,{} Njij TV,,{} Ni GA Entropy: S TTp V If you know the free energy, you know everything! G A V p p V T T G H G TS H G T T p G G U H pV U G T p T p p T G A U TS A G p p T S 2 G U C T C T C p T p T 2 V T p p V ,N Spontaneous Processes Conditions that determine the direction of spontaneous change and the conditions for equilibrium dG 0 dA 0 dS 0 pT,, N VTN,, H, p dU 0 dH 0 VSN,, pSN,, Chemical Equilibria products reactants 0 0 0 0 0 0 Grxn Hrxn TSrxn Grxn Nii N j j i j K exp(G0 / RT ) eq rxn 6 Statistical Thermodynamics The partition functions play a central role in statistical mechanics. All the thermodynamic functions can be calculated from them. Microcanonical Ensemble (N,V,E) All microstates are equally probable: P = 1/Ω where Ω is the number of degenerate microstates. Skln Boltzmann entropy in terms of degeneracy Canonical Ensemble (N,V,T) Average energy E is fixed. Canonical Partition Function: Q Q eEkTi / i edEkT/ E For one particle in n dimensions: 1 Q eH (,)/qp kT dq n dp n n h The classical partition function for N non-interacting particles in 3D: 1 33NN Qdpdpe 333(,)/NNNHpqkT 3N h 3333N 3 qqqq,2 Nii where qxyz,i , i If the kinetic and potential energy terms in the Hamiltonian are separable as Tp33NN Vq , then 3NVqkT (33NN )/ 1 3NTpkT ( )/ Qdqe dpe 3N h 3N 1/2 where hmkT2 /2 . So 3 N 1 3N Vq / kT Qdqe 3N 7 Microstate Probabilities eEi /kT PE ii Q or describing the probability of occupying microstates of an energy Ej, which have a distinguishable degeneracy of g(Ej): E j /kT gE()j e PEjj Q eE()/rkT Pr or as a probability density: Q Internal Energy 1 E j /kT U E Pj E j E je j Q j Helmholtz Free Energy Q(N,T,V) leads to A(N,T,V). Using dA pdV SdT dN : A A U TS U T T V ,N A kT lnQ 8 All the other functions follow from Q, U and A U 1 A A AT lnQ k T 2 T T T 2 T T V ,N V ,N V ,N 2 lnQ U kT T V ,N U CV T V ,N A U lnQ S k lnQ kT T T T V ,N S Cp T T p A lnQ p kT V T ,N V T ,N A lnQ kT N N T ,V T ,V Entropy Entropy in terms of microstate probabilities and degeneracies E E S U A i i Ei kT Pi lnQ But lne , so k kT i kT kT S P lneEi kT lnQ i k i E kT e i ln Pi i Q ln P Pi i i S kP i ln Pi Gibbs equation: S in terms of microstate probabilities i 9 Ensemble Averages Other internal variables (X) can be statistically described by N eEkTi / XPXPE ii i i i1 Q For our purposes, we will see that translational, rotational, and conformational degrees of freedom are separable Q Q Q Q ... trans rot conf Grand Canonical Ensemble (μ, V, T) Average energy E and average particle number Nj are fixed. Thermodynamic quantities depend on microstates (i) and particle type (j). Pji ( ) is the probability that a particle of type j will occupy a microstate of energy Ei: E ( j)/k T N ( j)/k T e i B e j B P( j) i U E Pi ( j)Ei ( j) and N Pi ( j)N( j) j i j i The grand canonical partition function is j N ( j)/kBT Q(N j ,V ,T )e j Stirling’s Approximation For large N: N ! N ln N N N N 1 or N ! 2 N N ! N ln N N ln(2 N ) e 2 First Law Revisited Let’s relate work and the action of a force to changes in statistical thermodynamic variables:2 The internal energy: 2.
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