Biological Physics

Intro-Biomolecules

Statistical-Physics-Basics

Basics

Microstate: M particles, N levels: \(\Omega = M^N\) Macrostate: Histogram of the number of particles per energy level Macrostate{3, 1, 5}:

• 3 particles are in level 1
• 1 particle is in level 2
• 5 particles are in level 3

Fundamental concept of statistical Physics

• Describe a system only as a MACROSTATE = Histogram
• It is not relevant which individual particle is in what level
• Compared to the exact description of a system (MICROSTATE), information is lost. A histogram contains LESS information than the data set that generated it
• Cannot construct the exact(MICROSTATE) of a system from the MACROSTATE because the MACROSTATE is just a distribution
• The amount of the missing information is measureed by the statistical weight \(\Omega\) of the Macrostate, or \(\ln(\Omega)\)

Stastical weight of a macrostate \(\Omega\)

How many microstates belong to the macrostate eg: {2,0,0,1,0} has 3 microstates

\(\Omega = N!\)(所有粒子能量均不相同)

Degeneracy

\({1,0,0,2}: \Omega = \frac{3!}{2!}\)

Entropy

\(S = k_B \ln \Omega, \Omega = \frac{N!}{n_0!n_1!\cdots}\)

Stirling's theorem \[N! \approx \frac{N^N}{e^N}\sqrt{2\pi N}\] \[\ln N! \approx N \ln N -N\]

Entropy S is a property of a macrostate of a system, gives the level of disorder of that macrostate

Boltzman-Free-Energy

Boltzman

\(\ln \Omega = \ln \frac{N!}{\prod_{i}n_i!} = N\ln N - \sum_{i}(n_i\ln n_i)\) Maximum of entropy occurs for \(dS = d(k_B \ln \Omega)=0\) \[d\ln\Omega=-\sum_{i}\ln n_i dn_i=0\] \(\sum_i n_i = N, \sum_i dn_i=0\) Use Lagrange's method of undertimined multipliers: \(n_i = \frac{N}{M}\)

Microcanonical ensemble

\[\sum_i \epsilon_in_i = E\] \(n_i = Ae^{-\beta\epsilon_i},A = e^{-\lambda}\) \(n_i = N\frac{e^{-\beta\epsilon_i}}{\sum_ie^{-\beta\epsilon_i}}\) \(p_i = \frac{e^{-\beta\epsilon_i}}{\sum_ie^{-\beta\epsilon_i}}\) \(Z = \sum_{i=1}^{M}e^{-\beta\epsilon_i}\) \(T=0,\beta = \infty,p_1=1,p_{i\geq 2}=0\) \(T=\infty,\beta=0,p_i = \frac{1}{M}\) \(E = \sum_i \epsilon_in_i=N\sum_i\epsilon_ip_i=\frac{N}{Z}\sum_i\epsilon_ie^{-\beta\epsilon_i}=-N\frac{\partial\ln Z}{\partial\beta}\)

Constant total energy of system depends on temperature

Connection

\(dQ=TdS\) \[E=U=\sum_i\epsilon_in_i\] \[dU = \sum_i\epsilon_idn_i+\sum_in_id\epsilon_i\] 因为\(\ln \Omega\)里面有\(dn_i\)，所以第一项指的是\(dQ\)，第二项是\(dW\)

Ensembles

\[S_{AB}=S_A+S_B\]

看看下面三个定义怎么推的

Temperature definition

\[\frac{1}{T}\equiv \frac{\partial S}{\partial E}\]

Pressure definition

\[\frac{P}{T}=(\frac{\partial S}{\partial V})_E\]

Chemical potential definition

\[\frac{\mu}{T}=-(\frac{\partial S}{\partial N})_{E,V}\]

Free energy

看ppt还有tutorial吧

Partition Function

Degeneracy

\(Z=e^{-\beta\epsilon_1}+2e^{-\beta\epsilon_2}\)

In equilibrium, E,S and F are connected t Z by simple equations

\(S=\frac{E}{T}+Nk\ln Z\) \(F =E-TS\) \(S=Nk(\ln Z +T\frac{\ln Z}{\partial T})\)

Partition function of the system

\(Z_N=Z^N\) \(Z_N=\sum_ie^{-\beta E_i}\)

Boltzmann Distribution derivation by system

\(p_i=\frac{1}{Z_N}e^{-\beta\epsilon_i}\)

Mixing-Osmosis

Mixing Entropy and dilute solutions

Lattice models: \(N=N_W+N_S\)

Entropy of Mixing

Polymer solutions-Mixing entropy

Polymer solutions

Entropy of mixing: \(\Delta S_{mix}=-k(A\ln \frac{A}{A+B}+B\ln \frac{B}{A+B})\)

看ppt还有tutorial吧

Osmosis

Entropy of mixing: \(\Delta S_{mix}=-k(N_S\ln \frac{N_S}{N_S+N_W}+N_W\ln \frac{N_W}{N_S+N_W})\approx -k(N_S\ln \frac{N_S}{N_W}-N_S)\)

Solvation energy = contribution per solute molecule multiplied by the total number of such molecules. \[dU=\epsilon_s dn_s\] First law of thermodynamics \(\epsilon_sdN_s+\mu_W^0dN_W=TdS+\mu_SdN_S+\mu_WdN_W\) 分别令\(dN_S,dN_W=0\), 有\(\mu_W=\dots, \mu_S = \dots\) 使用\(\Delta S_{mix}\)得到所需偏导 最后得到：\(\mu_S=\epsilon_S+kT\ln \frac{n_S}{n_W}\) \[\mu_S=\mu_0+kT\ln \frac{c}{c_0},\enspace c =\frac{n_S}{V},\enspace c_0=\frac{n_W}{V}\]

Osmotic Pressure

\[\Delta P=cRT\] \(R = k_BN_A\) if equation contains kB, quantities relate to one particle, otherwise relate to one mole

\[\Delta P=cRT,\enspace c(mol/L)\]

\[\Delta P=ck_BT,\enspace c(个/m^3)\]

Revere Osmosis

看tutorial吧

5. Binding

5.1 Equilibria Binding

Ligand-Receptor Binding

要看一下ppt和tutorial，推\(p_{bound}\) Hill function

Reactions

\[K_{eq} \equiv \prod_i c_{i-eq}^{\upsilon_i}\]

\(\Delta G = \sum_i\upsilon_i\mu_i\) \(\mu = \mu_0+kT\ln \frac{c}{c_0}\) \(\Delta G =0\) \[\Delta G_0=-k_BT\ln K_{eq}\]

\[\Delta G=k_BT\ln \frac{K}{K_{eq}}\]

5.2 Protein folding

全是叙述，看一下ppt，不看也没问题

6. Grand-canonical-cooperativity

6.1 Gibbs-Ensemble

finding a given state of the system(charaterized by an energy \(E_A\) and nuber of particles \(N_A\) is proportional to the number of states available to the reservoir \(R\)) when the system is in this state

\[p_A = \frac{Z}{e^{-\beta(E_A-\mu N_A)}}\] \[Z=\sum_ie^{-\beta (E_i-\mu N_i)}\]

看一下tutorial，怎么用公式

6.2 Cooperativity

使用上节公式，看一下tutorial，怎么用公式

7. Aminoacids-pKa-ATP-Phosphorylation

7.1 Aminoacids-pKa

Proteins, chirality, diversity, peptide bond formation, basic protein structure, from primary to quaternary structure, disease \(K_W=[OH^-][H_3O^+],pH\) \(pK_a=pH-\lg \frac{[A^-]}{[AH]}\) 得：\(\frac{[A^-]}{AH}=\dots\) \[p(A^-)=\frac{1}{1+10^(pK_a-pH)}\]

\[p(HA)=\frac{1}{1+10^(pH-pK_a)}\]

7.2 ATP

看一下tutorial ATP怎么算

7.3 Phosphorylation

看一下tutorial ATP怎么算

8. Diffusion

8.1 Mean Variance SEM

mean: \(\overline{x}=\frac{1}{N}\sum_{i=1}^{N}x_i=\sum_ip_ix_i=\int_{-\infty}^{\infty}xp(x)dx\) \(\sigma:\) 三种形式

推一下ppt概率公式？

8.2 Diffusion

Divergence theorem: \(\int_V\nabla \cdot \mathbf EdV =\oiiint \mathbf E\cdot d\mathbf A\) Continuity equation: \(\nabla\cdot\mathbf j(\mathbf r,t)=-\frac{\partial \rho(\mathbf r,t)}{\partial t}\) Flux: \(\mathbf j \equiv \frac{Substance}{Area \times Time} = \rho \mathbf v\) Fick's First Law: \[j = -D\frac{d\rho}{dx},\enspace \mathbf j=-D\nabla \rho\]

Fick's Second Law: \[\frac{\partial\rho}{\partial t}=D\nabla^2\rho\] Solution: \[\rho(x,t)=\frac{\rho_0}{\sqrt{4\pi Dt}}e^{-\frac{(x-x_0)^2}{4Dt}}\]

\[\rho(\mathbf r,t)=\frac{\rho_0}{\sqrt{4\pi Dt}}e^{-\frac{(\mathbf r-\mathbf r_0)^2}{4Dt}}\] mean: \(\overline{x}(t)=0\) variance: \(\sigma_x^2=2Dt\)

Brownain motion microscopic model: Random walk

\(\overline{X}=0\) But what counts is not the mean position, but the spread \(\sigma_x^2=N(\overline{x^2}-\overline{x}^2)=N\overline{x^2}=Nl^2=N\overline{v}l\tau=\overline{v}lt\) 又：\(\sigma _x^2(t)=2Dt\) \(D=\frac{1}{2}\overline{v}l\) Different equations for different situations are in ppt

\[x = \sigma_x = \sqrt{2D}\sqrt{t}\]

Diffusion with external forces

\(v_D=\mu F=\frac{1}{\gamma}mg\) Down-flux: \(j_1=\mu mg\rho\) Up-flux: \(j_2=-D\frac{d\rho}{dh}\) \(j_1=j_2\) \[\rho(h)=\rho_0e^(-\frac{\mu mg}{D}h)\]

Compared to Boltzman distribution: \[\rho(h)=\rho_0e^{-\beta mgh},\enspace E=mgh\]

Einstein Relation: \[D=\mu kT\]

Special case for Stoke's Law

Frictional force (drag force) exerted on spherical objects with very small Reynolds numbers (e.g., very small particles)in a continuous viscous fluid Stoke's Law: \[F=6\pi R\eta v,\enspace \mu=\frac{v}{F}\]

\[D = \frac{kT}{6\pi R\eta}\]

Flux with external force: \(\mathbf{j}=-D\nabla \rho +\mu F\rho\) Diffusion equation with external force: \(\frac{\partial \rho}{\partial T}=D\nabla^2\rho-\mu F\nabla\rho\)

9. Electro

9.1 Poission-Boltzmann

Electronics for Salty Solutions

Magnetic fields play (almost) no role \[\nabla \cdot\mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\varepsilon_0}\] Poisson's equation: \(\nabla^2\Phi(\mathbf{r})=-\frac{\rho(\mathbf{r})}{\varepsilon_0}\) Laplace's equation: \(\nabla^2 \Phi(\mathbf{r})=0\)

\(p_{Ion}(\mathbf{r})\propto e^{-\beta E(\mathbf{r})}, \enspace E(\mathbf{r})=q\Phi(\mathbf{r})\) Charge density of the ion cloud: \[\rho_i(\mathbf{r})=q_in_i^{\infty}e^{-\beta q_i\Phi(\mathbf{r})}\]

total charge density: \[\rho_{Ions}(\mathbf{r})=\sum_iq_in_i^{\infty}e^{-\beta q_i\Phi(\mathbf{r})}\]

Poisson-Boltzman equation \[\nabla^2\Phi(\mathbf{r})=-\frac{e}{\varepsilon}\sum_{i=1}^nz_ic_{i,\infty}e^{-\beta z_ie\Phi(\mathbf{r})}-\frac{\rho(\mathbf{r})}{\varepsilon}\]

看一下tutorial

9.2 Debye-Huckel theory

Bjerrum length

Balance the ekectrostatic interaction energy: \[\frac{e^2}{4\pi \varepsilon_0Dl_B}=k_BT\]

\[l_B=\frac{e^2}{4\pi\varepsilon_0Dk_BT}\]

Debye-Huckel theory

Debye-Huckle equation: \[\nabla^2\Phi(\mathbf{r})=\frac{e^2\beta}{\varepsilon}(\sum_iz_i^2n_i^{\infty})\Phi(\mathbf{r})\]

Debye length \[\lambda_D=(\frac{\varepsilon k_BT}{e^2\sum_iz_i^2n_i^{\infty}})^{\frac{1}{2}}\]

Ionic strength \[I\equiv \frac{1}{2}\sum_iz_i^2n_i^{\infty}\]

\[\nabla^2\Phi(\mathbf{r})=\frac{1}{\lambda_D^2}\Phi(\mathbf{r})\]

看一下tutorial中公式怎么用的