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

10. Polymers-CentralLiit

10.1 Intro-Polymers

10.2 Binomial distribution

\[ p(k) = C_n^k p^k(1-p)^{n-k} \]

Special case p=0.5

Expectation value:

If n=1, p(1)=p. \(\langle x \rangle = \sum_{i=1}^1 1\cdot p = p\) $x+y = x y $ therefore \(\langle x \rangle = np\)

Poisson limit theorem

n较大, p较小时，二项分布可以近似为泊松分布

10.3 Polymers-Random walk Central limit theorem

Atoms don't always matter.

Random walks: freely jointed chain (FJC)

几段等长的segment连接，每一段都可以是任意方向，each segment 的长度叫做 Kuhn length (a)

Chain: 1D random walks

The expected value of the walker's distance from the origin, R, after N steps is 0. Similar to diffusion mean = 0, variance = \(Na^2\) R为end-to-end distance

Central limit theorem

For a set of N random variables {x_n} with finite mean and variance, the sum X of {x_n} will tend towards a Gaussian distribution regardless of the distribution of x_n

10.4 Polymers - Random walk

Sharpness of the Gause curve: \[ \sigma_x = \sqrt{N} \]

\[ \frac{\sigma_x}{N} = \frac{1}{\sqrt{N}} \]

As N becomes larger, the RMS distance increases, but the relative (to N) RMS distance becomes tiny, so it is extremely unlikely to be far (as compared to N) from the pub door.

Polymer end‐to‐end vector (1D)

\(\langle x^2 \rangle = Na^2\) \(p(x,N) \frac{1}{2\pi Na^2}e^{-\frac{x^2}{2Na^2}}\)

3D case

\[ P(R;N) = P(R;N)(1D case)^3 \]

Similar to diffusion

End-to-end distance R

\(P_{3d}(N,R)4\pi R^2dR\) 3D Maximum of distribution NOT at zero! Different from the 1D case.

Limitation of Guassian model

Wrong for \(R>Bb\)

\(P(R;N)\) is sharply peaked at R=0

11. FJC-Kuhn_Rg

11.1 FJC

\(C_n=\frac{1}{n}\sum_{i=1}^nC_i'\): Flory's characteristic ratio \(\langle R^2 \rangle = l^2\sum_{i=1}^nC_i'=C_nnl^2\)

\(C_{\infty}\): Flory's characteristic ratio, for large chain

Freely rotating chain model (FRC)

different from FJC: \(\theta\) is const. same for all

\[ \langle R^2 \rangle = nl^2 + 2nl^2\frac{\cos \theta}{1- \cos \theta} = nl^2\frac{1+\cos \theta}{1-\cos \theta} \]

\[ C_{\infty} = \frac{1+\cos \theta}{1-\cos \theta} \]

Persistence length

\[ s_p = -\frac{1}{\ln(\cos \theta)} \]

For rigid model: \(L_p=a\) For Continuous model: tangent correlation function \[ \langle t(s)\cdot t(u) \rangle = e^{-|s-u|/\zeta _p} \] For Worm-like chain(WLC) modle：DNA: small \(\theta, \ln(\cos \theta)\approx -\frac{\theta ^2}{2}\) \[ l_p = s_pl = l\frac{2}{\theta ^2} \]

\[ C_{\infty} = \frac{1+\cos \theta}{1-\cos \theta} \approx \frac{4}{\theta ^2} \]

\[ b=l\frac{C_{\infty}}{\cos (\frac{\theta}{2})}\approx l \frac{4}{\theta ^2}=2l_p \]

Kuhn lebgth b is twice the persistence length.

11.2 Polymers - Radius of Gyration(回转半径)

The radius of gyration is used because it can be easily measured experimentally.

Radisu of gyration determined via scattering experiments

Tutorial 证明 看一下tutorial中公式怎么用的

12. EntropicForce-AFM

12.1 Entropic Force

\[ \Omega (N,x) \propto p(N,x), S = k_B\ln \Omega \]

\[ \Delta S = -\frac{k_B}{2Na^2}x^2 \]

\[ \Delta G = \Delta U - T\Delta S \]

The ideal chain has no internal energy U

\[ \therefore \Delta G = -\frac{k_BT}{2Na^2}x^2 \]

\[ f = \frac{\partial G(N,x)}{\partial x} = \frac{k_BT}{Na^2}x \]

Hooke's law: entropic spring constant \[ k = \frac{k_BT}{Na^2} \]

It's easier to stretch polymers with:

• large numbers of monomers N
• large monomer size a
• at lower temperature T

The entropic nature of elasticity in polymers distinguishes them from other materials.

The ideal chain can be thought of as an entropic spring and obeys Hooke's law for elongations much smaller than the maximum elogation. 可以把上面那个x换成end-to-end vector R

等会看ppt吧，写不下去了

12.2 DNA Properties and genetics

structure and packing Polymerase chain reaction

13. DNA-Genetics-Bending

13.1 DNA-Genetics

GENETICS NETWORKS: DOING THE RIGHT THING AT RIGHT TIME maybe 再看一下ppt上概率那一张

13.1 Bending

Three modes of deformation of a beam:

• stretching
• bending
• twisting

if a bond stretches by \(\Delta a\), the beam stretches by \(\Delta L = \frac{L_0}{a_0}\Delta a\)

\[ \epsilon = \frac{\Delta L}{L_0} = \frac{\Delta a}{a_0} \]

看ppt上公式吧，还有tutorial

14. DNA-Looping-packing-Viruses

14.1 DNA-Looping

\[ E_{bend} = \frac{EIL}{2R^2}, L = 2\pi R \]

\[ \therefore E_{loop} = \frac{\pi EI}{R} = \frac{2\pi^2EI}{L}, \zeta_p / x_p = \frac{EI}{k_BT} \]

\[ E_{loop} = k_BT2\pi^2\frac{x_p^2}{L} = k_BT2\pi^2\frac{x_p^2}{d\cdot N_{bp}}, L = d \cdot N_{bp} \]

\[ \Delta S_{loop} = k_B \ln p_o = k_B(-\frac{3}{2}\ln N_{bp}+const) \]

\(p_o\) is the probability of loop formation, \(p_o \propto N_{bp}^{-\frac{3}{2}}\)

\[ \Delta G_{loop} = \Delta E_{loop} - T\Delta S_{loop} \]

14.2 DNA packing, Viruses

Viruses as Charged Spheres \(q = -2e\) per base pair

\[ U_{el} = \int dU = \int _0^R V(r)dq \]

DNA in solution: \(\lambda _D\) DNA in capsid

\[ W = NkT \ln (\frac{V_{cloud}}{V_{capsid}}) \]

ppt上单分子技术

15. Reactions

看一下tutorial中公式怎么用的或者pdf推导

Some examples

Cytoskeleton polymerization

\[ \frac{dn}{dt} = k_{on}(c_)-\frac{Mn(t)}{V}-k_{off} \]

16. Membrane-Surfaces

16.1 Membranes-Intro

• 2D fluid: Lateral diffusion
• MP can also diffuse
• Lipid flip-flop

Lipids self assemble

The effective shape of a lipid molucule is described by a packing parameter:

\[ P = \frac{v}{al} \]

Membrane Permeability

\[ j = P\Delta c \]

P为Permeability coefficient

All membranes indergo spontaneous shape changes and fluctuations due to thermal energy or application of external forces

Membrane fusion and budding

16.2 Surfaces Math Background

看一下tutorial中公式怎么用的, ppt 要是有时间也可以看看吧

\[ dE_{surface} = [\frac{\kappa}{2}(\frac{1}{R_1}+\frac{1}{R_2}-\frac{2}{R_0})^2+\frac{\kappa G}{R_1R_2}]dA \]

16.3 Membrane deformation and curvature

bending curvature: height function

\[ \kappa = \frac{1}{R} = \frac{\partial ^2h}{\partial x^2} \]

对角化Hessian matrix, 本征值即为两个方向上的曲率

\[ G_{bend} = \frac{K_b}{2}\int da [\kappa _1+\kappa _2]^2 \]

\[ G_{thickness} = \frac{K_t}{2} \int (\frac{w-w_0}{w_0})^2da \]

\[ G_{stretch} = \frac{K_a}{2} \int (\frac{\Delta a}{a_0})^2 da \]

17. Membrane-Potential

17.1 MPs

Membrane proteins

MP structure determination

17.2 Membrane Potential

\[ c(x) \sim e^{\beta E(x)} = e^{-\beta z eV(x)} \]

\[ \Delta V = V_2 -V_1 = \frac{k_BT}{ze} \ln \frac{c_1}{c_2} \]

\[ \Delta \epsilon = \Delta \epsilon _{conf} - p \frac{V_{mem}}{d} \]

\[ p_{open} = \frac{e^{-\beta \Delta \epsilon}}{1+e^{-\beta \Delta \epsilon}} \]

Nerst equation relates chemical potential to electric potential

18. Membrane-PatchClamp

18.1 Vesicles

For a vesicle of radius R:

\[ GG_{bend} = 8\pi K_b \]

For a Cylinder:

\[ G_{bend} = K_b\pi \frac{L}{R} \]

18.2 Surface Tension

表面处：\(E = -\frac{z}{2}\epsilon\) 内部：\(E = -z\epsilon\)

\[ \Delta E = +\frac{z}{2}\epsilon \]

Surface tension \(\gamma\): Energy per unit area of the surface = surface energy density

\[ \gamma = \frac{F}{2L} = \frac{dW}{dA} \]

\(dA\) is total area change, 2 sides

Laplace-Young law:

\[ \Delta P = \frac{2\gamma}{R} \]

看tutorial怎么用

18.3 Membranes: Patch clamp

看tutorial怎么用