Statistical mechanics of biological processes 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. 1 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. 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. 2 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. 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. 3 Modeling biological processes Boltzmann’s distribution: key equations 1 −Ei kBT p(Ei ) = e Z Z = ∑e−Ei kBT i 1 F = − ln Z β 1 −Ei kBT ∂ E = ∑Eie = − ln Z Z i ∂β ... 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. 4 Microstates in biology Lattice model very useful to describe statistical mechanics of molecular recognition events (concentration arises naturally as key variable). • 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. 5 Microstates in biology 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. 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. 6 Microstates in biology DNA (or any other polymer) in solution Definition of microstates not absolute, but depends on problemShape of ofpolymer interest! (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. 7 Modeling biological processes Boltzmann’s distribution: examples of two-states systems 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. 8 Modeling biological processes Boltzmann’s distribution: examples of two-states systems 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. 9 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: ∑ 1bound states pbound = ∑ 1bound + ∑ Lunbound states 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. 10 Ligand-receptor binding • Numerator statistical weight of having 1 ligand bound and L-1 in solution 1 W 1bound e−βεb L 1 ∑ bound = microstates = × ∑ ( − )unbound states states −β L−1 ε Ω! −β L−1 ε L −1 = e ( ) sol = e ( ) sol ∑ ( )unbound ∑ % ' states states (L −1)!&Ω−(L −1)(! • Denominator configurational partition function for L ligands swimming in solution with no binding to receptor: −βLεsol Ω! −βLεsol ∑ Lunbound = ∑ e = e states states L!(Ω− L)! 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. 11 Ligand-receptor binding • If Ω L as often happens, approximation holds: Ω! L ≈ Ω (Ω− L)! • Introducing energy difference, concentration c and concentration standard c0 (L = Ω ): Δε = εb −εsol c = L ΩVbox c0 =1 Vbox • Probability can be written after rearrangement of equation as: −βΔε Langmuir adsorption isotherm (c c0 )e or pbound = −βΔε 1+(c c0 )e Hill function with coefficient 1 -βΔε Weights to have zero or one ligands bound are 1 and c/c0 e respectively. 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. 12 Ligand-receptor binding Langmuir adsorption isotherm Estimate of parameters by choosing 3 3 Vbox = 1 nm c0 = 1 molecule/nm = = 1024 molecules/l = 24 = 10 /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. 13 Ligand-receptor binding Langmuir adsorption isotherm Estimate of parameters by choosing 3 3 Vbox = 1 nm c0 = 1 molecule/nm = = 1024 molecules/l = 24 = 10 /NA M ~ 1.66 M 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. 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. 14 Modeling biological processes Boltzmann’s distribution: examples of two-states systems 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. 15 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. 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. 16 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. 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