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## Protein folding dynamics and more

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**Protein folding dynamics and more**Chi-Lun Lee (李紀倫) Department of Physics National Central University**For a single domain globular protein (~100 amid acid**residues), its diameter ~ several nanometers and molecular mass ~ 10000 daltons (compact structure)**Introduction**Modeling for folding kinetics • N = 100 # of amino acid residues (for a single domain protein) • n = 10 # of allowed conformations for each amino acid residue • For each time only one amino acid residue is allowed to change its state • A single configuration is connected to Nn = 1000 other configurations**Concepts from chemical reactions**Transition state theory Transition state F DF* Unfolded Folded Reaction coordinate Arrhenius relation : kAB ~ exp(-DF*/T)**For complex kinetics, the stories can be much more**complicated Statistical energy landscape theory unfolded folded r (order parameter)**Energy surface may be rough at times…**• Traps from local minima • Non-Arrenhius relation • Non-exponential relaxation • Glassy dynamics**Cooperativity in folding**Peak in specific heat vs. T c T Resemblance with first order transitions (nucleation)?**Theory : to build up and categorize an energy landscape**• Defining an order parameter r • Specifying a network • Assigning energy distribution P(E,r) • Projecting the network on the order parameter continuous time random walk (CTRW) Generalized master equation**Random energy model**Bryngelson and Wolynes, JPC 93, 6902(1989) – e0 ,when the ith residue is in its native state. a Gaussian random variable with mean –e and variance De when the residue is non-native. ei = De – e non-native de – e0 native**Random energy model**• Another important assumption : random erergy approximation (energies for different configurations are uncorrelated) • This assumption was speculated by the fact that one conformational change often results in the rearrangements of the whole polypeptide chain.**Random energy model**• For a model protein with N0 native residues, E(N0) is a Gaussian random variable with mean and variance order parameter**Random energy model**Using a microcanonical ensemble analysis, one can derive expressions for the entropy and therefore the free energy of the system:**Kinetics : Metropolis dynamics+CTRW**Transition rate between two conformations ( R0 ~ 1 ns ) Folding (or unfolding) kinetics can be treated as random walks on the network (energy landscape) generated from the random energy model**Random walks on a network (Markovian)**after mapping on r One-dimensional CTRW (non-Markovian) Two major ingredients for CTRW : • Waiting time distribution function • Jumping probabilities**can be derived from statistics of the escape rate :**And can be derived from the equilibrium condition equilibrium distribution:**Let us define**probability density that at time t a random walker is at r probability for a random walker to stay at r for at least time t probability to jump from r to r’ in one step after time t**Therefore**0 jump 1 jump 2 jumps or Generalized Fokker-Planck equation**Results : second moments**long-time relaxation Poisson**Results : first passage time (FPT) distribution**0 < a < 1 Lévy distribution**Locating the folding transition**folding transition**Results : a dynamic ‘phase diagram’**(power-law decay) (exponential decay)**A ‘toy’ model : Rubik’s cube**3 x 3 x 3 cube : ~ 4x1019 configurations 2 x 2 x 2 cube : 88179840 configurations**Metropolis dynamics (on a 2 x 2 x 2 cube)**Transition rate between two conformations**Energy : -(total # of patches coinciding with their**central-face color)**A possible order parameter : depth r (minimal # of steps**from the native state) r**Two timing in the ‘folding’ process : t1 , t2**Anomalous diffusion Rolling along the order parameter ‘downhill’ : R1 >>1 ‘uphill’ : R1 <<1**Summary**• Random walks on a complex energy landscape statistical energy landscape theory (possibly non-Markovian) • Local minima (misfolded states) • Exponential nonexponential kinetics • Nonexponential kinetics can happen even for a ‘downhill’ folding process (cf. experimental work by Gruebele et al., PNAS 96, 6031(1999)) Acknowledgment : Jin Wang, George Stell**T**U t1 , t2 t3 , t4 t1 , t2 U F F • If T is high (e.g., entropy associated with transition state ensemble is small) exponential kinetics likely • If T is low or there is no T nonexponential kinetics**Waiting time distribution function**long-time scale : power-law decay short-time scale : exponential decay**Results : diffusion parameter**Lee, Stell, and Wang, JCP 118, 959 (2003)