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Protein folding dynamics and more Chi-Lun Lee ( 李紀倫 ) Department of Physics National Central University
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For a single domain globular protein (~100 amid acid residues), its diameter ~ several nanometers and molecular mass ~ 10000 daltons (compact structure)
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Introduction N = 100 # of amino acid residues (for a single domain protein) = 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 N = 1000 other configurations Modeling for folding kinetics
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Concepts from chemical reactions Transition state theory F Reaction coordinate Unfolded Transition state Folded F* Arrhenius relation : k AB ~ exp(- F*/T)
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foldedunfolded (order parameter) For complex kinetics, the stories can be much more complicated Statistical energy landscape theory
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Energy surface may be rough at times… Traps from local minima Non-Arrenhius relation Non-exponential relaxation Glassy dynamics
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Peak in specific heat vs. T c T Resemblance with first order transitions (nucleation)? Cooperativity in folding
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Defining an order parameter Specifying a network Assigning energy distribution P(E, ) Projecting the network on the order parameter continuous time random walk (CTRW) Theory : to build up and categorize an energy landscape Generalized master equation
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Random energy model i = – 0, when the ith residue is in its native state. a Gaussian random variable with mean – and variance when the residue is non-native. – 0 native – non-native Bryngelson and Wolynes, JPC 93, 6902(1989)
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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.
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Random energy model For a model protein with N 0 native residues, E(N 0 ) is a Gaussian random variable with mean and variance order parameter
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Random energy model Using a microcanonical ensemble analysis, one can derive expressions for the entropy and therefore the free energy of the system:
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Kinetics : Metropolis dynamics+CTRW Transition rate between two conformations Folding (or unfolding) kinetics can be treated as random walks on the network (energy landscape) generated from the random energy model ( R 0 ~ 1 ns )
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Random walks on a network (Markovian) One-dimensional CTRW (non-Markovian) Two major ingredients for CTRW : Waiting time distribution function Jumping probabilities after mapping on
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can be derived from statistics of the escape rate : And can be derived from the equilibrium condition equilibrium distribution :
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probability density that at time a random walker is at probability for a random walker to stay at for at least time probability to jump from to ’ in one step after time Let us define
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0 jump 1 jump 2 jumps Therefore or Generalized Fokker-Planck equation
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Results : mean first passage time (MFPT)
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Results : second moments Poisson long-time relaxation
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Results : first passage time (FPT) distribution 0 < < 1 Lévy distribution
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Power-law exponents for the FPT distribution
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Locating the folding transition folding transition
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cf. simulations (Kaya and Chan, JMB 315, 899 (2002))
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Results : a dynamic ‘phase diagram’ (power-law decay) (exponential decay)
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A fantasy from the protein folding problem…
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A ‘toy’ model : Rubik’s cube 3 x 3 x 3 cube : ~ 4 x 10 19 configurations 2 x 2 x 2 cube : 88179840 configurations
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Metropolis dynamics (on a 2 x 2 x 2 cube) Transition rate between two conformations
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Monte Carlo simulations
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Energy : -(total # of patches coinciding with their central-face color)
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A possible order parameter : depth (minimal # of steps from the native state)
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Funnel-like energy landscape
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Free energy
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Energy fluctuations (T=1.3)
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A strectched exponential relaxation
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Two timing in the ‘folding’ process : 1, 2 Anomalous diffusion Rolling along the order parameter ‘downhill’ : R 1 >>1 ‘uphill’ : R 1 <<1
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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
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U F 1, 2 T F 3, 4 U 1, 2 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
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short-time scale : exponential decay long-time scale : power-law decay Waiting time distribution function
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Results : diffusion parameter Lee, Stell, and Wang, JCP 118, 959 (2003)
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