Computational Biophysics and Systems Biology
Computational Biophysics and Systems Biology CSE 60531
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This 0 page Class Notes was uploaded by Mrs. Damaris Hyatt on Sunday November 1, 2015. The Class Notes belongs to CSE 60531 at University of Notre Dame taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/232757/cse-60531-university-of-notre-dame in Computer Science and Engineering at University of Notre Dame.
Reviews for Computational Biophysics and Systems Biology
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Date Created: 11/01/15
Review Questions Section 14 1 Isolated systems with constant energy and volume 2 The probability density function for microcanonical ensemble is given by where P de nes the phase space 3 pP will have a constant nonzero probability 9 in the region 7 Elge and 0 otherwise 4 Instantaneous temperature is based on the kinetic energy for a given set of momenta The kinetic temperature is based on the average kinetic energy of the system 5 Let P5 be an energy shell de ned by E 7 e lt HP lt E 6 where E is a small positive value The ergodic hypothesis states that for almost every choice of initial values7 a trajectory P Ht samples the enegetically accessible part of energy shell uniformly Ergodic hypothesis concludes that ensemble average of some observable of a trajectory 0P can be computed as a time average as t 0rprdrthm 0rt dt 2 H00 0 Review Questions Section 15 1 No 2 System with constant volume and temperature Often7 it is part of a larger system which is NVE 3 The probability density function for Canonical ensemble is given by HW P957P0lt59510 W 7 3 where z are positions and p are the momenta 4 The probability that an atom 239 will have positive x component of velocity pm is given by f oppzdpm 1 oo 7 4 fem wdpz 2 ln microcanonical ensemble the probability of each point in the phase space is uniform Now each atom can have either positive x component of velocity with probability Therefore there are 2N choices for N atoms Out of these only one possible outcome will give us positive z component for all N atoms Therefore the probability of ln canonical ensemble the probability of for a given atom to have positive z component of velocity is given by 71 fgoenfm m pg wdpim 1 7 5 fiooo6pi m1piwdpim 2 For all N atoms N 1 Z mm 10m fgoezp i1 dpm dpMm N 1 v 6 Z mm 10m fiooo6p 111 dez which can be simpli ed to considering the fact that momenta are independent fgompi mipiwdpm f owp miflpi wdpm i i 7 7 l 1 2 7 l 1 2 N fiooomp ml plrdpm f ioexp mm erde 2 Therefore the probabilities are the same for both canonical and microcanonical ensemble 5 Newton7s Equation of motion for NVE Lagevin Equation for NVT Exercise Section 15 a The canonical probability distribution function is given by pz7poltexp 7 8 Since H pTM lp Ux It can also be written as M 7PTM 1P pltzpgtoltexp exp lt9 Therefore z and p are independent Note that SN 2 2 2 T 71 10m pig Pu M 7 7 7 10 p p mimimi Therefore plmp1yplz p3NZ are independent 2 Exercise Section 15 b Note that pay is the y component of the momenta of atom 239 We already showed that momenta of atoms are independent in canonical ensemble The probability distribution of pay is given by 2 FM my olt kW 11 Therefore7 pay has Gaussian distribution with mean 1 0 and standard deviation 039 Exercise Section 15 c Multiply Z with the standard deviation of pm to generate pay Also you need to use the standard deviation to modify the normalization constant 27m Chandler 36 Helmholtz free energy is given by A 7111 Q 12 Therefore7 BA 7 ln Q 13 Then 66A 61 Q n 7 7 7 14 66 a Nyv H which gives us 36A E 15 lt 36 NV Question 2 Total time 9 100315 A total of 18 simulations did not fold Time 9 considering the rst passage times7 9 30918 715 So 71 82 Therefore folding rate k1 g 082715 16 The standard deviation 0k1 g 009055ns 17 With rst passage time7 the folding rate kf Q 2652715 18 9 and standard deviation 0kf 502937 15 19 The folding time estimators are given by 1 lt t1 gt 7 12195 715 lt k1 gt and with rate from rst passage time is given by lt t gt 1 0 377 715 f lt kf gt 20 21