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# 634 Class Note for PHYS 597A with Professor Albert at PSU

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Date Created: 02/06/15

PHYS 597A Graphs and Networks in Systems Biology Optimization based Network Modeling and Analysis of Stochastic Processes Srid har Ranganathan Dept of Industrial Engineering Course Project Presentation Presentation Overview Motivation for the Project Network Evolution Concepts in the Project Literature Review Stochastic Simulator by DT Gillespie Research Interests Optimization of Network Processes Chemical Reaction NetWorks Complex Supply Chain Networks Motivation How are the real networks modeled What is the process governing the specific connectivity of the real world networks Can the network be modeled in an optimization based approach What are the procedures to model the network Using optimization of processes How can such a network compare with the real world network Network Evolution Theory andom Network Models Erdos Ren39yi Connection probability dependence on the connectivity Real World Network modeling Growth Preferential Attachment Aging Network Rewiring Node Removal random preferential Network Evolution Theory Algorithm Based modeling of evolution 1 Connection probability 2 Number of nodes entering the network 3 Timebased process Mathematically incorporates all that is required 39for a realnetwork Can the network evolved based on algorithm represent a particular network How can a network represent a particular real world network to a high degree of similarity Network Evolution Theory Relative dependence of network statistics in large networks 1 Connection Probability vs Degree Distribution PWFCNEW For a Random Graph 2 Growth HI flt k gt Hy No of nodes entering the system 1 Directly proportional to average degree in the network 2 lnversely proportional to distance between a selected node pair Network Evolution Theory Aging 1 Based on distance from key nodes and time step 2 End of a sequence of optimization nodes that have distances greater than critical values are found 3 Ifthe node has lt kgt 2 kcritical the node will be eliminated Network Evolution Theory Connection Probability 1 Distance Dependent Probability PM e m 2 Degree Dependent Probability k7 1101 m M 3 Dependent Probability to incorporate large distance connectivity 5 PM dram a Network Evolution Theory Clustering Selected node39s constrained to have clustering cooeefficient above critical values Goal Programming approach Other Constraints 1 Pathle ngths 2 Upper limit on the degree of selected nodes Network Evolution Optimization Model Minimize Z 2d dj Sub to pNgtcN e 19N 11PM 1211M 1312041 11 12 13 1 Pa 13 ijdw PM m M 3 dlJ PAM Network Evolution Optimization Model In between sequential optimization kiltkgt 3 k K gt critical kcritical X ltkgt X 6i 60 Literature Review Exact Stochastic Simulation Summary Chemical Reaction Network Spatially Homogenous system If a fixed volume V contains a uniform mixture of N Chemical metabolites M Chemical reaction pathways X1 X2 Xn number of molecules of metabolite 12 N Can we predict the number of molecules in the system for each metabolite after any time t in the system Treat the system as a network of discrete variable values changing through stochastic processes Literature Review Exact Stochastic Simulation Conventional Methods Deterministic approach Assumptions interaction is continuous and predictable Consider the set of Chemical Reactions A 2C rgt B gt C A 28 gt F D F gt G B gt D D G gt F E gt D Literature Review Exact Stochastic Simulation A 2c gt B B gt 0 Ex Literature Review Exact Stochastic Simulation A 2c gt B B gt c A 2B gt F D F gt G B gt D E G Literature Review Exact Stochastic Simulation A 2c gt D B gt c A ZB gt F D F gt G B gt D D G gt F E gt D E V V A R D E G Literature Review Exact Stochastic Simulation The number of molecules of each metabolite can be expressed as an ordinary differential equation dXAdt fAXA XB xN dXBdt fBXA XB XN dXNdt fNXA XB XN f function of the connectivity and stoichiometric constants eg dXdt k1A32 k2CDF2 Literature Review Exact Stochastic Simulation Assumptions in deterministic approach to networks 1 Change in the concentration levels is continuous Mathematically the time evolution of the system is a continuous function The reaction rate equations are governed by continuous ordinary differential equa ons 2 The reactions in the network are deterministic in nature The temporal nature of the participation of the metabolites in different reactions are deterministic in nature Literature Review Exact Stochastic Simulation Assumptions in stochastic approach to networks 1 The time evolution of the system is not continuous Molecular population levels can change in integer levels 2 The interaction process is stochastic probabilistic in nature because it is impossible to predict the exact levels of concentration of the metabolites at any instant of time take into account to the exact levels of reaction rates and positions of the nodes in the network Literature Review Exact Stochastic Simulation Questioning the Assumptions 1 What is the acceptability of the assumption of discrete nature of the system Literature Review Exact Stochastic Simulation Physical Basis of the Stochastic Formulation of Chemical Kinetics Stochasticity The collisions of molecules in itself is a random process Molecular Collision Consider two molecules S1 and 39in the system Let r1 and r2 be the molecular radii v1 and v2 be the velocities of the molecules in the system Then r12 r1 r2 and v12 v1 v2 ln interval dtv volume swept cho Tr39r12 2 v12 dt Collision occurs in time interval t tdt Literature Review Exact Stochastic Simulation Stochastic Reaction Constant Cu Reaction Probability per unit time analogous with reaction rate S1S2 quotquot39gt c1 dt average prob that a 12 pair will react according to the reaction R1 X1 X2 molecules in the system gt X1X2 01 bit is the probability that reaction R1will occur in the system in infinitesimal time interval dt Literature Review Exact Stochastic Simulation Calculating the Stochastic time evolution Cu dt Probability that the reaction Ru will occur in the system in infinitesimal time interval clt fundamental hypothesis in the model Questions given X1 X2 XN 1 When will the next reaction occur 2 What reaction will it be Reaction Probability Density function Pr u Ddr Probability that given the state X1 X2 XN at time t the next reaction in V Will occur in the infinitesimal time interval tr trdt and the reaction will be Ru Literature Review Exact Stochastic Simulation hu cu dt Probability that reaction Ruwill occur in infinitisemal time interval t tdt given that the system is in the state X1 X2 XN at timetfor u 1 2 M P0 239 eXp Zau dry P 23111 au eXpaO 2 0 otherwise Thank you Questions

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