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

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Date Created: 02/06/15

Discrete dynamic modeling of biological systems The functional form of regulatory relationships and kinetic parameters are often unknown Increasing evidence for robustness to changes in kinetic parameters bistability two steady states Hypothesis the kinetic details of individual interactions are less important than the organization of the regulatory network Discrete dynamic models assume that nodes can be characterized by only a few minimum two discrete states Discrete models can handle larger networks than continuous models Boolean modeling of biological systems Main assumption components have two main states Expressed or not expressed active or inactive open or closed ion Channel high or low level Denote these states by ON 1 or OFF 0 The changes in state are given by discrete logical rules The future state of a regulated node the output depends on the current state of its regulators inputs which may or may not include its own current state eg lf transcription factor is active gene will be transcribed gene will be expressed in the next time step Boole logic based on the operators NOT AND OR Can be defined based on set intersection and union or inputoutput relations gates truth tables Truth tables for Boolean operators NOT AND OR n1 n2 Out n1 n2 Out In Out 0 O O O O O 0 1 O 1 O 0 1 1 1 o 1 O O 1 O 1 1 1 1 1 1 1 Out NOT In Out n1 AND n2 Out n1 OR n2 In Out quot 1 quot 2 Out ln1 ln2 Out 0 1 0 0 0 o o o 1 o 0 1 0 o 1 1 1 O O 1 o 1 Out NOT In 1 1 1 1 1 1 Outn1 AND n2 Outn1 OR n2 Ex 1 Give examples for the realization of these Boolean rules in a gene regulatory network Ex 2 Consider a transcription event activated by a transcription factor Compare the continuous and Boolean description of this process From doseresponse curves to a dXdt with xi Boolean switches VII vl X mRNA Y transcriptional activator If v is large the doseresponse curve becomes a switch If YgtKY dXdtgt0 If YltKY dXdtlt0 The activation threshold is KY If activation is weak mRNA can decay Boolean simplification X Y Activation If YON XON Decay If Y OFF XOFF Hybrid models Boolean regulation combined with continuous decay Each node is characterized by both a continuous and a Boolean variable d A 28 X X X dt 1 2 z o X is defined by the threshold rule 0 if X lt 05 1 1in gt05 Comparedto Tmpg W V x thisassumes it Kr HJr constant activation threshold05 maximal synthesis rate decay rate 1 L Glass 8 Kauffman J Theor Biol 39103 1973 Implementing time in discrete models 1 Synchronous models The state of each node is updated simultaneously at multiples of a common timestep Thus the future state means the state at the next timestep Underlying assumption the timescales of all synthesis and decay processes are similar 2 Asynchronous models The state of each node is updated individually Implementations k Different update time for each node 7 k7i Select a random update order in each timestep Tlquot1 Nk W i where is a permutation of the nodes Synchronous models have deterministic state transitions asynchronicity introduces stochasticity update order dependence in the dynamics Boolean models of signaling networks Start with a known or reconstructed network The directed edges in the network indicate regulator target pairs Assume that the state of each node can be 0 or 1 o The rule giving the new state of each node is determined by a Boolean function of the states of the nodes that regulate it Choose between synchronousasynchronous update Start with a known or assumed initial condition The state of the whole network changes in time Identify the attracting states or behaviors of the system Ex 3 Construct a network of three nodes such that their indegree is one or two Associate a Boolean rule to each node Assume that each node s state changes at the same time synchronous update Start with an initial state and update the state of the nodes 10 times What is happening to the state of the network Start from a different initial state Is the final behavior be the same How many different fina statesbehaviors can the network have I I I o 1101 o o o I I I D39I 0 0391 0 o I o o mm o I o I 1 0 0 1 o o I I o I I o suomsueJi 91218 edwex3 Concepts in Boolean network dynamics Attractor a set of states that repeats itself in afixed sequence can be periodic or a fixed point Fixed point Future State Current State Previous State All states lead to or are part of an attractor Basin of attraction all states leading to a given attractor In a network of N nodes the maximum possible length of a periodic attractor is the total number of states 2N In practice the period length of the attractor is much shorter than this maximum Cause many nodes become frozen due partly to canalizing functions an attractor state shown in detail transient tree and sub trees Andy Wuenche wwwddabcom Canalizing forcing functions At least one of the inputs has the property that the output is fixed if this input has one particular value eg a AND b is canalizing because a0 implies a AND b 0 Ex 4 How many twoinput Boolean functions are there How many of them are canalizing Ex 5 Consider a network of four nodes Node A is the signal the Boolean rules of the other three nodes are the following BAorC CAand notD DBandC SetA0 a Assume that each node s state changes at the same time synchronous update Start with an initial state and update the state of the nodes 5 times What attractor did you find b Now start from the same initial state but update the nodes one by one such that each node is updated in each step in a different order Is the result the same Q How can you determine the fixed points of a Boolean model without performing updates In the fixed point time does not matter thus the transfer functions become equations BAorC CAandnotD SolutionA0B0C0D0 DBandC Asynchronous models have the same fixed points as synchronous ones Ex 7 Consider a the same network of four nodes BAorC CAand notD DBandC Set A 1 Assume that each node s state changes at the same time synchronous update Start with an initial state and update the state of the nodes 10 times What attractor did you find Is it the same as for A 0 Attractors for synchronous and asynchronous models an w Synchronous The analog of a periodic orbit in a synchronous model is a strongly connected component in state space in an asynchronous model Harvey T Bossomaier Proc ECAL97 67 1997 Integrating the Boolean rules into the network The future expression of a node depends on a combination of the expression of other nodes hh EN and not CIR CIR II J E Introduce complementary nodes gt PET OCR hh hh CIR 5 not CIR Associate pseudonodes to node combinations E5 5 EN and CIR The future expression of nodes depends on the expression of pseudonodes hh ECR CIR 9 Ex 8 construct the augmented network for Ex 5 Use different styles for edges ending in pseudonodes Boolean modeling of gene regulatory networks in the absence of data Cell differentiation is based on differential gene expression Genes regulate each other s expression Stuart Kauffman 1965 Ideas genes can be modeled by onoff switches the structure of the gene regulatory networks is unknown the regulatory functions are unknown network states correspond to cell types The Kauffman NK model Construct a network where each node s indegree is K Assume that the state of each node can be 0 or 1 o The state of each node is updated at each timestep The rule giving the new state of each node is determined by a random Boolean function of the states of its regulators Find the attractors of the network states The number of attractors corresponds to the number of possible cell types How does the number and type of attractors change with N and K Attractors in Kauffman networks For K1 networks are frozen median number of attractors is close to 2N median cycle length close to 1 o For Kgt5 networks are chaotic few attractors median cycle length close to 2N For K2 interesting level of order median number and length of attractors both scale as N This is fairly similar with the number of cell types in different organisms Stability of Kauffman networks What is the effect of a mutation changing the state of a randomly selected node If the final number of changed nodes is small frozen network Percolating changes chaotic network The threshold between order and chaos is K2 One can bias the Boolean functions so there are more of Os or 1s 1 Kc 2Q1Q Then the threshold varies with the bias Q as Ordered behavior for kltKC Does the threshold behavior apply to non regular networks Order 2Q1 QK lt 1 This relation is maintained if the 03 underlying network is ER with as ltkingtK Q 04 Q How does this compare with the threshold of a large connected 02 component 0 0 20 100 Does the threshold behavior apply to non regular networks 1 For scalefree networks a With PkZk71 m 06 the condition becomes Q DA V Chamic M B h fi r gutmu 2Q1 Q 20 1 lt 1 Z 7 1 I15 1 115 239 25 3 35 4 39lquot Scalefree networks with y gt 25 l are robust to random perturbations M Aldana P Cluzel PNAS 100 8711 2003 As we find out more about gene regulatory networks it is not necessary to assume random topologies and regulatory functions anymore It is still interesting to see how successful an ONOFF framework and Boolean logic can be as compared to chemical kineticsbased models Example Boolean modeling of the segment polarity gene network Continuous G von Dassow et al Nature 406 188 2000 Synchronous Boolean R Albert H G Othmer Journ Theor Biol 223 1 2003 Asynchronous Boolean M Chaves R Albert E Sontag Journ Theor Bio 235 431 2005 Continuous Boolean hybrid M Chaves E Sontag R Albert IEE Proc Systems Biology 2006

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