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

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 15 views.

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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 organiza ion oflhe regulatory network Discrete dynamic models assume that nodes can be characterized by only a few minim um two discrete states Discrete models can handle larger networks than continuous models Boolean modeling of biological systems Main assump ion components have two main states Expressed or not expressed ac ive or inactive open or closed ion channel high or low level Denote tnese states by ON 1 or OFF 0 The changes in state are given by discrete logical rules The future state ofa regulated node the output depends on the current state of its regulators inputs which may or may not include its own current state eg lftranscription factor is active gene will be transcribed gene will be expressed in he next ime step Boole logic based on the operators NOT AND OR Can be denned based on set intersection and union or inputroutput reiations gates truth tables Truth tables for Boolean operators NOT AND OR n1 ln2 Out in Out 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 Out NOTln Outln1 ORan Out ln1 AND ln2 in Out W l 0 1 ln2 Out 0 1 0 0 0 o o 1 o 0 1 0 1 1 1 o o o i Out NOTln i i i 1 Outln1ANDln2 Outln10Rln2 Ex 1 Give examples for tne reaiization ofthese Boolean rules in a gene regulatory network Ex 2 Consider a transcription event activated ov a transcription factor Compare tne continuous and Boolean description ortnis process From doseresponse curves to Boolean sWItches XmRNA Y transcriptional activator lfv is large the doseresponse curve becomes a switch so fYgtKV dXdtgt0 fYltKV dXdtlt0 The ac ivation threshold is Ky lf activation is weak mRNA can decay rimtit with x Boolean simpli ca ion X Y Activation fYON X ON Decay fY OFF X OFF Hybrid models Boolean regulation combined with continuous decay Each node is characterized by both a continuous and a Boolean variable dX B X X a X dz 2 X is de ned by the threshold rule 7 0 if 2lt05 1 ifXgt05 I Compared to zwpx i this assumes I Ity Y H constant activation threshold05 maximal synthesis rate decay rate 1 L Glass S Kauf 39nan J Theor Biol 39103 1973 Implementing time in discrete models 1 Synchronous models The state of each node is updated simultaneously at multiples ofa common timestep Thus he future state means the state at he next imestep Underlying assumption the timescales of all synthesis and decay processes are similar 2 Asynchronous models The state of each node is updated individually lmplementa ions k Different update time for each node 7 k7t Select a random update order in each timestep 7 Nk quoti where h is a permutation of he nodes Synchronous models have determinis ic 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 he network indicate regulator target pairs Assume that the state ofeach node can be 0 or 1 The rule giving the new state of each node is determined by a Boolean function ofthe states ofthe nodes hat regulate it Choose between synchronousasynchronous update Start with a known or assumed ini ial condi ion The state ofthe whole network changes in time Identify the attracting states or behaviors of the system Ex3 Construct a network ofthree nodes such that heir indegree is one or two Associate a Boolean rule to each node Assume that each node s state changes at he same time synchronous update Start with an initial state and update the state ofthe nodes 10 times What is happeningto the state ofthe network Start 39om a different ini ial state Is he nal behavior be the same How many different nal statesbehaviors can he network have Example A States Transitions mum quot 39 mler u ILLm umm i mum lLlLLII illIXl4 Concepts in Boolean network dynamics Attractor aset of states hat repeats itselfin a xed sequence can be perlodic or a xed polnt 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 ofN nodes the maximum possible length ofa periodic attractor is he total number ofstates 2 In practice the period length ofthe attractor is much shorter than this maximum Cause many nodes become frozen due partly to canallzlng functions an attractor state shown In detail transient tree and subrtrees Andy Wuenche WWW dtllab corn Canailzing forcing functions At least one ofthe inputs has the property that the output is fixed ifthis 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 ofthem are canalizing Ex 5 Consider a network offour nodes Node A is the signal the Boolean rules ofthe 0 her hree nodes are the following BAorC CAandnotD DBandC SetA0 a Assume that each node39s state changes at the same ime synchronous update Stan wi h an ini ial state and update the state ofthe nodes 5 times What attractor did you nd 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 he result the same Q How can you determine the xed points ofa Boolean model without performing updates In he xed point time does not matter thus he transfer func ions become equations BAorC CAandnotD SolutionA0B0C0D0 DBandC Asynchronous models have he same xed points as synchronous ones Ex 7 Consider a the same network of four nodes BA orC CAand notD DBandC Set A 1Assume that each node39s state changes at the same time synchronous update Stan wi h an i 39ial state and update the state of he nodes 10 times What attractor did you nd Is it the same as forA 0 Attractors for synchronous and asynchronous models synenrdndus Asynch rm The analdg gt a penddle drpltln a synenmndus mddel ls a strungly ednneeted edmpdnert ln state space ln an asynenrdndus mddel l HanEMT EussumalEVvauc ECALBIN lam Integrating the Boolean rules into the network Thefumre Expresslun gt a nude depends an a edmplnatldn drtne eltpressldn dr dtner nudes hh39 EV andnntcm quot lntmduee drnplernentary nudes we 4 and CIR 7 Assdelate pseuddnddestd ndde edmplna ldns Ea EN and a Thefuture Expresslun dr nudes depends dntne eltpressldn dr pseuddnddes hh ECR Ex 8 unstructthe augmented netwurk turEx 5 Use dlYfErEnt styles rdredges endlng ln pseuddnddes Boolean modeling of gene regulatory networks in the absence of data cell dlrrerentlatldn ls pased dn dlnerennal gene expressldn Genes regulate eaen d has Expresslun Stuart Kaumnan laei ldeas genes can pe mddeled by unmsztenes tne structure at ne gene regulatdry netwurks ls unknuwn tne regulatdry runendns are unknuwn netwurk states ednespdnd tel cell types The Kauffman NK model Cunstrut a netwurk Where eaen nude s lnrdegree ls K Assume tnattne state at eaen ndde can be U ml ThE state at eaen ndde ls updated ateaen nmestep ThE rule glylng ne new state dreaen nude ls detennlned by a randum addleanrunetldn drtne states at lts regulaturs FlndthE attracturs pr ne netwurk states The number at attracturs ednespdnds tel tne number at pdsslple cell types Huw dues ne numper and type of attraetdrs enange Wltn N and K v Attractors in Kauffman networks FurK1 neIWurKs are rmzen meman numberufattractms rs erusetu 2N meman eyere rengm tusetut Juno netwurks are charm rew attracturs meman eyere rengm erase m 2 Fur rnterestrng ever at under meman numberand reng n er attracturs bath scare as N39 ans rs famy srmHarWrth the number er 2 types m mrrerem urgamsms Stability of Kauffman networks What rs the erect er a muta run hangmg the state at a rammeer sereeteu nude rrmerrnar numberufchanged nudes rs smaH ermzen netwurk Pemutanng changes eenau r netwurk The mresnure between under and them rs K2 One can brasthe auureanmneuuns su here are mere EIst ur 1s Thenmemresnum vanes wr hthe mas o as K 1 quot ZQUQ Omered nehamnrrner Does the threshold behavior apply to non regular networks Order 1Q1 QKlt1 Thrs rera run rs marmarned r he uneerryrng netwurws ER wrm ltKquotgtK 0 Hum duesthrs umpare Wrth memresnere er a arge cunnected umpurvervt Does the threshold behavior apply to non regular networks Fur staterfree nemws 39 W m Pk2rk r m as the eenmuenneeemes QM Chum o Buhaviar Rahm 2Q1 Qz 1 lt1 Behavior 27 1 05115225335 7 Scatefreenewmrkswrth p25 are mbustturandum pemrbatmns M mama D HUD cw mu am pm As we nd out more about gene regulatory networks it is not necessary to assume random topologies and regulatoryfunctions 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 ofthe segment polarity gene network Con inuous G von Dassow et al Nature406 188 2000 Synchronous Boolean R Albert l l e omineiuoum Theor Biol 223 i 2003 Asynchronous Boolean M Cnaves R Albert E Sontag Journ Theor Bio 235 43 2005 Continuous Boolean hybrid M Cnaves E Sontag R Albert EE Proc Systems Biology 2000

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