Comp Functional Genomics
Comp Functional Genomics ECS 234
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This 31 page Class Notes was uploaded by Ashleigh Dare on Tuesday September 8, 2015. The Class Notes belongs to ECS 234 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/187796/ecs-234-university-of-california-davis in Engineering Computer Science at University of California - Davis.
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Date Created: 09/08/15
I u L Colurecl bonus indium past gastruBr donulns of expression gne 3 Gsx z HZ Fr Mesodeum Enuoderm 397quot K111 7 m N P u 39l I N F LIJ I39 mphv Flnlzvm recount meww a mxod A mnsmrag lcs39mu panegm cream ECS 234 Gene Regulation Modelling Continuous Models 5M0 Transcriptional Regulatory Systems Cis re ulato elements DNA sequence speci c sites promoters enhancers silencers Trans regulatory factors products of regulatory genes generalized speci c Zinc nger leucine zipper etc Known properties of real gene regulatory systems cistrans speci city small number of trans factors to a cis element 810 cis elements are programs regulation is event driven asynchronous regulation systems are noisy environments ProteinDNA and proteinprotein regulation regulation changes with time ECS 234 Gene Networks models of measurable properties of Gene Regulatory Systems Gene networks model functional elements of a Gene Regulation System together with the regulatory relationships among them in a computational formalism Types of relationships causal binding specificity proteinDNA binding protein protein binding etc ECS 234 Simple Genetic Circuits a mmrellinu 5 s f 0 V P comb atur 00mm MrAdams andArkm 31111998 505234 Cell cycle network in S Cerevisiae LI Fangtlng et al 2004 Proc Natl Acad 5c ECS 234 Segment polarity network in Drosophila 1 Albert and Otmer JTB 2003 Nodes mRNA round protein rectangle protein complex octagon Edges biochemical interactions or regulatory relationships ECS 234 Gene network of endomeso development in Sea Urchin Davidson 61 al Science 2002 ECS 234 Logic of Cis regulation Module B Module A BP CY CB1 Ul R CBZ 061 P OTX Z CG2 SPGCF1 CGS CG4 11 0Y8 CB1 i1 1 if i50 i7 OTX1 else i1 05 else i7 0 i8 i6 i7 i2 11 Ull ifForEorDCampZ i91 else is 0 n R is 0321 else i3 koBzm if i91 no o 1ltklt2 else no is ifFampCG1ampCBZ 142 i40 i1CG2ampCG8ampCG4 1112 else i11 1 ii UltgtihresholdampFlampi4O 15 1 2 i110i10 else is 0 i6 i4i2i3 ECS 234 Cisregulatory input of lacZYA operon in E coli Salty 1 u nus mu Ecsm Modeling Formalisms Combinatorial Qualitative Physical Quantitative 0r Continuous Static Graph Models Boolean Networks Weight Matrix Linear Models Bayesian Networks ECS 234 Stochastic Models Difference Differential Equation Models ChemicalPhysical Models Concurrency models Continuous Models of Gene Regulation Outline Quantitative Modeling Discrete vs Continuous Modeling problems Models ODE PDE Stochastic Conclusions ECS 234 Quantitative Modeling in Biology State variables concentrations of substances eg proteins mRNA small molecules etc Knowing a system means being able to predict the concentrations of all key substances state variables Quantitative Modeling is the process of connecting the components of a system in a mathematical equation Solving the equations yields testable predictions for all state variables of the system ECS 234 Discrete vs Continuous Here we will talk about continuous models Where values of variables change continuously in time andor space On a molecular scale things are discrete but on a macro scale they blend in and look continuous Next class we ll discuss discrete models ECS 234 Why Continuous Continuous models are appealing because they allow for instantaneous Change Continuous models let us express the precise relationships between instantaneous states of variables in a system c dA 1 2A 6 VS 61 3 051 B dt dC ECS 234 Problems When modeling with differential equations we face all the same problems as in the discrete models Posing the equations This presumes we understand the underlying phenomenon Data Fitting How do we learn the model from the data Solving the equations Means we can do the math Model Behavior Analyzing the fitted model to understand its behavior ECS 234 Recall the Modeling Process Knowledge Modeling Objectives Construct and Revise Models Model behavior and predictions hPJNE U Compare to new data 6 Better Models goto 3 7 Learn ECS 234 1 Ordinary Differential Equations Rate equation dx xl iSn dt f where X X1 Xn is a vector of 11 concentrations fl x R a R is a function Systems of ODEs There are n such equa ons Solving the rate equations depends onf but what is the form of the functionf The answer is as simple as possible ECS 234 The Rate Function and Regulation The rate function specifies the interactions between the state variables Its input are the concentrations and the output is indicative ie a function of the change in a gene s regulation The regulation function describes how the concentration is related to regulation x hxem w 19quot 8 1 I with 9 gt 0 the threshold for the regulatory in uence of xi on a target gene and m gt O a steepness param39eIte The function rangesfrom 0 to 1 and increases as x gt 00 so that an increase in x Will I I was we 5 f 1 cmquot fut mwss e e x1 decreases the expressqiri t m iEllt llon agyt gt eregularon unc NH 9 m s ep eed by shame 19W V6 39ii l liquot 39 Wquot expe a usl m examples of regulation functions that will be discussed in more detail in a later section Due to the nonlinearity of analytical solution of the rate equations 5 is not normally possible In special cases qualitative properties of the solutions such as the number and the stability of steady 1 hrj9m 1 511151 1 omen6 1 Hi I J I L ECS 234 Nonlinear ODEs The rate function is nonlinear Eg Sigmoidal Nonlinear additive Summarizes all pair Wise and nothing but pair Wise relationship Xm Z Tz39jfj X1 J Nonlinear nonadditive Summarizes all pairs and triplets of relationships Xm E 2 zjkfjltXjgtfkltXkgt2 TaitXi jk j ECS 234 Solving 39 In general these equations are dif cult to solve analytically when x are nonlinear 39 Numerical SimulatorsSolvers work by numerically approximating the concentration values at discretized consecutive timepoints Popular software for biochemical interactions DBsolve GEPASI MIST SCAMP 39 Although analytical solutions are impossible we can learn a lot from general analyses of the behavior of the models which some of the packages above provide ECS 234 Model Behavior Feedback is essential in biological systems The following is known about feedback negative feedback loops system approach or oscillate around a single steady state positive feedback loops system tends to settle in one of two stable states in general a negative feedback loop is necessary for stable oscillation and a positive feedback loop is necessary for multistationarity ECS 234 Data Fitting Fitting the parameters of a nonlinear system is a difficult problem Common solution nonlinear optimization scheme explore the parameter space of the system for each choice of parameters the models are solved numerically e g Runge Kutta the parameterized model is compared to the data with a goodness of fit function It is this function that is optimized Genetic Algorithms and Simulated Annealing with proper transition functions have been used with promising results ECS 234 Linear and Piecewise Linear ODEs Linear These are much easier to deal with if the input variables are limited by a constant they can be solved and learned polynomially depending on the amount of data available alel39 dt 2 W ij One way to learn them is by approximating them with linear weight models ECS 234 Piecewise linear Approximating the sigmoid regulatory function with a step function dX X xlSiSn dt g r giX 2 kilbilx2 0 EL Here the function bi is a function of n variables defined in terms of sums and products of step functions A 1 6139 0k This amounts to subdividing ndimensional space into orthants and in each of the orthants the PLODEs reduce to ODEs ECS 234 I m D mm ngmmmmpln Il quotN947 8 I 6 101I 515w21621 71931 Wt 5E2 Hz1 519u5a6393172 quot1 quotquot3 is 53 S3917912 K4 13 932 Va 373 a b FIG 9 m L c A L I x1 x2 and x3 represent protein or mRNA concentta ons respectively K1 K4 production constants n ya A A aquot 91 I i1 N1 7111 93932 7212 is K31 73 It A a b FIG lot a The phase Space box of the model in Fig 9 divided into 2 3 3 18 onhants by Lhe threshold planest b The state equau39onsfor the orthant o 5 x lt 921 91 lt x g quot2172 and 933 lt x3 5 W3 the uthant demarcated by bold lines de long JCB 2002 ECS 234 2 PDES ODEs count on spatial homogeneity In other words ODEs don t care Where the processes take place But in some real situation this assumption clearly does not hold Diffusion Transcription factor gradients in development Multicelular organisms ECS 234 Example Reaction Diffusion Equations ha 7 T from 5 EMU 24 24 U 1515 n 1 lt1 lt 4 16 Noti letwork is the l T ey can be gene addition the EDIEEE mm with s as well perhaps quotm mm um um MMquot NW mu m um a H J m an mm 1 neneoua onsa lgouagdescubeslt1 change immatorm vernal WWW legriisrnsanw Win V aesx CH 35 CO SO 0 39 p wdmmberaomgg sis large qthis e e r s laus u ns mm axi 3 x 2 fix5afz 05lsilsi5n 17 If it is assumed that no diffusion occurs across the boundaries 1 0 and l k the boundary conditions become 32 a 31275 1 0 and maxim t 0 18 These equations were first introduced in the study of developmental phenomena and pattern formation by Turing Direct analytical solutions are impossible even for two variables n2 ECS 234 Drosophila Example These PDE models have been used repeatedly to model developmental examples in the fruit y Instances of the reactiondiffusion equations only more speci c have been used to model the striped patterns in a drosophila embryo ECS 234 3 Stochastic Master Equations Deterministic modeling is not always possible but also sometimes incorrect Assumptions of deterministic continuous models Concentrations of substances vary deterministically Conc of substances vary continuously On molecular level both assumptions may not be correct Solution Instead of deterministic values accept a joint probability distribution similar to the one discussed in the Bayesian Network lectures ECS 234 E qu an on species em The time evolution of the function pX t can now be speci ed as follows pXtAtpXt lizaim Zejm 21 11 jl where m is the number of reactions that can occur in the system aj At the probability that reaction j will occur in the interval t t At given that the system is in the state X at t and mm the probability that reaction j will bring the system in state X from ancther state in tt At Gillespie 1977 1992 Rearranging 21 and taking the limit as Al vgt 0 gives the master equation van Kampen 1997 8 m 5mm 203 apXi r 22 i1 These equations are very difficult to solve and simulate Simeon smhgm mums Suhstraba uni named nnzsuusnn39lsanlzsnxsnnsnz c Jason Kastner and Caltech ODE vs Stochastic solutions ECS 234
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