genenets cont models

8/10/2019 Genenets Cont Models

1/32


2/32

ECS 234

TranscriptionalRegulatory Systems

Cis regulatory elements: DN se!uence "speci#ic sites$ promoters%

en&ancers% silencers% Trans regulatory #actors: products o# regulatory genes

generali'ed speci#ic "(inc #inger) leucine 'ipper) etc*$

+no,n properties o# real gene regulatory systems:

cis-trans speci#icity small num.er o# trans #actors to a cis element: /-01 cis elements are programs

regulation is e ent dri en "async&ronous$ regulation systems are noisy en ironments rotein-DN and protein-protein regulation regulation c&anges ,it& time


3/32

ECS 234

Gene Networks: models of measurableproperties of Gene RegulatorySystems.

Gene net,or s model #unctionalelements o# a Gene Regulation Systemtoget&er ,it& t&e regulatory relations&ipsamong t&em in a computational#ormalism*

Types o# relations&ips: causal) .indingspeci#icity) protein-DN .inding) protein-

protein .inding) etc*


4/32


5/32

ECS 234

Li, Fangting et al. (2004) Proc. Natl. Acad. Sci.

Cell cycle net,or in S*

Cere isiae


6/32

ECS 234

Segment polarity

net,or in Drosop&ila

Nodes : mRN "round$) protein "rectangle$) protein comple6 "octagon$Edges : .ioc&emical interactions or regulatory relations&ips

l.ert and 7tmer) 8T9 2113


7/32

ECS 234

Gene net,or o# endomesode elopment in Sea rc&in

Da idson et al* Science 2112


8/32

ECS 234

;ogic o# Cis-regulation


9/32


10/32

ECS 234

Modeling =ormalisms

Static Grap&Models

9oolean Net,or s

>eig&t Matri6

";inear$ Models 9ayesian Net,or s

Stoc&astic Models Di##erence ?Di##erential E!uationModels

C&emical? &ysicalModels

Concurrency models

Combinatorial(Qualitative

!"ysi#al(Quantitative or

Continuous


11/32


12/32

ECS 234

7utline @uantitati e Modeling Discrete s* Continuous

Modeling pro.lems Models:

A 7DE

A DE A Stoc&astic

Conclusions


13/32

ECS 234

@uantitati e Modelingin 9iology

State aria.les: concentrations o#

su.stances) e*g* proteins) mRN )small molecules) etc*

+no,ing a system means .eing a.leto predict t&e concentrations o# all

ey su.stances "state aria.les$ @uantitati e Modeling is t&e process

o# connecting t&e components o# asystem in a mat&ematical e!uation

Sol ing t&e e!uations yields testa.le predictions #or all state aria.les o#t&e system


14/32

ECS 234

Discrete s* Continuous

Bere ,e ,ill tal a.out

continuous models) ,&erealues o# aria.les c&angecontinuously in time "and?orspace$

7n a molecular scale t&ingsare discrete) .ut on a macroscale t&ey .lend in and loocontinuous

Ne6t class ,e ll discussdiscrete models


15/32


16/32

ECS 234

ro.lems>&en modeling ,it& di##erentiale!uations ,e #ace all t&e same

pro.lems as in t&e discrete

models:

A osing t&e e!uations* T&is presumes,e understand t&e underlying

p&enomenon A Data =itting* Bo, do ,e learn t&e

model #rom t&e data A Sol ing t&e e!uations* Means ,e

can do t&e mat& A Model 9e&a ior* naly'ing t&e

#itted model to understand its .e&a ior


17/32


18/32

ECS 234

0* 7rdinary Di##erentialE!uations

Rate e!uation:

#unctionais:$"

J6))K6

,&ere

0 $)"

ionsconcentrato# ectoraisn0

R R

$

$

=

=

ni

n

ii

x f

ni f dt dx

Systems o !"#s$ %&ere are n s'c&e 'ations

Solving t&e rate e 'ations de ends on f ,*'t +&at is t&e orm o t&e 'nction f

%&e ans+er is$ as sim le as ossi*le.


19/32

ECS 234

T&e rate #unction speci#ies t&e interactions .et,een t&e state aria.les*

Lts input are t&e concentrations) and t&e output

is indicati e "i*e* a #unction o#$ t&e c&ange in agene s regulation T&e regulation #unction descri.es &o, t&e

concentration is related to regulation

T&is is a typical regulation #unction) called asigmoid) .ello, compared to similar ones

T&e Rate =unction and

Regulation


20/32

ECS 234

Non-linear 7DEs

T&e rate #unction is nonlinearEg*0* Sigmoidal2* Nonlinear) additi e* Summari'es all

pair ,ise "and not&ing .ut pair ,ise$relations&ip

3* Nonlinear) non-additi e* Summari'esall pairs and triplets o# relations&ips

+= j

j jij jk

k k j jijk i X f T X f X f T

dt dX

$"$"$"

= j

j jiji X f T

dt dX $"


21/32

ECS 234

Sol ing Ln general) t&ese e!uations are di##icult to

sol e analytically ,&en f i( $ ) are non-linear Numerical Simulators?Sol ers ,or .y

numerically appro6imating t&econcentration alues at discreti'ed)

consecuti e time-points* opular so#t,are#or .ioc&emical interactions:

A %&solve A GE!'S A ) S* A SC')!

lt&oug& analytical solutions areimpossi.le) ,e can learn a lot #rom generalanalyses o# t&e .e&a ior o# t&e models),&ic& some o# t&e pac ages a.o e pro ide


22/32

ECS 234

Model 9e&a ior:

=eed.ac is essential in .iologicalsystems* T&e #ollo,ing is no,n a.out#eed.ac :

A negati e #eed.ac loops: systemapproac& or oscillate around a singlesteady state

A positi e #eed.ac loops: system

tends to settle in one o# t,o sta.lestates A in general: a negati e #eed.ac loop

is necessary #or sta.le oscillation)

and a positi e #eed.ac loop isnecessary #or multistationarity


23/32

ECS 234

Data =itting

=itting t&e parameters o# a non-linearsystem is a di##icult pro.lem*

Common solution: non-linear

optimi'ation sc&eme A e6plore t&e parameter space o# t&e

system A #or eac& c&oice o# parameters t&e models

are sol ed numerically "e*g* Runge-+utta$

A t&e parameteri'ed model is compared tot&e data ,it& a goodness o# #it #unction*Lt is t&is #unction t&at is optimi'ed

Genetic lgorit&ms and Simulatednnealing) ,it& proper transition#unctions &a e .een used ,it& promisingresults


24/32

ECS 234

;inear and iece,ise ;inear

7DEs;inear A T&ese are muc& easier to deal ,it&:

i# t&e input aria.les are limited .ya constant) t&ey can .e sol ed andlearned polynomially) depending ont&e amount o# data a aila.le

A 7ne ,ay to learn t&em is .yappro6imating t&em ,it& linear,eig&t models

= j

jiji X w

dt dX


25/32

ECS 234

iece,ise linear

ppro6imating t&e sigmoid regulatory#unction ,it& a step #unction

Bere t&e #unction b il is a #unction o# naria.les) de#ined in terms o# sums and

products o# step #unctions:

T&is amounts to su.di iding n-dimensionalspace into ort&antsO) and in eac& o# t&eort&ants t&e ;7DEs reduce to 7DEs

=

=

Ll il il i

iiii

bk g

x g dt

dX

1$"$"

ni0 )$"

$$

$

0

-

j

0

-

k


26/32

ECS 234

de 8ong) 8C9 2112


27/32

ECS 234

2* DES

7DEs count on spatial

&omogeneity Ln ot&er ,ords) 7DEs don t

care ,&ere t&e processes ta e place

9ut in some real situation t&isassumption clearly does not&old

A Di##usion A Transcription #actor gradients inde elopment

A Multicelular organisms


28/32


29/32

ECS 234

Drosop&ila E6ample T&ese DE models &a e .een

used repeatedly to modelde elopmental e6amples in t&e

#ruit #ly Lnstances o# t&e reaction-di##usion

e!uations "only more speci#ic$&a e .een used to model t&estriped patterns in a drosop&ilaem.ryo


30/32

ECS 234

3* Stoc&astic MasterE!uations

Deterministic modeling is not al,ays possi.le) .ut also sometimes incorrect

ssumptions o# deterministic)continuous models: A Concentrations o# su.stances ary

deterministically A Conc* o# su.stances ary continuously

7n molecular le el) .ot& assumptionsmay not .e correct

Solution: Lnstead o# deterministicalues) accept a oint pro.a.ility

distri.ution) similar to t&e onediscussed in t&e 9ayesian Net,orlectures*


31/32

ECS 234

E!uation:

%&ese e 'ations are very di ic'lt to solve and sim'late

!"# vs. Stoc&astic sol'tions

(c) 1ason astner and Caltec&


32/32

genenets cont models

Documents