1 Modeling and control of gene regulatory networks Madalena Chaves BIOCO 2 RE (Biological control of artificial ecosystems)
2 Math 640 Topics in control theory
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5 Genetic networks: transcription and translation DNA mRNA Protein (1-2 copy /cell) (10 3 in E. coli) (10 6 in E. coli 10 9 mammalian) Transcription Translation (RNApolymerase) (Ribosomes) 1 min to transcribe 2 min to translate 2-5 min mRNA lifetime 10 3 polymerase/cell 10 4 ribosomes/cell
6 Genetic networks: some common interactions Binding event Signaling event + (eg., MAPK cascade) Activation of transcription (A M) Translation (M P) A Repression (X M) X X
7 Experimental data (“data rich/data poor” Sontag 2005 ) Expression of gene Microarray wingless, relative changes fly embryo (red: expression (dark: higly increased) expressed) Cdc2, cyclin B, Pomerening, Kim & Ferrell, Cell 2005
8 Genetic networks: questions and challenges Modeling Understanding the system; dynamics; predictions Model and experiments: available data different mathematical formalisms give different information Parameters calibration of models; robustness
9 Genetic networks: questions and challenges (Too) many components: model reduction techniques Two well-known modules: interconnection of two systems Control How to find feedback laws? How to implement? Synthetic biology: assembling components; re-wiring a network State estimation, observers
10 Genetic networks: how to model A Activation of transcription (A M) Concentration of mRNA in terms of activator A n dM A = n − M M dt n A Repression (X M) Concentration of mRNA in terms of repressor X X n dM n − M M = dt n X Concentration of protein Translation (M P) dP = M − P P dt
11 Example: drosophila segment polarity network Model: concentrations of mRNA and proteins , for a group of 5 genes responsible for generating and maintaining the segmented body of the fruit fly Goal: reproduce the observed pattern of expression for these 5 genes Expression of gene wingless
12 A model using ordinary differential equations Drosophila segment polarity genes von Dassow et al, Nature 2000
13 Parameters and dynamical behavior Drosophila segment polarity genes von Dassow et al, Nature 2000 About 180 eqs. Randomly try 200,000 sets of parameters About 0.5% yield “correct” gene pattern
14 Alternative frameworks: qualitative models CIR Boolean models: logical rules; 0/1 or ON/OFF states hh k 1 = EN k and not CIR k hh EN Robustness of the model to perturbations in the environment? Fluctuations in the mRNA/protein concentrations; Different timescales in biological phenomena; Degradation and synthesis rates
15 A Boolean model of the segment polarity network 0, if i ∈{ 1,2 } = SLP i 1, if i ∈{ 3,4 } = CIA i and SLP i and not CIR i or [ wg i and CIA i or SLP i and not CIR i ] wg i = wg i WG i = WG i − 1 or WG i 1 and not SLP i en i Albert & Othmer = en i J Theor Biol 2003 EN i = EN i and not CIR i hh i = hh i HH i = CIA i and not EN i and not CIR i ptc i = ptc i or PTC i and not HH i − 1 and not HH i 1 PTC i = not EN i ci i = ci i CI i = CI i and [ not PTC i or HH i − 1 or HH i 1 or hh i − 1 or hh i 1 ] CIA i = CI i and PTC i and not HH i − 1 and not HH i 1 and not hh i − 1 and not hh i 1 CIR i
16 The model exhibits multiple “biological” equilibria Wild type No segmentation Broad stripes wg expression wg WG en EN hh HH ptc PTC ci CI CIA CIR en mutants ptc mutants, (lethal phenotype) heat shocked genes
17 How to study Boolean models? Dynamics: synchronous or asynchronous algorithms? k and not CIR T CIR k 1 = EN T EN k hh T hh Chaves, Albert & Sontag, JTB 2005 Piecewise linear models - Glass type CIR CIR dhh c CIR c = − i hh c F hh dt with: F hh t = EN t and not CIR t hh EN hh hh = { EN { 0, if hh c 0.5 hh c EN c 1, if hh c 0.5
18 Boolean models: updates and dynamics ∆ t k T Synchronous All variables simultaneously updated. ⇒ Deterministic trajectories in a directed graph. Positive loop Synchronous transition graph O11 O01 A B 000 101 100 111 C 110 010
19 Boolean models: updates and dynamics k T 1 k T 2 Asynchronous ... k T N Each variable updated at its own pace: perturbed time unit (1+ r ) T , r in [-ε, ε] ⇒ NOT deterministic
20 Boolean models: updates and dynamics k T 1 k T 2 Asynchronous ... k T N Each variable updated at its own pace: perturbed time unit (1+ r ) T , r in [-ε, ε] Follow one of many possible trajectories in the asynchronous transition graph, O11 O01 A B 000 101 100 111 C 110 010
21 Totally asynchronous and random order updates Starting from same initial state, percentage of simulations that converge to each steady state ----- low robustness ... 56%, Wild type 24%, Broad stripes 15%, No segmentation 4%, Wild type variant 1%, Ectopic and variant
22 Random order updates + Timescale separation First, update all protein nodes; then, update all mRNA nodes Any permutation among protein nodes followed by any permutation among mRNA nodes Theorem : Trajectories diverge from the wild type steady state if and only if the first permutation among proteins satisfies the following order, in the third cell CIR 3 CI 3 CIA 3 PTC 3 CI 3 CIR 3 CIA 3 PTC 3 [CI-PTC] CI 3 CIA 3 CIR 3 PTC 3 and all other proteins may appear in any of the remaining sites. Chaves, Albert & Sontag, JTB 2005
23 Random order + Timescale separation 87.5%, Wild type Markov Chain with two absorbing states Increased 12.5%, robustness Broad stripes
24 Piecewise linear systems: Glass-type model Timescale of node X i Synthesis of gene/protein X i (ON/OFF) dx i = i F i X 1 , , X n − x i dt Steady states: same as in with: F i X 1 , , X n = Boolean rule for node X i Boolean model and: X i = { } 0, if x i i 1, if x i i Based on: Glass& Kauffman, 1973; Edwards and Glass, 2000
25 Some simulations Four cells in each parasegment; periodic boundary conditions Initial Cell 1 Cell 2 Cell 3 Cell 4 Final (stage 8) (stages 9-11)
26 Timescale separation: 100% convergence to WT Assumption I: protein 2 mRNA Assumption II: 1 = i ≤ 0.5 Assumption III: PTC 3 CI 3 Theorem : Under these assumptions the Glass-type model always converges to the wild type steady state Chaves, Sontag & Albert 2006
27 Analysis of Boolean models and beyond Robustness and fragility of Boolean models for genetic regulatory networks, Chaves, Albert and Sontag, 2005 : Paper was in JTB top 10 most cited (of the last 5 years) “Timescale separation” leads to “Priority classes” (Bioinformatics: GINsim software Chaouiya, Thieffry, etc.) Further work: asynchronous transition graphs and the dynamical behavior of “large” networks Further work: piecewise linear systems
28 Piecewise linear systems: qualitative framework x = f x − x ˙ n , n ×ℝ ≥ 0 n , x ∈ℝ ≥ 0 f : ℝ ≥ 0 = diag 1, , n 1 ⋯ i r i M i Thresholds: 0 i Function f is a sum of products of step functions X , s X Refs: Casey, de Jong & Gouz é , 2006
29 Piecewise linear systems k i x i i k i 1 Regular domains: B k 1 , ,k n , k i ∈{ 0, r i } , i l , Switching domains: D l , x i = i for some i − 1 f k 1 , ,k n − x = 0 ⇒ k 1 , ,k n = k 1 , ,k n Focal points: ˙ x = f Example: − x 2 , 2 − 1 x 1 x 1 = 1 s ˙ − x 1 , 1 − 2 x 2 x 2 = 2 s ˙
30 Measurements and control Qualitative measurements: x i , i r ∈ { 0,1 } s Know only: position of variables with respect to thresholds (either “ weakly expressed ” or “ strongly expressed ”) Qualitative inputs : u piecewise constant (in each regular domain) n u: ℝ ≥ 0 ×ℝ ≥ 0 { u min , 1, u max } Can only implement three values . Inputs can act on degradation or synthesis rates (inducers) Chaves & Gouz é, Automatica 2011
31 Control of simple biological motifs: the bistable switch − x 2 , 2 − 1 x 1 , − x 1 , 1 − 2 x 2 x 1 = 1 s x 2 = 2 s ˙ ˙
32 Control of the bistable switch − x 2 , 2 − 1 x 1 , − x 1 , 1 − 2 x 2 x 1 = u 1 s x 2 = u 2 s ˙ ˙ Problem : using only qualitative control laws, is it possible to drive the system to either of its stable steady states? Control : relocate focal points
33 Control to steady state P1 1 x ∈ B 11 ∪ B 10 u = u min u x = x ∈ B 01 u min x ∈ B 00 u max u = u max Theorem : Assume that Φ ( ϴ 1 )< ϴ 2 . The system with this control law converges to point P1. Chaves & Gouz é, Automatica 2011
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