Graph-based Modeling of Biological Regulatory Networks : - PowerPoint PPT Presentation

Graph-based Modeling of Biological Regulatory Networks : Introduction of Singular States Adrien Richard, Jean-Paul Comet, Gilles Bernot LaMI : Laboratoire de M´ ethodes Informatiques. e d’´ UMR 8042, CNRS & Universit´ Evry

Motivations Dynamics of regulatory networks ⇓ Differential systems ⇓ Discretization ✗ ✔ ✗ ✔ | 0 | | 1 | | 0 , 1 | | 1 | | 1 | | 1 | | 1 , 2 | | 1 | | 2 | | 1 | ✖ ✕ ✖ ✕ ✎ ☞ • 0 1 1 1 2 1 ✍ ✌ ✗ ✔ | 0 | | 0 , 1 | | 0 , 1 | | 0 , 1 | | 1 | | 0 , 1 | | 1 , 2 | | 0 , 1 | | 2 | | 0 , 1 | • ✖ ✕ 0 0 1 0 2 0 | 0 | | 0 | | 0 , 1 | | 0 | | 1 | | 0 | | 1 , 2 | | 0 | | 2 | | 0 | Thomas’ formalism Introduction of singular states The introduction of singular states in Thomas’ formalism allows us to check by computer temporal properties concerning them

Summary 1. Dynamics of regulatory networks with singular states ⊲ Quantitative description ⊲ Qualitative description 2. Selection of models : how find satisfactory models ? ⊲ Feedback circuit functionality ⊲ Temporal logic and Model checking 3. SMBioNet : a tool for the selection of models 4. Conclusions

Biological regulatory networks Regulatory networks ⇒ oriented labelled graphs : ⊲ vertices ⇒ biological entities ⊲ edges ⇒ regulations (activations/inhibitions) dependent on thresholds A regulation u → v is labelled by a sign α uv and a threshold θ uv High abstraction level : + , θ uv prot ARNm ✗ ✔ ✗ ✔ ✓ ✏ ✓ ✏ gene u gene v u v ✒ ✑ ✒ ✑ ✖ ✕ ✖ ✕ prot ARNm − , θ vu

Quantitative description Differential equation systems : dx v I α uv ( x u , θ uv ) − λ v x v � = for all v dt u → v = synthesis rate − degradation rate I + ( x u , θ uv ) I − ( x v , θ vu ) k uv k vu x u x v 0 0 θ uv θ vu ✎ ☞ ✎ ☞ ✎ ☞ ✎ ☞ + − u v v u ✍ ✌ ✍ ✌ ✍ ✌ ✍ ✌ ⊲ No analytic solution ⊲ To much unknown parameters

Quantitative description Piecewise linear approximation : I + ( x u , θ uv ) I − ( x v , θ vu ) k uv k vu x u x v 0 0 θ uv θ vu ✎ ☞ ✎ ☞ ✎ ☞ ✎ ☞ + − u v v u ✍ ✌ ✍ ✌ ✍ ✌ ✍ ✌ ⊲ Uncertain effect on thresholds : I − ( θ uv , θ uv ) =]0 , k uv [ ⊲ On singular states, the equation system becomes an inclusion system : dx v I α uv ( x u , θ uv ) − λ v x v , � dt ∈ for all v u → v

Quantitative description Piecewise linear approximation : ⊲ Analytic solution on each domain where the synthesis rates are constant x v A ( x ) θ uv ✓ ✏ ✓ ✏ u v A ( x ) = lim t →∞ x ( t ) θ vu ✒ ✑ ✒ ✑ x θ vu x x u 0 0 θ uv ⊲ Towards a qualitative description

Qualitative description : discretization A gene u with n targets has 2 n − 1 qualitative expression levels : I + ( x u , θ uv ) ✎ ☞ v ✍ ✌ + , 1 x u ✎ ☞ u I − ( x u , θ uw ) ✍ ✌ θ uv θ uw ✎ ☞ − , 2 w ✍ ✌ x u d u ( x u ) → | 0 , 0 | | 0 , 1 | | 1 , 1 | | 1 , 2 | | 2 , 2 | The qualitative variable x u = d u ( x u ) can have tow kinds of values : ⊲ regular value : if x u = | q, q | = | q | then u regulates q of its targets ⊲ singular value : if x u = | q, q + 1 | then u regulates the same q targets and regulates uncertainly an other one

Qualitative description : resources At the qualitative state x : ⊲ the regular resources R v (x) of a gene v are the genes which induce the increase of synthesis of v ⊲ the singular resources S v (x) of a gene v are the genes which regulate v uncertainly ✎ ☞ R v (x) = ∅ ∅ { u } { u } { u } v ✍ ✌ S v (x) = ∅ { u } ∅ ∅ ∅ + , 1 ✎ ☞ u ✍ ✌ { u } { u } { u } ∅ ∅ R w (x) = ✎ ☞ − , 2 S w (x) = ∅ ∅ ∅ { u } ∅ w ✍ ✌ | 0 , 0 | | 0 , 1 | | 1 , 1 | | 1 , 2 | | 2 , 2 | x u = | 0 | | 0 , 1 | | 1 | | 1 , 2 | | 2 |

Qualitative description : attractors At the state x, the variable x v evolves towards the attractor A v (x) according to its regular and singular resources : A v (x) = | K v, R v (x) , K v, R v (x) ∪ S v (x) | Where the qualitative parameters associated v are such that : ⊲ if ω 1 ⊆ ω 2 then K v,ω 1 ≤ K v,ω 2 ⊲ K v,ω ∈ { 0 , ..., n } if v has n targets A v (x) gives the tendency of x v : ⊲ if x v < A v (x) then x v tends to increase ( ր ) ⊲ if x v > A v (x) then x v tends to decrease ( ց ) ⊲ if x v ⊆ A v (x) then x v is steady ( � )

Qualitative description : example   + , 1 K u, ∅ = 0  K u,v = 1  ✓ ✏ ✓ ✏ ,   u v Model =   ✒ ✑ ✒ ✑   K v, ∅ = 0     K v,u = 1 − , 1 Tendencies x u x v A u (x) A v (x) A u (x) A v (x) of x u of x v | 0 | | 0 | | K u,v | | K v, ∅ | | 1 | | 0 | ր � | 0 | | 0 , 1 | | K u, ∅ , K u,v | | K v, ∅ | | 0 , 1 | | 0 | ր ց | 0 | | 1 | | K u, ∅ | | K v, ∅ | | 0 | | 0 | ց � | 0 , 1 | | 0 | | K u,v | | K v, ∅ , K v,u | | 1 | | 0 , 1 | ր ր | 0 , 1 | | 0 , 1 | | K u, ∅ , K u,v | | K v, ∅ , K v,u | | 0 , 1 | | 0 , 1 | � � | 0 , 1 | | 1 | | K u, ∅ | | K v, ∅ , K v,u | | 0 | | 0 , 1 | ց ց | 1 | | 0 | | K u,v | | K v,u | | 1 | | 1 | ր � | 1 | | 0 , 1 | | K u, ∅ , K u,v | | K v,u | | 0 , 1 | | 1 | ց ր | 1 | | 1 | | K u, ∅ | | K v,u | | 0 | | 1 | ց �

Qualitative description : example Tendencies ⇒ asynchronous state graph : x u x v Tendencies | 1 | | 1 | | 0 , 1 | | 1 | | 1 | | 1 | | 0 | | 0 | ր � | 0 | | 0 , 1 | ր ց ✛ ✘ ⇒ | 0 | | 1 | ց | 0 , 1 | | 0 | | 0 , 1 | | 0 , 1 | | 1 | | 0 , 1 | � ✚ ✙ | 0 , 1 | | 0 | ր ր | 0 , 1 | | 0 , 1 | � � | 0 | | 0 | | 0 , 1 | | 0 | | 1 | | 0 | . . . . . . . . . . . . Without singular states ⇒ Thomas’ formalism : x u x v Tendencies | 1 | | 1 | | 1 | | 1 | | 0 | | 0 | ր � ⇒ | 0 | | 1 | � ց | 1 | | 0 | � ր | 0 | | 0 | | 1 | | 0 | | 1 | | 1 | ց �

Qualitative description : properties u ∈ ω k uv For a given network, if for all K v,ω we have K v,ω = d v ( � λ v ) then for all state x such that d ( x ) = x we have A(x) = d ( A ( x )) x v ✛ ✘ A ( x ) | 1 | | 1 | | 0 , 1 | | 1 | | 1 | | 1 | ✚ ✙ x v | 0 , 1 | | 0 | | 0 , 1 | | 0 , 1 | | 1 | | 0 , 1 | θ vu | 0 | | 0 | | 0 , 1 | | 0 | | 1 | | 0 | x x x u x u 0 0 θ uv Consequently : ⊲ The qualitative description extracts the essential qualitative features of the continuous one ⊲ each continuous steady state corresponds to a qualitative steady state

Qualitative description : properties ⊲ Increase in the number of states but not in the number of qualitative parameters ⊲ The qualitative parameters are unknown but they can take a finite num- ber of possible values ⊲ We can use an exhaustive approach to model the dynamics of a network : Generate all the models associated to a network in the aim to select those which give a dynamics coherent with the experimental knowledge ⊲ Three approaches : – feedback circuit functionality = steadyness of singular states – temporal logic – model checking

Selection of models : circuit functionality Feedback circuits play a major role : ⊲ a positive circuit is a necessary condition for multistationarity ⊲ a negative circuit is a necessary condition for stable cycle Some singular states are circuit-characteristic states and when they are steady, the corresponding circuits are functional : ⊲ a positive circuit generates multistationarity (differentiation) ⊲ a negative circuit generates stable cycle (homeostasis) Differentiation and homeostasis are experimentally measurable ⇓ constraints on K for the steadyness of circuit-characteristic states

Selection of models : circuit functionality +,1 ✎ ☞ ✎ ☞ ⇒ u v +,2 functionality of both circuits ✍ ✌ ✍ ✌ − ,1 ⇓ ✛ ✘ ✛ ✘ ✓ ✏ | 0 | | 1 | | 0 , 1 | | 1 | | 1 | | 1 | | 1 , 2 | | 1 | | 2 | | 1 | 0 1 1 1 2 1 ✚ ✙ ✚ ✙ ✒ ✑ ✛ ✘ | 0 | | 0 , 1 | | 0 , 1 | | 0 , 1 | | 1 | | 0 , 1 | | 1 , 2 | | 0 , 1 | | 2 | | 0 , 1 | 0 0 1 0 2 0 ✚ ✙ Thomas | 0 | | 0 | | 0 , 1 | | 0 | | 1 | | 0 | | 1 , 2 | | 0 | | 2 | | 0 | With the singular states the multistationarity and homeostasis induced by the circuit functionality are more explicit

Selection of models : model checking Temporal logic and model checking can be used as an indirect criterion to constrain parameters K Undeterminism in biology : ⊲ temporal logic : Computation Tree Logic ⊲ model checker : NuSMV

SMBioNet : input/output Biological system ✗ ✔ ✗ ✔ Regulatory network Temporal properties ✖ ✕ ✖ ✕ Functional circuits All models CTL formulas Model checking ✗ ✔ All coherent models ✖ ✕ Empty Experimental plans

SMBioNet : example Network controling the immunity of bacteriophage λ : − ,1 ✗ ✔ ✓ ✏ cro +,1 cI − ,2 ✒ ✑ ✖ ✕ − ,1 ⊲ 1296 models ⊲ CTL formulas : (cI=0 & cro=0) → EF AG(cI=1 & cro=0) lysogeny initial state (cI=0 & cro=0) → EF AG(cI=0 & cro > 0) lyse ⊲ Functional circuit : steadyness of | 0 | | 1 , 2 | (functionality of the negative circuit on cro during the lytic way)

SMBioNet : example

SMBioNet : example

SMBioNet : example SMBioNet generates the 48 models which make the state | 0 | | 1 , 2 | steady and selects the 2 models which fullfil the CTL formulas lyse 0 2 1 2 0 1 1 1 cro ✓ ✏ 0 0 1 0 ✒ ✑ cI lysogeny