Black magic constant Vincent Danos CNRS Univ Paris

Black magic constant Vincent Danos CNRS & Univ. Paris 7

Bio Model building The point: Use a discrete model (MLT methode logique de Thomas) to constrain a continuous one (QPL =quasi piecewise linear) not a biology talk mainly i’ll go through an example 2

1. On the QPL side we start with a QPL model (with steep Hills, say n=10) which we first “sigmoidise” (ie, n=+infinity) we don’t suppose it is diagonal-PL (though this kills monotony the dynamics) 3

Hill toolkit max value steepness midpoint 4

a QPL system ma x midpoi nt Kaufman et al. , ACI vicanne, Par 5

Parameters. . . where do these parameters come from ? 6

How to find parameters ? is this really trial and error (black magic) ? parameters such that what ? such that a certain dynamical behaviour is obtained idea: use a qualitative model search for a qualitative behaviour 7

2. On the MLT side defines a finite search space: brute force shows a suitable behaviour (eg, in terms of strongly connected components) After M. Kaufman (ACI Vicanne meeting, Paris) 8

Logical sketch One first specifies V, E, levels (n), and edge thresholds (theta) 9

The MLT model Levels (n), thresholds (theta): Dynamic s. K 10

K: Behaviour

Information There is a lot more information in the discrete model at the moment ! this what obtained by brute force search Can we bring it to bear on the continuous one ? 12

3. Fitting the models necessary conditions to inherit the qualitative properties models need to be compatible fitting is in two phases partition fit (uses theta): orders the midpoints dynamic fit (uses K): constrains the other parameters 13

Models are compatible Connecting models: crunch and refine P L 14 ML T

3. 1 Partition fit no midpoint for mc C’s natural partition is: 4 x 1 x 2 x 3 D’s is: 3 x 2 x 2 x 2 There is a unique crunch & refinement that fits the two partitions p’s midpoints are now ordered 15

General crunch lambda labels C-influence graph theta labels D-influence graph pi is a graph morphism bijective on nodes (variables) This formula makes sense for any pi 16

3. 2 Dynamic fit For each discrete state we compute the symbolic steady state this works because the discrete partition refines the natural one Then we place steady states where MLT says they are: 17

Summary Glue models beyond discretisation: syntactic condition (mass action, Hill) compatibility condition (morphism pi) symbolic computation of steady states condition 18

Theorem ? There is the usual PL discretisation theorem if the dynamics is piecewise monotonic which is *not* the case in the example Else it is a heuristics 19