Dynamic Modeling Of Biological Systems

1. Dynamic Modeling Of Biological Systems

2. Why Model? • When it’s a simple, constrained path we can easily go from experimental measurements to intuitive understanding. • But with more elements often generates counter-intuitive behavior. • Counter-intuitive, but not unpredictable.

3. Why Model? • Knowledge integration • Hypothesis testing • Prediction of response • Discovery of fundamental processes

4. Predictions • Indicating we have identified a necessary amount of players, to a necessary precision → what is important to this process and what is not. • If our predictions fail, might indicate not qualitively understanding the underlying mechanism. • Feeds back to the experimental realm: pointing to where more data is needed

5. Creating the model • The model itself may be the explanation, i.e if you can intuitively understand the results of changing a parameter without solving the underlying equations. • Building the model requires thinking out the important questions that lead to qualitive understanding of the system: what we’re looking for. • The model can be a plausibility test

6. Scale • Biological pathways occur on radically different scales of time, complexity and space. • Choosing the right model scale is critical: • Coarse grained models rely on prior intuition and generalization, but less accurate causality. • Fine grained requires more detailed data, while the amount of precision provided may not be necessary.

7. Low Scale Model of Large Scale Phenomena • It can be easier create a computation-heavy modeling of molecular interactions, and see the emerging, expected high-level phenomena. • The model itself might not yield any insight on the principles of the mechanism. • But it does give us complete control of every parameter in the system

9. Interpretability • While translating biological data into a model may be relatively easy (and has a long history), how things translate back into the realm of biology is not always that clear. • With dynamic models which you allow to “run”, the model will lead to new states, what do these states represent?

10. Qualitative Networks

11. Boolean Networks: B(V,F) • A Boolean network is a directed, weighed graph, for which each component (=vertex), has a state: 0 or 1. • The effect of each component on the next is a function of its value and the edge weight.

12. Boolean Networks • For each state i at time t we have a function for the value of t in the next round.

13. w = -0.5 w = 1 A B w = 1 w= 1 w = -2 C w = -0.5 100 110 111 011 001 000 010 101 Attractors: states visited infinitely many times Boolean Networks: Example So: A=0 → B=0 A=1 → B=1 B=0 → C=0 B=1 → C=1 B,C =0,0 → A= A-1 B,C =0,1 → A= 0 B,C =1,0 → A= 1 B,C =1,1 → A= 0 We can represent a state at a given time as a triplet: (101): A=1, B=0, C=1

14. Qualitive Networks: Q(V,F,N) • We would like to allow the expression of a component to a finer detail than just ‘ON’ and ‘OFF’. • In a Qualitive Network, each component can have a value between 0 and N.The Qualitive Network Q(V,F,1) is in fact a Boolean Network.

15. Qualitive Networks • The transition function in a Qualitive Network defines for every component ci a targetifunction, of {0…N}|C|→{0…N}. • We allow changing the expression of a component by maximum of 1 each turn:

16. Qualitive Networks: Calculating targeti • Like in Boolean Networks, Inhibition and Activation are marked by negative and positive weights on the edges. • We will calculate the amount of activation on component i relatively to the maximum amount of activation it could receive: and symmetrically:

17. Qualitive Networks: Calculating targeti • So we get: • The second line entails a hidden assumption, that if ci gets no activation, it’s activation is not modeled.

18. Representing Unknown Interactions • We do not always know how each element behaves in a system. Also, many elements may be influenced by components external to our model. • Such components, with unknown behavior can be modeled by non-deterministic variables. • These variables may start at any value, but are still confined to changing by at most 1 at each turn. This is sufficient to capture any possible behavior of this component.

19. Non-Determinism • Instead of simulation (which can be exponentially hard), we’ll use model-checking tools to verify the specifications of the entire system when we have non-deterministic elements. • Because we have in fact checked for any possible behavior from the unknown components, we have shown that the specifications hold, independent of unknown component behavior.

20. Attractors: Infinitely Visited States • The attractors of a Qualitive Network correspond to the steady states of the biological system. Other states can be seen as unstable steps that will quickly evolve into an attractor. • When checking if a specification holds for the system, we do not insist that they hold for every state, only for the attractors.

21. Attractors: Infinitely Visited States • In the Qualitive model, we will often concern ourselves with the attractors in the model, specifically: • How many are there? • Which start positions lead to which attractors? • Instead of testing the exponentially many possible start positions, we will prune the number based on biological data and only test those that interest us.

22. ? Maintain cell in proliferating state Initiate differentiation Example: Crosstalk between Notch and Wnt Pathways • Pathways that play roles in proliferation and differentiation in mammalian epidermis.

23. Crosstalk between Notch and Wnt Pathways • Assumption 1) When GT1 > GT2 the cell is proliferating, when GT1 < GT2 the cell is differentiated. • Assumption 2) When the cell is more inclined to proliferation (GT1 is high or when GT2 is low) the cell is more sensitive to chemically induced carcinogenesis.

24. We’ll take 5 cells to represent the layers of the skin Low Wnt from the upper layers of skin High Wnt from the Dermis

25. Modeling • 5 identical cells. • 4 levels of activation: off, low, medium, high • All activation and inhibition have equal weight. • Each cell senses the Wnt and Notch ligand expressions of it’s two immediate neighbors.

26. Specifications H1) GT11 > GT21GT14-5 < GT24-5 H2) GT1i = GT2i → for j>i: GT1j ≤ GT2j GT1i < GT2i → for j>i: GT1j < GT2j Notch KO experiments show an increased proliferation as well as increased sensitivity to carcinogenesis, we’ll formulate these as: H3) Notch KO → GT14 > GT24 H4) Notch KO → GT11-5 increase or GT21-5 decrease.

27. Analysis 6561 infinitely visited states were found • All adhere to H1 and H2 (C1 proliferating, C4-5 differentiated) • Not all agree on levels of C2, perhaps indicating it is in transition. • KO of Notch starting from a steady state leads to satisfaction of H3 and H4 as well.

28. Analysis • Single cell analysis in which all external signals are non-deterministic refute the hypothesis that Notch-IC activates transcription of β-Cat:For no starting state do we arrive at an attractor for which GT1 > GT2; no cell could be proliferating. • This means there is another mechanism activating β-Cat.