Dynamic Modeling Of Biological Systems

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.

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

We discussed over the course an attempt to (mostly) statically analyze biological pathways. Our “cartoon” drawings were a sort of model; emphasizing who interacts and operates on whom, who is important and who isn’t. Also, these were of course abstractions and generalizations, which entail a qualitive understanding. These are models. We would like to create mathematical models, and test

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

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

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.

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

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?

Qualitative Networks

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.

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

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 A B C w = 1 w = -2 w = Attractors: states visited infinitely many times We can represent a state at a given time as a triplet: (101): A=1, B=0, C=1

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.

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

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: Qualitive Networks: Calculating target i and symmetrically:

So we get: The second line entails a hidden assumption, that if c i gets no activation, it’s activation is not modeled. Qualitive Networks: Calculating target i

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.

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.

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.

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.

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

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.

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

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.

Specifications H1) GT1 1 > GT2 1 GT1 4-5 < GT2 4-5 H2) GT1 i = GT2 i → for j>i: GT1 j ≤ GT2 j GT1 i i: GT1 j < GT2 j Notch KO experiments show an increased proliferation as well as increased sensitivity to carcinogenesis, we’ll formulate these as: H3) Notch KO → GT1 4 > GT2 4 H4) Notch KO → GT1 1-5 increase or GT2 1-5 decrease.

Analysis 6561 infinitely visited states were found All adhere to H1 and H2 (C 1 proliferating, C 4-5 differentiated) Not all agree on levels of C 2, perhaps indicating it is in transition. KO of Notch starting from a steady state leads to satisfaction of H3 and H4 as well.

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.