Dynamic Modeling Of Biological Systems
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Transcript of Dynamic Modeling Of Biological Systems
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Dynamic Modeling Of Biological Systems
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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.
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Why Model?
• Knowledge integration
• Hypothesis testing
• Prediction of response
• Discovery of fundamental processes
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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
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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
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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.
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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
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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?
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Qualitative Networks
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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.
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Boolean Networks• For each state i at time t we have a function
for the value of t in the next round.
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Boolean Networks: Example So: A=0 → B=0
A=1 → B=1B=0 → C=0B=1 → C=1
B,C =0,0 → A= A-1
B,C =0,1 → A= 0B,C =1,0 → A= 1B,C =1,1 → A= 0
A B
C
w = 1
w = 1
w= 1w = -2
w = -0.5
w = -0.5
101
100 110 111 011 001 000
010
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
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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.
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Qualitive Networks
• The transition function in a Qualitive Network defines for every component ci a targeti function, of {0…N}|C|→{0…N}.
• We allow changing the expression of a component by maximum of 1 each turn:
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• 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 targeti
0
0
ji
ji
aji
ajji
i a
ca
act
and symmetrically:
0
0
ji
ji
aji
ajji
i a
ca
inh
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• So we get:
• The second line entails a hidden assumption, that if ci gets no activation, it’s activation is not modeled.
Qualitive Networks: Calculating targeti
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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.
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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.
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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.
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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.
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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
?
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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.
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We’ll take 5 cells to represent the layers of the skin
High Wntfrom the Dermis
Low Wntfrom the upper layers of skin
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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.
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Specifications
H1) GT11 > GT21
GT14-5 < 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.
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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.
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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.