Model Building and Model Checking for Biochemical Processes

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Model Building and Model Checking for Biochemical
Processes

• Presented by Danny C. Cheng

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Topics to be covered

• Brief background
• S-Systems
• XS-Systems
• Definitions 1-5
• Temporal Logic
• Software Solutions
• Conclusions

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Graphical Representation
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Graphical Representation
The reaction between X1 and X2 requires coenzyme
X3 which is converted to X4
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S-Systems

• Dependent Variables Xi(t), i1,,n, 0 5 t.
• System is described in terms of the temporal
  changes in dependent variables
• E.g., Instantaneous product formation in response
  to changes in the exogenous substrate, inhibitor
  or enzyme concentration
• Kinetic Laws Relate a reaction rate to
  concentrations.
• Reaction Rate Instantaneous temporal rate of
  change in concentration of substrate or product.

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Systems of Differential Equations

• dXi/dt (instantaneous) rate of change in Xi at
  time t Function of substrate concentrations,
  enzymes, factors and products
• dXi/dt f(S1, S2, , E1, E2, , F1, F2,, P1,
  P2,)
• E.g. Michaelis-Menten for substrate S product
  P
• dS/dt - Vmax S/(KM S)
• dP/dt Vmax S/(KM S)

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General Form

• dXi/dt Vi(X1, X2, , Xn) Vi-(X1, X2, , Xn)
• Where Vi() term represents production (or
  accumulation) rate of a particular metabolite and
  Vi-() represent s depletion rate of the same
  metabolite.
• Generalizing to n dependent variables and m
  independent variables, we have
• dXi/dt
• Vi(X1, X2, , Xn, Xn1, Xn2, , Xnm)
• Vi-(X1, X2, , Xn, Xn1, Xn2, , Xnm)
• These n differential equations are called the
  systems equations, or the system description or
  Kirchhoffs node equation

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XS-Systems
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Currently

• Researchers will often model the underlying
  biological and biochemical mechanisms with sets
  of relatively simple differential algebraic
  equations (DAE), each one representing a
  reversible chemical reaction, a degradation
  process, a synthesis process or a reaction
  modulated by an enzyme or a co-enzyme.

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Problem

• As the model complexity of the biological systems
  increases, the sets of numerical traces become
  increasingly difficult to interpret and the
  traditional biological reasoning process fails to
  scale beyond a handful of genes and relatively
  small and coarsely-modeled pathways.

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Thus

• To cope with this problem, an approach was given
  that first summarizes the numerical traces into
  an automaton with distinguishable biological
  states and a deterministic set of rules to
  transition from state-to-state and checks the
  automaton model for its ability to satisfy
  various temporal logic statements

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What is a XS-Systems?

• An XS-system is simply a list of expressions
  describing the rate of change of a given quantity
  in a model (say the concentration of a compound),
  plus a set of equations describing some
  constraints on the relationships among some of
  the parameters characterizing the model. Each of
  the expressions describing a rate has a very
  simple form as well it is simply a difference
  between two algebraic power-products (or
  monomials) one representing synthesis and the
  other, dissociation.

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Definition 1
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Definition 2
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Definition 3
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Collapsing States
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Therefore

• As a consequence, once this automaton has been
  constructed, it provides many different avenues
  for
• Performing a qualitative analysis on the temporal
  evolution of the S-system, and
• Studying in parallel (within a single structure)
  multiple evolutions and experiments differing in
  rate constants and kinetic orders.

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Temporal Logic

• Temporal Logic (TL) 8,10 has been studied in
  depth in the context of systems whose behavior
  change in time, for instance, computer hardware,
  network protocols and engineering systems.
• Fundamental to a temporal logic is the notion
  that time-dependent terms from natural language,
  such as eventually and always, can be given a
  precise meaning (semantics) in terms of the
  abstract behavior of a system under discourse.

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English to TL

• The translation from English to TL is rather
  straightforward. Simple conjunctions (ands),
  disjunctions (ors) and negations (nots) can
  be expressed directly. The corresponding
  prepositional logic is then augmented with
  temporal modes Always and Eventually.

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Example

• Now, suppose we wish to determine if (1) our
  system reaches a steady state and (2) the level
  of GTP is less than k after a certain instant.
• steady_state and Eventually(Always(GTP lt k)).

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Repressilator using MATLAB
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Simpathica
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Conclusion

• Reactions Models
• Hybrid Systems
• Spatial Models
• State Space
• Communication
• Hierarchical Models