Kinetic Logic: What it is, why its useful

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Kinetic LogicWhat it is, why its useful

• James S. Cavenaugh

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Example of non-trivial behavior chaotic dynamics

• Consider a simple 3 element system

which has Jacobian matrix

from Thomas, R. and Kaufman, M. Conceptual tools
for the integration of data,C. R. Biologies 325
(2002) 505-514.
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Advantages of qualitative modeling

• Conducive to human understanding of complex
  systems
• Typical of how biologists naturally talk, think
• Example MHC-I expression is upregulated
  following viral infection.
• Avoids potentially obfuscating details
  complexities of random variables
• Useful when more precise data are unavailable
• Avoids illusion of precision in ODEs when data
  are too sparse (similar to too many significant
  figures)
• Useful for time-dependent, qualitative data

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Nice quotations

• Self-evidence it is more informative to be
  imprecise than wrong, qualitative and accurate
  than quantitative and inaccurate.
• From Eric Fimbels website http//lesia.ele.etsmt
  l.ca/web/ericFimbel/research.html
• All models are wrong, some are useful. Box

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What is it?

• Levels of modeling
• verbal ? pictures with arrows ? logical
  description ? stochastic, or ODEs or PDEs
• 2 improvements over Kaufmans Boolean modeling
• Non-Boolean
• allows asynchronous time steps

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Naive kinetic logic

• Boolean but with asynchronous (and unspecified)
  time slots
• Logical variables x, y, z, etc.
• Logical functions X, Y, Z, etc.
• The variables are for the state of the system
  the functions are for the tendency of the system.
• Example Gene product x can be present or absent,
  and gene X can be turned on or off.

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Naïve kinetic logic, cont.

• Example
• x y z
• Q What does the state (0,0,0) imply?
• A X 1 (z 0, so X is ON),
• Y 0 (x 0, so Y is OFF),
• Z 1 (y 0, so Z is ON).

• In this circuit,
• product x is made iff z is absent y is made iff
  x is present z is made iff y is absent i.e.
• X NOT z
• Y x
• Z NOT y

-

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In other words, the state vector (0,0,0) implies
the evolution vector (image vector in Thomas
terminology) (1,0,1).
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Naïve kinetic logic, cont.

• The resulting state table
• x y z X Y Z
• 0 0 0 1 0 1
• 0 0 1 0 0 1
• 0 1 0 1 0 0
• 0 1 1 0 0 0
• 1 0 0 1 1 1
• 1 0 1 0 1 1
• 1 1 0 1 1 0
• 1 1 1 0 1 0

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Naïve kinetic logic, cont.

• (000/101) is equivalent to (000).
• Logical equivalent of time derivative dx/dt is X
  x.
• This is a positive feedback loop!
• choice of 2 stable states (multistationarity)

• The resulting state table
• x y z X Y Z
• 0 0 0 1 0 1
• 0 0 1 0 0 1
• 0 1- 0 1 0 0
• 0 1- 1- 0 0 0
• 1 0 0 1 1 1
• 1- 0 1 0 1 1
• 1 1 0 1 1 0
• 1- 1 1- 0 1 0

For proper parameter values any positive feedback
loop can generate a choice between 2 stable
states.
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Naïve kinetic logic, cont.

• Positive feedback circuits
• have an even parity for negative interactions
• are involved in multistationarity
• in biology, as differentiation and epigenetic
  attributes
• Negative feedback circuits
• homeostasis
• damped oscillations
• Ex. NF-kB (Hoffman et al. Science (2002))

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Naïve kinetic logic, cont.

• Which trajectory will be actually followed?
• Ex. (000) ? (100) if tx lt tz
• ? (001) if tx gt tz
• Stochastic aspect enters with time delays,
  distributions
• Some discussion exists for induction going from
  data to models.

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Generalized kinetic logic

• Allows gt 2 values. If a variable has n distinct
  actions, we provide it with n thresholds and
  treat it as an (n1)-level variable.
• Logical parameters give closer agreement to
  differential (ODE) analyses.
• Depending on the parameters values, different
  qualitative behaviors arise.
• Example X Y 1x 2y

Without logical parameters, this circuit is
as shown above. x 0,1, y 0,1,2 since x
only acts on y but y acts on both x and itself.
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x y


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With logical parameters
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Logical parameters

• K1, K2, and K3 are real numbers.
• K1, K2, and K3 are integers. The dx and dy
  operators discretize according to the scale of
  variable x or y, respectively. x can have values
  (0, 1) y can have values (0, 1, 2).
• Consider Y. The expression in parentheses can
  only have the following real and discretized
  values
• 0 (if 1x 2y 0) 0
• K2 (if 1x 1, 2y 0) K2
• K3 (if 1x 0, 2y 1) K3
• K2 K3 (if 1x 2y 1) K23 (K23 ? K2 K23 ?
  K3)
• Why the discretization operator? Were not really
  interested in the real values, but rather in
  their location in the scale of the variable
  considered.
• Whats the point? Different possible
  trajectories can arise from different values of
  logical parameters.

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Generalized kinetic logic

• Upshot is that logical parameters give better
  agreement with differential description.
• One can then find all steady states of the
  system, write in matrix form, analyze for steady
  states, etc.
• Reference René Thomas, e.g. J. Theor. Biol. 153,
  1-23 (1991).

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Summary of kinetic logic

• Any regulatory net can be decomposed into a
  well-defined set of simple feedback loops that
  usually interact with each other.
• A feedback loop is either positive or negative,
  depending on the even vs. odd parity of the
  number of negative control units.
• Positive loops each element exerts a positive
  influence on its own later expression, and v.v.
• Negative feedback loops generate homeostasis with
  or without periodicity.
• Positive feedback loops (i.e., direct or indirect
  autocatalysis) generate multiple steady states m
  independent loops can generate up to 3m steady
  states, of which 2m can be stable (i.e., m binary
  choices).

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Areas of application

• Cellular differentiation
• flower formation T helper cells lytic vs.
  lysogenic choice in bacteriophage embryogenesis
• Homeostasis
• Simplified analysis of more general physical
  systems
• Areas of dynamic but qualitative variables (e.g.,
  psychology)

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