Sam Pastva,
with Eva Šmijáková, David Šafránek, and Luboš Brim
Gene X @DNA
Gene X @mRNA
Protein X
Gene Y @DNA
Gene Y @mRNA
Protein Y
\(\oplus\)
\(\ominus\)
Gene X
Gene Y
Gene Regulatory Network (GRN)
Gene X
Gene A
Gene B
Gene C
Do we need both A and C, or just one? Does B interact with A and C equally or not?
A
C
B
Boolean Network (BN)
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Synchronous update: all updates happen at the same time.
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Asynchronous update: better captures unknown time-scales of processes.
What "outcomes" are possible in our model? (Attractors - bottom SCCs)
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Assumption: chemistry is fair. Only terminal states/components can be attractors.
Different attractors can represent different biological functionality (phenotype).
Gene X
Gene Y
Input A
Exponentially many "parametrisations" for logical parameters, double exponential for functional parameters.
Benes, N., Brim, L., Pastva, S., Polácek, J., & Šafránek, D. (2019). Formal Analysis of Qualitative Long-Term Behaviour in Parametrised Boolean Networks. ICFEM.
Target
Target can mean slightly different things, depending on biological context, model, etc...
Optionally, source can be specified too: control must work in all source states.
Source
Regardless, target must admit some trap set in which the model can stay indefinitely.
One-step perturbation (e.g. short-term treatment)
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Temporary perturbation (e.g. long-term treatment)
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Permanent perturbation (e.g. mutation)
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Perturbation forces a subset of variables to a constant high/low state.
Target
Source
Baudin, Alexis, et al. "Controlling large Boolean networks with single-step perturbations." Bioinformatics 35.14 (2019): i558-i567.
Target
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×
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Su, Cui, Soumya Paul, and Jun Pang. "Controlling large Boolean networks with temporary and permanent perturbations." FM 2019.
Source
Target
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Su, Cui, Soumya Paul, and Jun Pang. "Controlling large Boolean networks with temporary and permanent perturbations." FM 2019.
Source
To minimize biological "burden" on the system, a perturbation must be as small (simple) as possible.
In practice, minimal perturbations are often small!
Additional steps are needed to asses biological viability of perturbations!
Whether a perturbation works depends on the valuation of logical and functional parameters.
A small perturbation that works only sometimes
may be too unreliable.
A large perturbation that works always
may not be biologically viable.
# parameter valuations where perturbation p achieves control
# parameter valuations that admit the desired target
robustness(p) =
A good perturbation is not only small, but also robust.
Brim, Luboš, et al. "Temporary and permanent control of partially specified Boolean networks." Biosystems 223 (2023): 104795.
Parametrisation+State of a BN is a bit-vector. Sets/relations of these can be encoded via Boolean functions.
Set operations correspond to logical operations on functions (and graph operations on BDDs)
Since Boolean network is described by Boolean functions, computing successors/predecessors (including parameters) is relatively straightforward.
Benes, N., Brim, L., Pastva, S., Polácek, J., & Šafránek, D. (2019). Formal Analysis of Qualitative Long-Term Behaviour in Parametrised Boolean Networks. ICFEM.
A perturbation is treated as another type of parameter:
"Perturbation parameters" need to be differentiated from "uncertain" parameters (i.e. during robustness assessment).
"Naive" encoding uses two "bits" per variable. Here we show that algorithms also exist for a simpler "one bit" encoding.
Brim, Luboš, et al. "Temporary and permanent control of partially specified Boolean networks." Biosystems 223 (2023): 104795.
Target
Strong basin
Target
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Strong basin
Perturbed strong basin
(of the strong basin)
Source
Result: A symbolic relation (a BDD) with all viable perturbations for all viable parametrisations.
We can "search" this relation to find minimal perturbations, robust perturbations, or both.
*up to 10^9 parameter valuations/model
Algorithm runs symbolically for all perturbations and parametrisations.