CAV 2014
Finding Instability in
Biological Models
Byron Cook1,2, Jasmin Fisher1,3, Benjamin A. Hall1,
Samin Ishtiaq1, Garvit Juniwal4, Nir Piterman5
1Microsoft
Research
3University
of Cambridge
5University
of Leicester
2University
4UC
College London
Berkeley
1
Biological Signaling Networks
• Network of chemicals (usually proteins) regulating each others’
concentration, evolving the overall state.
• Qualitative Networks offer a good level of abstraction for
modeling and analysis. [BMC Syst Biol ’07] Schaub et. al.
• Have been applied to blood cell differentiation, skin homeostasis
and cancer cell development.
2
Homeostasis
• Homeostasis (or stability) is a natural requirement for
most of these systems. It represents the ability to stay at
robust equilibrium.
• Model of a naturally occurring phenomenon  stability
desired.
 starting from every state, the same stable state reached.
• Required sanity check during development stage.
• Central problem: Enabling biologists to be able to quickly
check stability and examine instability.
3
Bio Model Analyzer (BMA)
http://biomodelanalyzer.research.microsoft.com/
[CAV’12] Visual Tool for Modeling and Analyzing Biological Networks.
Benque et. al.
4
Contribution
• Previous: BMA’s stability algorithm from [VMCAI’11 Cook
et. al.]
• Problem: takes too long (~ 2 hours) for certain classes of
models.
• This work: divide/conquer based algorithm to tackle hard
cases.
• Benefit: Up to 3 orders of magnitude of speed up. Can solve
all existing models in matter of seconds making BMA truly
interactive.
5
Outline
• Definitions – Qualitative Networks and stability
• Summary of previous algorithm and its issues
• New divide/conquer algorithm
• Evaluation and conclusion
6
Qualitative Networks (QNs)
• Variables
X, Y, Z ∈ 0, 𝑁
𝐹: 0, 𝑁 × 0, 𝑁 → [0, 𝑁]
• finite domain – [0, N]
• Dependencies
• Target functions
• Z moves towards F(X,Y) in
increments/decrements of 1
• Typically monotonic
• Synchronous updates
• Unique next state
X
Y
Z
Z′:=
𝑍 + 1 𝑖𝑓 𝐹 𝑋, 𝑌 > 𝑍
𝑍
𝑖𝑓 𝐹 𝑋, 𝑌 = 𝑍
𝑍 − 1 𝑖𝑓 𝐹 𝑋, 𝑌 < 𝑍
7
Typical Size of a QN
•
•
•
•
Variables (|V|) -- 50
Domain size (N+1) – 4
Average in-degree – 3
State space – 450
8
Stabilization Behavior in QNs (1/2)
For a QN with variables V
• State Space ∑ = {0, …, N}|V|
• Transition Function δ: ∑→∑ (defined via target functions)
• All states are initial
• A state s said to be recurring if it is possible to reach from s
to itself with finite applications of δ.
• Unique recurring state  Stabilizing QN
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Stabilization Behavior in QNs (2/2)
State
Stable
Self-loop state
Cycle
Unstable
Cycle (length >=2)
Cyclic Instability
(CI)
Multiple self-loops
Bifurcation
(BF)
10
Proving Stability
[VMCAI’11 Proving Stabilization of Biological Systems. Cook et. al.]
• Goal: compute the set of recurring states
• Idea: over-approximate via rectangular abstraction
• If over-approximation has single state - STABLE
• Else:
1) Check for multiple self-loop states by encoding to a SAT query
2) Check for cycles of increasing length starting at 2 (BMC)
3) If no cycles found for a length > diameter – STABLE
(using a naïve over-approximation of diameter)
12
Over-Approximation Technique
• Keep track of an interval per variable
• Start with the set of all states 0, 𝑁 × ⋯ × 0, 𝑁
• For a variable, given the current intervals of its inputs and the
target function, is it possible to tighten its interval?
X
Y
[0,2]
[0,9]
X
Y
[0,2]
[1,3]
𝐹(𝑌) = 𝑋 + 1
• Order of updating variables doesn’t matter
 Pick arbitrarily from a work list.
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Pitfalls
A
0
1
2
B 0
1
2
3
Trivially Stable (TS)
Bifurcation (BF)
SAT !!
Cyclic Instability (CI)
Computationally prohibitive to
distinguish these two cases
Non-trivially Stable (NTS)
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Main Contribution:
Finding Instability
• New instability finding algorithm to distinguish between
Cyclic Instability (CI) and Non-Trivial Stability (NTS)
• Rectangular abstraction. A Cartesian product of intervals is
called a region.
For example, [1, 4] × [0, 2]
• Uses two new generic procedures SHRINK and CUT.
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SHRINK
• SHRINK: region  region.
Given a region ρ, it returns another region ρ’ contained in
ρ s.t. all cycles and self-loops in ρ are within ρ’
A
0
1
2
B 0
1
2
3
ρ
ρ'
• The previous interval update technique is one way to
implement SHRINK
• The old algorithm can be thought of as a single application
of SHRINK, now we use it within a recursive procedure.
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CUT
• A cut of a region ρ is a pair (ρ1, ρ2) of regions s.t. ρ1∪ρ2 = ρ.
• A frontier of a cut (ρ1, ρ2) is a pair of sets of states around the
cut.
• A frontier can be two/one/zero-way.
ρ
frontier (two-way)
ρ2
ρ1
cut
• CUT: region  cut × frontier.
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FINDINSTABILITY
// distinguish between CI and NTS
// returns either stable or a cycle
FINDINSTABILITY(ρ):
ρ  SHRINK(ρ)
if ρ contains single state then
return stable
else
(ρ1, ρ2)  CUT(ρ)
res1  FINDINSTABILITY(ρ1)
if res1 is cycle then return res1
res2  FINDINSTABILITY(ρ2)
if res2 is cycle then return res2
return (cycle) FINDCYCLEACROSSCUT(ρ1, ρ2)
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Concrete Implementation of SHRINK
• Reduce region while retaining all contained cycles;
update intervals
• Issue: Due to cuts, might get outgoing transitions causing
intervals to grow
• Fix: Change target functions by limiting with a min/max
value.
• TB’ = max(min(TB, 1), 0)
B
0
1
2
3
19
Concrete Implementation of CUT
• Split the interval of one of the |V| variables.
• Variables change by at most 1  the slice around the split
point is a frontier.
• Enumerate over N*|V| choices until a zero/one-way frontier
is found because no loops can exist across such frontiers.
• Zero/one-way property can be checked via a SAT query.
cut
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FINDCYCLEACROSSCUT
• If the frontier is zero/one-way
Return none
• Else enumerate over states in the frontier and run
simulations of the transition function.
Stop when a cycle of length >= 2 is found.
• Works well because
• SHRINK is effective in reducing the search space.
• One-way cuts are prevalent due to simple and monotonic target
functions.
CYCLE!!
21
Examples
one-way frontier
`
Non-trivially
Stable (NTS)
Cyclic
Instability (CI)
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Benchmarks and Evaluation
Model
Variables
Depend N+1 BMA(old)
encies
(ms)
BMA(new)
(ms)
Speed up
Dicty Population
35
71
2
60066
2514
23.6x
Firing Neuron
21
25
21
105
2
4
218
43934
458
9865
0.5x
4.5x
SSkin 1D
57
40
92
46
3
5
4497
T.O.
446
132350
10.1x
>6.8x
SSkin 2D 2 layers
40
64
5
T.O.
2706
>322.6x
Ion Channel
10
7
2
499
173
2.9x
Lambda Phage
8
13
2
3113
197
15.8x
Resting Neuron
21
28
2
T.O.
244949
>3.7x
E. Coli Chemotaxis
9
10
5
T.O.
250
>3600x
L Model
Leukemia
T.O. = 15 minutes
http://www.cs.le.ac.uk/people/npiterman/publications/2014/instability/
23
Conclusion
• SHRINK and CUT work effectively for systems under
consideration.
• Running time down from hours to milliseconds
• The added capabilities make the tool clinically relevant for
industrial biomedicine.
• Found stability and instability results for previously untractable and biologically important models:
• bacterial chemotaxis
• dictyostelium discoideum
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Thank You!
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