Finding Instability in Biological Models - PowerPoint Presentation

Slide1

Finding Instability in Biological Models

Byron Cook1,2, Jasmin Fisher1,3, Benjamin A. Hall1, Samin Ishtiaq1, Garvit Juniwal4, Nir Piterman5

1Microsoft Research 2University College London3University of Cambridge 4UC Berkeley5University of Leicester

1

CAV 2014Slide2

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.2Slide3

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.3Slide4

4

Bio Model Analyzer (BMA)http://biomodelanalyzer.research.microsoft.com/[CAV’12] Visual Tool for Modeling and Analyzing Biological Networks.

Benque et. al.Slide5

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.5Slide6

Outline

Definitions – Qualitative Networks and stabilitySummary of previous algorithm and its issuesNew divide/conquer algorithmEvaluation and conclusion6Slide7

Qualitative Networks (QNs)

Variablesfinite domain – [0, N]Dependencies Target functionsZ moves towards F(X,Y) in increments/decrements of 1Typically monotonicSynchronous updates

Unique next stateXZY:=

7Slide8

Typical Size of a QN

Variables (|V|) -- 50Domain size (N+1) – 4Average in-degree – 3State space – 4508Slide9

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 initialA 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 9Slide10

Stabilization Behavior in QNs

(2/2)

Stable

Unstable

Cycle (length >=2)

Multiple self-loops

S

tate

Cycle

Self-loop state

10

Bifurcation

(BF)

Cyclic Instability

(CI)Slide11

Proving Stability (hidden slide)

Compute the set recurring statesLet ∑ be the set of states that can be reached after i transition steps.All state are initial  ∑0

∑1∑2∑-1 = ∑Stabilizing Network: ∑ is singleton 11[VMCAI’11 Proving Stabilization of Biological Systems. Cook et. al.]Slide12

Proving Stability

Goal: compute the set of recurring statesIdea: over-approximate via rectangular abstractionIf over-approximation has single state - STABLE

Else:Check for multiple self-loop states by encoding to a SAT query Check for cycles of increasing length starting at 2 (BMC)If no cycles found for a length > diameter – STABLE (using a naïve over-approximation of diameter) 12[VMCAI’11 Proving Stabilization of Biological Systems. Cook et. al.]Slide13

Over-Approximation Technique

Keep track of an interval per variableStart with the set of all states

For a variable, given the current intervals of its inputs and the target function, is it possible to tighten its interval?Order of updating variables doesn’t matterPick arbitrarily from a work list. 13XY 

X

Y

 Slide14

Pitfalls

14

0

1 2 3

0

1

2

A

B

Trivially Stable (TS)

Bifurcation (BF)

Cyclic Instability (CI)

Non-trivially Stable (NTS)

Computationally prohibitive to

distinguish these two cases

SAT !!Slide15

Main Contribution:Finding

InstabilityNew 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. 15Slide16

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’The previous interval update technique is one way to implement ShrinkThe old algorithm can be thought of as a single application of Shrink, now we use it within a recursive procedure.16

0

1 2 3

0

1

2

A

B

ρ

ρ

'Slide17

Cut

A

cut of a region ρ is a pair (ρ1, ρ2) of regions s.t. ρ1ρ2 = 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. 

17

cut

frontier

(two-way)

Cut:

region

 cut

× frontier

.

ρ

ρ

1

ρ

2Slide18

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

(

ρ

1

)

if

res

1

is

cycle

then

return

res

1

res

2



FindInstability

(

ρ

2

)

if

res2 is cycle then return res2 return (cycle) FindCycleAcrossCut(ρ1, ρ2)18Slide19

Concrete Implementation of

ShrinkReduce region while retaining all contained cycles; update intervalsIssue: Due to cuts, might get outgoing transitions causing intervals to growFix: Change target functions by

limiting with a min/max value.TB’ = max(min(TB, 1), 0)19

B

0

1 2

3Slide20

Concrete Implementation of

CutSplit 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.20

cutSlide21

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 becauseShrink is effective in reducing the search space.One-way cuts are prevalent due to simple and monotonic target functions. 21

CYCLE!!Slide22

`

Examples

22

o

ne-way frontier

Cyclic Instability (CI)

Non-trivially Stable (NTS)Slide23

Benchmarks and Evaluation

23T.O. = 15 minuteshttp

://www.cs.le.ac.uk/people/npiterman/publications/2014/instability/ModelVariablesDependenciesN+1BMA(old)(ms)BMA(new)(ms)

Speed

up

Dicty

Population

35

71

2

60066

2514

23.6x

Firing Neuron

21

21

2

218

458

0.5x

L

Model

25

105

4

43934

9865

4.5x

Leukemia

57

92

3

4497

446

10.1x

SSkin

1D

40

46

5

T.O.

132350

>

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

>

3600xSlide24

Conclusion

Shrink and Cut work effectively for systems under consideration.Running time down from hours to millisecondsThe added capabilities make the tool clinically relevant for industrial biomedicine.Found stability and instability results for previously un-tractable and biologically important models:bacterial chemotaxisdictyostelium

discoideum24Slide25

Thank You!

25Slide26

Future Work (hidden slide)

Repair and synthesis of unstable QNs26

ModelVariablesDepend--enciesN+1ResultBMA(old)(ms)BMA(new)(ms)Speed up

Dicty

Population

35

71

2

CI(5)

60066

2514

23.6x

Firing Neuron

21

21

2

CI(6)

218

458

0.5x

SSkin

2D 2 layers

40

64

5

CI(18)

T.O.

2706

>322.6x

Lambda

phage

8

13

2

NTS

3113

197

15.8x

E. Coli

chemotaxis

9

10

5

NTS

T.O.

250

>3600x