Synthesizing Stochasticity in Biochemical Systems In partial fulfillment of the requirements for a...

Synthesizing Stochasticity in Synthesizing Stochasticity in Biochemical SystemsBiochemical Systems

In partial fulfillment of the requirements for a master of electrical engineering degree

Brian FettMarc Riedel

Computation and BiologyComputation and Biology

• Cell Chemistry modeled with reaction sets• Cell assumed to have a starting set of concentrations

– Given the size of a cell, this implies quantities• Modeling can be done in two ways

– ODEs – quick, deterministic, approximate– Gillespie – Monte Carlo, Markov chain, exact

Design ScenarioDesign Scenario

• Anderson/Arkin• Engineering Bacteria that will seek/destroy cancer• Multi-stage system

– Inactive, detect cancer, protection from host– Active, attack cancer, vulnerable

• The inactive form will be carried by blood stream– What if there is a cancer downstream?

Two Cancers DeterministicTwo Cancers Deterministic

Blood flow

Two Cancers StochasticTwo Cancers Stochastic

Blood flow

Discrete KineticsDiscrete Kinetics

“States”

A B C

4 7 5

2 6 8

22 0 997

S1

S2

S3

A reaction transforms one state into another:

21 1SS

Re.g.,

BCA

ACB

CBA

2

3

2 3

2

k1

k2

k3

R1

R2

R3

Track discrete (i.e., integer) quantities of molecular types.

S1 = [5, 5, 5]

S2 = [4, 7, 4]R1 R2 R3

S3 = [2, 6, 7]

S4 = [1, 8, 6]

Discrete KineticsDiscrete Kinetics

State [A, B, C]

Stochastic KineticsStochastic Kinetics

The probability that a given reaction is the next to fire is proportional to:

• Its rate constant (i.e., its ki).

• The quantities of its reactants.BCA

ACB

CBA

2

3

2 3

2

k1

k2

k3

R1

R2

R3

See D. Gillespie, “Stochastic Chemical Kinetics”, 2006.


Choose the next reaction according to:

jj

iiR

)Pr(

Ri kXnXn 2211

let

2

2

1

1

n

X

n

Xki

For each reaction


[1, 1, 9] [1, 5, 4] [4, 4, 0] [4, 0, 5]

[2, 2, 6] [2, 6, 1] [5, 1, 2]

p1p2

p3

p4p5

p6p7 p8 p9

[3, 3, 3]start

[1, 1, 9] [1, 5, 4] [4, 4, 0] [4, 0, 5]

[2, 2, 6] [2, 6, 1] [5, 1, 2]

p1p2

p3

p4p5

p6p7 p8 p9

[3, 3, 3]start

[4, 0, 5]

Probabilistic Lattice

Designing SystemsDesigning Systems

Stochastic and Deterministic

Modular SchemeModular Scheme

Module

inputs outputs

QuantitiesQuantities/

Probabilities

Modular SchemeModular Scheme

Module Module

Module

Module

DeterministicDeterministic

Module

inputs outputs

Quantities Quantities

Functional ModulesFunctional Modules

• Subtraction

Subtraction

y

xy


• Subtraction• Fan-out

Fan-outx

y2

y1

yn


• Subtraction• Fan-out• Linear Scaling

Linearx y


• Subtraction• Fan-out• Linear Scaling• Multiplication Multiply

z

xy


• Subtraction• Fan-out• Linear Scaling• Multiplication• Exponentiation

Expx y


• Subtraction• Fan-out• Linear Scaling• Multiplication• Exponentiation• Logarithmic

Logx y


• Subtraction• Fan-out• Linear Scaling• Multiplication• Exponentiation• Logarithmic• Power

Power

p

xy


• Subtraction• Fan-out• Linear Scaling• Multiplication• Exponentiation• Logarithmic• Power• Isolation

Isolationy

Composing ModulesComposing Modules

• Leading module must complete– Modules starting prematurely yield poor results– Rate separation between modules– Rate separation multiplies as modules are chained

• Module Locking– All modules can operate at similar pace– Subsequent modules locked– Key is generated when a module finishes– Key is required by ‘leading’ reaction of module

Module LockingModule Locking

Mult

Mult

Linear

Locking within a ModuleLocking within a Module

• Multi-reaction modules often loop• Most loops have 4 stages

– Initiate– Calculate– Terminate– Reset

• Keys must be destroyed at end of stage• Initiate creates Loop type

– very much like a key– Terminate destroys


Log

Initiate Calculate

Reset Terminate


Log

Initially 4 2 = Log2(4)

Stochastic ModuleStochastic Module

StochasticModule

Prob. 0.2 Prob. 0.8 Choose between multiple outcomes

Functions:• Choice is made in one reaction• Choice is quickly reinforced• Outputs are produced

Features:• PDF set by quantities of types• Any number of outcomes• Arbitrarily low error

Basic DesignBasic Design

• Initialize – Make the choice• Reinforce – Push forward with choice• Stabilize – Remove possibility of ‘contamination’

– Prevent subsequent ‘choices’• Purify – Remove contamination• Work – Do what was ‘chosen’

Initializing Reactions

Reinforcing Reactions

Stabilizing

Purifying

Working Reactions

where

Inside the Stochastic ModuleInside the Stochastic Module

ik

i dei i :

ik

ii dedi i 2:'

ik

ji dedij i''

:

iik

ii odfdi i ''''

:

''''''''''ijijiii kkkkk

'''

: ikji ddij

Initializing Reactions

Inside the Stochastic ModuleInside the Stochastic Module

For all i, to obtain di with probability pi, select E1, E2,…, En

according to:

j jj

iii kE

kEp

Use as appropriate in working reactions:

(where Ei is quantity of ei)

iik

ii odfdi i ''''

:

ik

i dei i :

Error AnalysisError Analysis

Let

for three outcomes (i.e., i, j = 1,2,3).

Require

Performed 100,000 trials of Monte Carlo.

2'''''''''' ,,1 ijijiii kkkkk

''''''''''ijijiii kkkkk

Locked Stochastic ModuleLocked Stochastic Module

• Lock Initializing reactions– All require the same key– Key generated by keysmith– Keysmith generated slowly

• Replace Purifying reactions– Destroy keysmith in presence of d– Preventative reactions instead– Linear growth rather than quadratic

Locked Stochastic ModuleLocked Stochastic Module

An ounce of prevention…

…worth a pound of cure.

Functional ProbabilitiesFunctional Probabilities

Make choices based on inputs

StochasticModule

Prob. 0.2 Prob. 0.8

DeterministicModule

Using IP/LPUsing IP/LP

• IP and LP solvers require– Set of constraints (inequalities)– An expression

• Solve for set of inputs– Meets constraints– Minimizes or maximizes expression

• Rewrite our problems in the form required

LP/IP TricksLP/IP Tricks

• Replace equality statements– Rounding errors break equality– Add error term– Minimize the error term

• Require a at least one molecule– Prevents solutions from devolving to zero

Sample the Input SpaceSample the Input Space

• Number/location arbitrary• More points

– More accurate fit– More time calculating

• User defines desired IO– Table of points– Set of inputs– Desired output PDF at

inputs

X1

X2

Solve each pointSolve each point

• Space is quantities of e• Solution is a ray• Solve minimal magnitude• Outside unit hyper-sphere

e2

e3

e1

Bringing Points TogetherBringing Points Together

• Pick a function– Solve for coefficients

• Scale point solutions• Minimize difference• Minimize coefficients

x

e

IP/LP ComputationIP/LP Computation

• Value space is whole number sets, requires IP• IP is NP-hard, LP is in P• Scales solutions are also solutions• Tricks

– Solve points with LP, bring together with IP– Solve coefficients with LP, scale solution

• Use IP to scale optimally (weigh scaling, error)• Scale by hand

Past Future workPast Future work

• 2 year Hiatus• Jiang / Riedel DSP• Builds on this work

– Fan-out– Addition– Linear Scaling

• Adds Clock and Delay Module

Asynchronous ClockAsynchronous Clock

• Clock– 3 phase– Asynchronous– Uses a locking mechanism

• Delay– Each delay locks clock– Keeps clock in a phase– Allows clock to proceed when phase completes

Clock Delay MechanismClock Delay Mechanism

Delay ElementDelay Element

• Extension of system– Shows strength of modular design

• Elegant asynchronous clock design• Possible improvement for intra-module locking

– Previous design – modified inter-module lock– Likely need more than 3 phases

• 4 phases – two calculating, two resetting• 6 phases – three of each

– Local/global clock

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