local-to-global algorithms in biology - amazon s3

Local-to-Global Algorithms in Biology:Drosophila and Beyond (maybe ...)

MIT Synthetic Biology Working Group

Dan Yamins

2007.12.05

spatial multi-agent systems

local information and processing

globally defined tasks

a space with agents embedded in the space

engineering:

McLurkin iRobot SwarmSensor Networks

Butera’s “paintable computer” concept

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Saul Griffith’s self-folding structures

biology:

drosophila embryo

• What agent resource capacities are required to solve a given task?

• What does thinking of natural spatial multi-agent systems qua computers tell us scientifically?

• Can we then turn around this knowledge to build a better bug?

outline

1. Drosophila background

II. Multi-agent systems Model of Drosophila

III. An information bound

IV. The Radius/State Tradeoff

V. Designing Local Rules

The first four sections study a known system, showing how multi-agent systems reasoning can help us understand features of local-to-global systems design.

The fifth section takes this knowledge, and applies it to develop the beginnings of a protocol for designing novel systems.

Drosophila Background

(Actually ... one stage in early Drosophila, utterly simplified)

Drosophila

Drosophila melanogaster embryoabout 100 minutes post-fertilization

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PROTEIN mRNA

Let’s think of the cell as a ....

Drosophila

.... bag of biomolecules

HIghlighted here are 13 substances that play are involved in the stripes in the picture on the previous slide

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Cubitus Interruptus (mRNA)Cubitus Interruptus repressor HedgeHog Cubitus Interruptus (protein)

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Considering cells along the A-P axis:

... there is a periodic pattern in the concentration of each of various substances depicted on the previous slide:

The concentration profile pattern at the molecular levelcorresponds the observed stripe pattern.

Drosophila

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HH⇒PROTEIN mRNA

Positive Induction

Negative Inhibition

Basal Production

The bag of biomolecules ... ... is actually a Gene

Regulatory Network

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Some components of the GeneRegulatory Network in one cell ...

... influence those in

neighboring cells.

. . .. . .

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(Si

1, Si2, . . . , S

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1 , Si−12 , . . . , Si−1

N , Si+11 , Si+1

2 , . . . Si+1N

)

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It’s a spatially coupled Gene Regulatory Network.

Such a network can be modeled as a coupled ODE:

And the ODE will (a la Garrett Odell) have the pattern as a stable nondegenerate steady state:

(a sketchy multi-agent model of drosophila)

a multi-agent model of drosophila

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... a simple “1-D” organism, each cell of which has some internal state (0,1,2,3, etc...)

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multi-agent model

2 0 1 11 0 1 2 13

2 0 1 31 0 1 2 13

action of local rule, on radius-2 ball, changing 1 --> 3

local rules by which the agent integrates information about states of nearby agents

and takes a differentiation action

any radius-1 (nearest neighbor) local ruleis allowed.

0 0 0 11 0 0 1 00 0 0

0 0 0 11 0 00

0 0 01 size 4 instance

size 8 instance

size 12 instance

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multi-agent model

Agent with unstructured internal state.

Structured “multi-register” internal states.

Multislot dynamics

Add “slots.”

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!→

multi-agent model

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multi-agent model

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⇒

multi-agent model

the gene regulatory network as a nearest-neighbor local rule

the “brains” of the cellular-agent

An informational

bound

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Answer: No. Because the with a radius 1 rule, 000 would have to be a fixed state.

0 0 01 0 0 0 . . . .

Local Checkability

Consider the repeat pattern:

Can this pattern be solved robustly with a nearest-neighbor rule?

A function Θ : Br → {0, 1} is a local check scheme for a pattern T if∧

i≤|X|

(Θ(Br(i, X)) = 1) ⇒ X ∈ T

0 0 0 11 0 . . . .0 0

local checkability; multi-agent reasoning about local equilibria

This means: if all cells think “criterion Theta” is true ... then actually the organismal pattern is correct

All repeated patterns are locally checkable.

1000 has a radius-2 check scheme.

Any pattern thus has a minimal check radius, a lower bound on how far local rules have to see to form the pattern robustly.

ENgrailed

Local check radius of this pattern is = 2, and in fact to actually make a rule radius 5/2 is required.

But the coupled gene network rules are nearest neighbor (i.e. r = 1).

Hey. Wait.

dSij

dt= fj

(Si

1, Si2, . . . , S

iN , Si−1

1 , Si−12 , . . . , Si−1

N , Si+11 , Si+1

2 , . . . Si+1N

)

Put another way, no equation of the form:

can have a nondegenerate stable steady state of the form * if engrailed is decoupled.

*

The system is “trying” to make a 4-coordinate....

local checkability; multi-agent reasoning about local equilibria

The Radius/State Tradeoff


0 0000 0 001

0 0000 0 001

0 0000 000

1 0000 0 000

1

A generic “cut-and-shift” procedure for implementing the Radius -> tradeoff:

High State, Low Radius High Radius, Low State

Θ

#slots = to go from radius r to radius k. !(2r + 1)/(2k + 1)"

have r = 4 want r = 1

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0 001

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1 000

1. . . . . .

ENgrailed

Let’s apply radius/state tradeoff “cut-and-shift” procedure.

To get a make the trade, the algorithm adds two extra “slots.”


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EnGrailed

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1. . . . . .

Interesting superficial resemblance:

Focus on the three proteins: Sloppy, Wingless, Engrailed.


Much stronger form of validation comes by isolating a “path” within the gene regulatory network that stabilizes the three proteins in a spatial feedback sequence:

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The dynamically activated subnetworks:

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Conclusion: the coordination of multiple input-output state relations that unfold over space and time allow cellular agents with nearest-neighbor views to generate long-range coordinate patterns.

Experimental: look in related drosophilids for modifications (DePace lab)

Consolidating the intermediates:

BeyondDrosophila

(kinda, sorta, maybe)

• What agent resource capacities are required to solve a given task.

• What does thinking of natural spatial multi-agent systems qua computers tell us scientically?

• Can we then turn around this knowledge to build a better bug?

It tells about local information requres, helping understand:

But the things we learned (i.e. cut and shift) were general.

so that we could, for example, make general patterns?

In theory.

For example: 1000-repeat pattern (which has an r = 2 check scheme),

Otherwise,

Repeat patterns have a well-defined gradient.

∇Θ(B)+ =

{i, if B ◦ i is consistent with ΘB(2R), otherwise

1 10 0 0 0 00

∇Θ(100)+ = 0

∇Θ(000)+ = 1 ∇Θ(010)+ = 0 ∇Θ(b) = b(3)

∇Θ(001)+ = 0

Goal: Design local rules to make given patterns.

Designing Local Rules

Now simply define:¬Θ

Θ

¬Θ Θ

Θ

¬Θ Θ

¬Θ

Gradient waves ...

F (B) = ∇+Θ(B(1 : 2r))

Multiple “waves” all at once in actuality ....

2r(Θ)B(1 : 2R)∇+Θ

Not hard to implement in higher dimensions.


Gradient “spreads correctness.”

Resulting rule is 100% robust to:

• Initial condition perturbations.

• Timing issues.


But: the rule to solve a repeat pattern of size |q| takes a radius of |q|.

Can run a radius-minimization algorithm.

And then, couple in the radius/state tradeoff.

A) B)

Anterior Posterior Anterior Posterior


Designing Local RulesSimilar principles can be applied to develop a language for building a diverse array of patterns.

Pattern

Local representation

Minimize radius

Tradeoff

Dynamic Rule

Network

ΘΘmin

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!→

???


• Is anything here doable? even remotely?

• Some VERY simple ideas in multi-cellular synthetic biology might be ``understandable” (or at any rate, imagineable) and connectable to cellular programs

• What is the simplest system that could possibly exploit some of these ideas?

• What I’d really like to explore is .... evolution.


local-to-global algorithms in biology - amazon s3

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