2006-4-26

Highly designable phenotypes and mutational buffers emerge

from a systematic mapping between network topology and

dynamic output

Nochomovitz & Li

The idea• Map network topologies to dynamical phenotypes

The designability of a phenotype is defined as the number of topologies mapped onto it.

Designability = 2

Designability = 1

Dynamic phenotype

Network topology

A 5-cycle phenotype(0001 0010 1001 0111 1010)

A limit cycle passing through a specific series of states or a particular fixed point.

Updating rules

Rule 1:

Rule 2:

The time scales of the decay process and the activating or inhibitory processes are of equal order.

The decay process occurs on a much slower time scale than the activating or inhibitory processes.

Phenotypes that are neither excessively rare nor excessively simple

Comparison of the two

rules

Under both rules, the strong, intermediate, and weakly designable phenotypes emerge consistently.

2

22

ˆ( )

( )

i ii

ii

y yR

y y

y

Designing larger cycles is exponentially hard.

Designability spectrum of 4-cycles for 4-node networks

Mean designability of 4-cycles

0001 0010 0101 0011

Designability spectrum of 5-cycles for 4-node networks

Mean designability of 5-cycles

Designability spectrum of 6-cycles for 4-node networks

Mean designability of 4-cycles

What are the dynamical features of highly designable dynamical phenotypes?

Case 1: (2 3 6 11)

Two classes of connections: fully conserved connections

• Correspond to a subnetwork module that is necessary to encode the disignable, stable phenotype.

variable connections• Control the transient dynamical flow.

Network with fully conserved connections

0001 0010 0101 1010

Designability = 2500

Case 2: (2 3 5 9 10)

(0001 0010 0100 1000 1001)

Designability = 250

Designability is independent of robustness

Robustness of a phenotype is measured by the number of transient states that flow toward the stable phenotype.

Mutational buffering:

The incorporation of a fourth node bearing specific connections to a set of genetically distinct three-node networks would trigger the three-node networks to acquire the same dynamical phenotype, despite possessing differing dynamical phenotypes as isolated three-node systems.

A scaffold refers to a fixed manner by which a fourth node interacts with a three-node subnetwork.

Buffering strength reports the ratio of designability of a dynamical phenotype observed on three-node subnetworks attached via scaffold to a fourth node, to that phenotype’s designability on isolated three-node networks.

An example of mutational buffer

Buffering strength = 11

Scaffold

Summary

• A small fraction of dynamical phenotypes possesses atypically high designabilities.

• Highly designable dynamical phenotypes arise via the presence of a core of fully conserved network links.

• No correlation exists between the stability of a dynamical phenotype and its designability.

• Identification of mutational buffers, which act to suppress phenotypic variation through specific dynamical relationships to other genes.