Biology Reference
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Figure 1 A directed graph representation of a gene-control network with 20 genes and
20 connections output by the program.
Nodes are genes, and a directed arrow indicates action of the gene at the tail and the expression
of the gene at the head of the arrow. Mutations act on connections, and may randomly reassign
the gene at the head or tail of the arrow. With 20 connections, it thus has 40 mutable sites.
This particular graph was produced by 100 mutation events acting on a closed-loop model gene
system of 20 genes and 20 connections. It is indistinguishable in generic properties from the one
constructed at random. (From Schank and Wimsatt, 1988, p. 51. Figure copyright retained by the
author.)
simulations of generative entrenchment in Stuart Kauffman's (1985) model of
the evolution of gene-control networks (Schank & Wimsatt, 1988; Wimsatt &
Schank, 2004). I co-opt it here to illustrate differential entrenchment. 17
In Fig. 1 (Schank & Wimsatt, 1988) the connection from 5 to 3 has no fur-
ther consequences (no arrows leave 3), but the connection from 16 to 13 has
many. From node 16, we can travel: 16 13 19 14 17 5 3,
with other divergent paths along the way. It is rare - essentially impossible
for robust reasons - to find interesting networks in which all nodes have equal
influence. Differential generative entrenchment of different nodes in a network
is a generic property in Kauffman's sense - virtually all networks will have
it. But it is even more powerfully anchored, for it is doubly robust - generic
17 Directed graphs provide one kind of measure of generative entrenchment, but not the only one and not always
the best. Thus Wagner (2005) presents cases where volume throughput of a product at a node correlates better
with evolutionary stability than the number of nodes which are reachable (downstream) in a directed graph.
He also argues that the necessity of a particular product is not well represented by its topological connectivity.
Many cases of the first type (where production capacity matters and is accomplished by duplicating like parts,
as with multiple copies of the DNA to make ribosomal RNA, or multiple liver cells) may represent k-out-of-m
structures like those discussed in Wimsatt and Schank (1988). In such cases the whole k-out-of-m structure
should be treated as a single unit for evaluating entrenchment.
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