The ``NK'' Adaptive Landscape Model

Kauffman's ``NK'' adaptive landscape model [8] will be used to illustrate the effects of constructional selection because it explicitly shows the epistatic structure of the genotype-phenotype map. The following is a generalized version of the NK model, a map between a set of genes and a set of fitness components. This is illustrated in Figure 2.

1.

The genome consists of n binary-valued genes, that exert control over f phenotypic functions, each of which contributes a component to the total fitness.

2.

Each gene controls a subset of the f fitness components, and in turn, each fitness component is controlled by a subset of the n genes. This genotype-phenotype map can be represented by a matrix,

$$\mbox{$\mbox{\boldmath\(M\)\unboldmath }= \left\Vert m_{ij} \right\Vert$ , $i = 1 \ldots n$ , $j=1\ldots f$ ,}$$

of indices $m_{ij} \in \{0,1\}$, where m_ij = 1 indicates that gene i affects fitness component j;

3.

The columns of $\mbox{\boldmath\(M\)\unboldmath }$, called the polygeny vectors, $\mbox{\boldmath\(g\)\unboldmath }_j = \left\Vert m_{ij}\right\Vert$, $i = 1 \ldots n$, give the genes controlling each fitness component j;

4.

The rows of $\mbox{\boldmath\(M\)\unboldmath }$, called the pleiotropy vectors, $\mbox{\boldmath\(p\)\unboldmath }_i = \left\Vert m_{ij}\right\Vert$, $j = 1 \ldots f$, give the fitness components controlled by each gene i;

5.

If any of the genes controlling a given fitness component mutates, the new value of the fitness component will be uncorrelated with the old. Each fitness component $\phi_i$ is a uniform pseudo-random function¹ of the genotype, $\mbox{\boldmath\(x\)\unboldmath }\in \{0,1\}^n$:

$$\phi_i (\mbox{\boldmath\(x\)\unboldmath }) = \Phi(\mbox{\bold... ...x{\boldmath\(g\)\unboldmath }_i) \sim \mbox{uniform on } [0,1],$$

where $\Psi:\{0,1\}^n\times\{1,\ldots,n\}\times\{0,1\}^n\mapsto[0,1]$, $\circ$ is the Hadamard product ( $\mbox{\boldmath\(x\)\unboldmath }\circ \mbox{\boldmath\(g\)\unboldmath }_j=\left\Vert x_i m_{ij}\right\Vert$, $i = 1 \ldots n$). Any change in i, $\mbox{\boldmath\(g\)\unboldmath }_i$, or $\mbox{\boldmath\(x\)\unboldmath }\circ \mbox{\boldmath\(g\)\unboldmath }_i$ gives a new value for $\Phi(\mbox{\boldmath\(x\)\unboldmath }\circ \mbox{\boldmath\(g\)\unboldmath }_i; i,\mbox{\boldmath\(g\)\unboldmath }_i)$ that is uncorrelated with the old;

6.

  If a fitness component is affected by no genes, it is assumed to be zero:

$$\Phi(\mbox{\boldmath\(x\)\unboldmath }\circ \mbox{\boldmath\(... ...math\(g\)\unboldmath }_i = \left\Vert \ldots 0 \right\Vert$ };$$

7.

The total fitness is the normalized sum of the fitness components:

$$w(\mbox{\boldmath\(x\)\unboldmath }) = \frac{1}{f}\sum_{i=1}^f \phi_i (\mbox{\boldmath\(x\)\unboldmath }).$$

  

Figure: Kauffman's NK model recast as a map between the genotype and a set of fitness components. Arrows indicate that the gene affects the fitness component. A new gene with effects on two fitness components is shown being introduced to the genome.