**Kauffman-Sheets**

The following
spreadsheets are based on Stuart Kauffman's 1995 book *At Home in the Universe*,
New York: Oxford University Press.

__Kauffman
10 x 10 Boolean Network__.
This simple 10 x 10 Boolean network (KAUF_10X_P3.xls) is based on
Stuart Kauffman's *At Home in the Universe* model on page 89 (where SK's
model is 24 x 24).
The edges and ends are mathematically joined to form a donut or torus
so that each cell has four neighbors.
On each iteration, the value in a cell is a Boolean function of its N,
S, E, and W neighbors.
The 100 (10 x 10) functions are chosen randomly in the ancillary
spreadsheet RANDP3.xls and then the values are pasted
(using "Paste Special") into this spreadsheet at the bottom.
Since the Boolean functions have four arguments (the 0s or 1s in the
four adjacent cells), there are 2^{4} = 16 cases each of which can
have a 0 or 1 value so there are 2^{16} possible functions of 4
arguments.
All the ones chosen have the property that there are either 3 or 13 (16
- 3) 1s among the 16 values. The ratio of the larger of the number of 1s or 0s
over 16 is the "P-value" of the function so all the functions have a
P-value of 13/16 = 0.8125.
By changing the P# from 3 to any other value from 0 to 8, one can
generate 100 functions with other P-values in the same RANDP3.xls spreadsheet
and then paste them into the model.
With a high P-value of 16/16 = 1, the model is rigid (crystalline
order) and at the other extreme with the P-value of .5 = 8/16, the model is
chaotic.
In between one can find the "sweet spot" P-value where the
order undergoes the phase transition from rigid to chaotic.
Right-click on the names (and choose "Save Target As…") to
download the files.

__Kauffman
N=6 K=3 Fitness Landscape__. This NK fitness landscape model
(NK_6_3.xls) is based on Stuart Kauffman's *At
Home in the Universe* model on page 172 with N = 6 genes each receiving
input from K = 3 other genes. In each of the 2^{(K+1)} = 16 genome cases (the gene
and the 3 connected to it being on or off), each gene is assigned a random
fitness contribution between 0 and 1 by the ancillary spreadsheet COUP_N6K3.xls
and then special-pasted (values only) into the array of fitness contributions
in NK_6_3.xls. If we lay the 6
numbers 0, 1,…,5 around a loop, each gene is assumed to receive input from
the K = 3 genes before it along the loop so gene 1 receives input from the
genes numbered 0, 5, and 4. There
are 2^{N} = 64 genotypes. The
fitness contribution of each genotype is computed as the average of the
fitness contributions of the 6 genes (which will depend on which of the 16
cases held for each gene). The
program seeks a local maximum of fitness by randomly starting at a vertex on
the 64 cornered hypercube. On
each iteration, it flips the on-off value of one of the six genes (i.e.,
considers an adjacent vertex on the hypercube) to test if the fitness
contribution is higher and then moves to that vertex if it is higher. Right-click on the names (and choose "Save Target As…") to
download the files.