The Royal Road Fitness Problem
11/13/03
Genetic computation - "Royal Road" problem - to illustrate the power of crossover.
-Evolve a population of bit strings in such a way as to improve the fitness of {best
individual, or the avg individual} by the # of blocks of all 1's achieved.
Genetic Algorithium –
Various variables –
M - the size of the population.
N - the number of blocks in the entire population/sequence (the block will be encoded
with 0's or 1's)
K - the number of bits in the block.
Encoded in a population of bit stream.
N
a b
c d
e
f
g
|-----|------|------|------|------|------|------|
1111 1011 0111 0111 0111 1000 1111
-This represents a sequence of DNA bases we have simply encoded them into 0 and 1, bit
representation.
-The N at the top of the diagram represents a block.
-The overall fitness of the sequence will be 2. This is because we measure fitness by the
number of block that are composed of all 1's. In this case blocks a and g will represent
blocks that contribute a fitness of one each.
-M will include all blocks from a to g.
-Any sequence with a 0 in its block will be considered to give an overall 0 contribution
to the fitness of the sequence.
-A biological analog of this example will be a population of earthworms.
-Various levels of fitness. (Strong worms and weak worms)
-Selection according to fitness.
-Mutation can allow you to get more 1's and also lose more 1's at the same time.
Mutation analysis Mutation will occur on every generation at a rate of q.
q - is the probability of a bit flipping (bit flip).
The probability that a mutation will occur in a size K sequence is q^K.
Crossover and its Genetic Significance -This is an extremely good way to pick up an extra level of fitness from two different
strands.
example:
|--------|-------| strand (a)
11111 11110
|--------|-------| strand (b)
00001 11111
-Both the above strands have a fitness of 1 respectively. If crossover occurred then a
possible resulting strand could be –
|--------|--------|
11111 11111
here the fitness will be 2, due to the 2 blocks that have each a fitness of 1.
-Random effects of mutation are reduced by averaging of the overall population with
respects to a reduced number of individual exhibiting the mutational effect.
-The variation will change due to the mixing power.
Mutation and Genetic Terms - Breeding better racehorses. Advantage is if winners are winning, does not mean that
the losing horses have all bad gene and will not contribute for the overall benefit of
the population.
Motivated by evolution of gene blocks in natural systems.
As an example of genetic computation.
(Silicon computers) =>
Example DNA computing & multiplexed DNA computation.
Kiwmas theory of "molecular evolution" - distribution of evolving populations.
-Fitness of an individual where you have to evolve the entire gene to make a progress in
fitness.
A 00
B 11
C 10
D 01
4 symbols can be in 2 bits.
-Kiwmas came up with way to bridge the 2.
-Mutation will change the fitness; it will change the sequence at a single point.
-Kiwmas came up with the way to bridge the 2.
-Mutation will change the fitness and therefore it will change the entire gene, not just a
single point.
-Kiwmas also asked how does mutation happen, does it change a single base. This single
base change is quite rare, with a probability of around 1/ million to 1/billion. The reason
for this is the structure of the gene and its inherent double helix format. Therefore a
single mutation will not change it, the entire meaning of a gene.
-However a bunch of mutations might allow you to get a better gene, overall.
- The region where proteins fit might be affected by a single point mutation. This is
because protein structure is more dependant upon structural fitting rather than base pair
matching.
-There are approximately 10^9 individuals of E-coli in a septic tank.
-Unrealizable population of >10^9.
If we have 10^14, then we can split this up into 10^6 populations each with size 10^8 - to
see how genome changes over time. With a population of 10^14 the initial size of
analysis is to large to formulate a feasible single experiment. Therefore we will break up
the gene population into multiple experiments. They are static; here we can keep a
million simultaneous assortments of populations.
- Evolution has one population based on chance events.
- Splitting upon the gene pool is better. We can multiplex DNA computation with
multiple populations processed in parallel. This is easy for us to do. As a model we have
a significant advantage.