In this third part of the "silicon-based" bacterial evolution, we move to the real action. I developed a program which perform evolutive selection based on mathematical criteria. The program has a set of rules to perform selection, but please note that most (if not all) of these rules are totally arbitrary. In this sense, it's nothing more than a toy, and I therefore warn against any scientific analysis of what we obtain. It's just for fun. Also, the code is pretty bad.

There are two classes in the program: Bacterium and Environment. A Bacterium can be fed, and its genetic code can mutate.The Environment is responsible for feeding the bacteria with the mathematical conditions, assess if they are worthy for survival and reap the unworthy. At every new generation, the Environment perform duplication for each bacterium in the pool, and apply mutations to the new entries. We try to keep the pool filled, so we always replenish it to 4000 bacteria at every generation, out from the available mutated. 4000 is also the limit for the number of bacteria simultaneously present. This is a very strong limitation, as it hinders the normal behavior of conventional bacteria: multiply exponentially. Unfortunately, we don't want to spend weeks in calculations, and in any case the computer memory is limited, so we have to find a proper and very harsh compromise.

Once the duplication/mutation step, the environment performs the selection (reaping). For each of the mathematical conditions, each bacterium is fed with the input value. The returned value is then compared against what the environment expects. A score is obtained by summing the relative errors for each condition, and then normalized on the number of conditions. The closest this score is to zero, the better is the response of the bacterium. Obtaining zero means that the bacterium genetic code is able to respond to the conditions exactly. In the more general situation, the error will be a positive floating point number which will be compared against a tolerance. If the result exceed the tolerance, then the bacterium is killed, otherwise it survives and is allowed to reproduce in the next generation. An additional condition tries to favor those bacteria having a more compact genome: a random number between 0 and 50 is extracted, and if the value is greater than the length of the genome of the bacterium, it will survive, otherwise it will be killed. This will prevent genomes longer than 50, and strongly favour short genomes, as it will be more probable that a random extraction is greater than their length, granting it survival.

In order to improve our genetic pool, the Environment defines an "epoch". An epoch is a number of generations where selection happens. During each generation of the epoch, the tolerance on the conditions will progressively decrease, meaning that, as time passes, the environment becomes more and more selective.

Enough general chatting. Let's see an actual example. In the program you can download, the following conditions are set for the environment


As you can see, the environment expects the bacteria to be able to solve the equation 2x+7. The line

e.epoch(1.0,0.0, 20)

Starts an epoch made of 20 generations. During these 20 generations, the tolerance will decrease from 1.0 to 0.0. In other words, the environment expects, at the end of the epoch, that the bacteria in the pool are able to solve exactly 2x+7 for all the specified points (x=0,2,4,5).

Suppose we have a bacterium in the pool with the following genetic code:

IncX -3             # a = n | x = -3 | y = 0
AddYtoA             # a = n | x = -3 | y = 0
Return              # returns a = n
LoadX 1             # Never reached
BranchXNotZero 2    #

As you can see, this bacterium will return the input value unchanged. This means that the result and relative error for each condition will be

Given  | Expected | Produced | abs(Relative Error)
0      | 7        | 0        | 7/7 = 1.0
2      | 11       | 2        | 9/11 = 0.81
4      | 15       | 4        | 11/15 = 0.73
5      | 17       | 5        | 12/17 = 0.70
                       Total = 3.24
                               / 4 (number of conditions
                             = 0.81

This bacterium (as it is, assuming no mutations) will therefore survive in the first two generations (where the tolerance is 1.0 and 0.9) but it will be killed in Generation 3, as the tolerance is now 0.8.

Let's see another example

MoveAtoY     # a = n   | x = 0 | y = n
IncA  3      # a = n+3 | x = 0 | y = n
MoveAtoY     # a = n+3 | x = 0 | y = n+3
Return       # returns a = n+3
LoadX 1      # Never reached

In this case, we get

Given  | Expected | Produced | abs(Relative Error)
0      | 7        | 3        | 4/7 = 0.57
2      | 11       | 5        | 6/11 = 0.54
4      | 15       | 7        | 8/15 = 0.53
5      | 17       | 8        | 9/17 = 0.53
                       Total = 2.17
                               / 4 (number of conditions
                             = 0.54

Much better. Of course, the best condition would be something like this (chose a long one, just for illustrative purpose)

MoveAtoY    # a = n    | x = 0 | y = n
IncX 3      # a = n    | x = 3 | y = n
AddYtoA     # a = 2n   | x = 3 | y = n
IncA 2      # a = 2n+2 | x = 3 | y = n
IncA 4      # a = 2n+6 | x = 3 | y = n
MoveAtoY    # a = 2n+6 | x = 3 | y = 2n+6
IncA 1      # a = 2n+7 | x = 3 | y = 2n+6
LoadY -3    # a = 2n+7 | x = 3 | y = -3
IncX -2     # a = 2n+7 | x = 1 | y = -3

This bacterium returns exactly what the environment expects. This as well

BranchXNotZero 2   # a = n    | x = 0 | y = 0 | no branch
LoadY 1            # a = n    | x = 0 | y = 1
MoveAtoY           # a = n    | x = 0 | y = n
IncA 3             # a = n+3  | x = 0 | y = n
AddYtoA            # a = 2n+3 | x = 0 | y = n
IncA 4             # a = 2n+7 | x = 0 | y = n

At the end of the run of the program, you will obtain bacteria like these. It is interesting to note that selection produced genetic code able to solve the equation outside its defined space of conditions. We never put the condition (10, 27), but these two bacteria are able to satisfy it.

In the current setup, I purposely introduced "noise" codons in the available genetic code. As you can see, the solution for 2n+7 can be obtained with a proper combination of MoveAtoY, IncA and AddYtoA. The remaining codons are not used, or if they are, their effect is neutral. You have two contrasting effects here: the need for the genetic code to be small (so to maximize its chance of survival against the length selection) and the need to have buffer codons that can mutate without particular trouble, in particular if they are after the Return codon. This reduces the chance that a mutation will ruin the achieved functionality, making the bacterium with a long genetic code less sensitive to mutation.

Of course, you would be tempted to try similar cases. I can assure you that simple equations will be properly satisfied. However, if you try to do x\^2, your bacteria will always die. Why ? X\^2 is a rather particular situation. First of all, there's an important codon which is not present : MoveAtoX. Once you have this codon, the space of the genetic code combinations allows you to potentially obtain the solution. This is one I wrote by hand:

LoadA 0
BranchXZero 5
IncX -1
BranchXNotZero -3

Obtaining this result from evolution is hard. In some sense, we face the issue of the so called Irreducible Complexity, an argument proposed to object evolution. Indeed this appears to be the case. The genetic code able to produce the square is irreducible. Either you take it as a whole, or you don't. From our toy program there's no "in-between" that satisfies the constraints and allows the generation of that code in steps. Although apparently a sound argument, there are many considerations to do on this point, which have a substantial effect against this position.

First, as I said the program shown here is a toy. You cannot put too much reasoning for proofs into it. We are running on a very restricted set of bacteria. Even if statistically improbable, the creation of the above genetic code when million, or even billion of bacteria are produced suddenly becomes more possible. Then, selection does its job by granting it full survivability, and therefore takeover of the population.

Second: the rules of chemistry are slightly more flexible than the rules provided here. In this sense, this program represents a situation more akin to a Ziggurat than an Egyptian pyramid. Electronic interaction of molecules allow a very refined, smooth and nuanced behavior, while our codons do not.

Third point is that we are assuming a single block of code to be able to produce a complex mathematical result. Biological systems do not work this way. Biological systems produce components, and make them interact. For example, a complex (but still pretty simple) biological process like the Krebs cycle is not performed by a single molecular übermachine. There are ten different enzymes involved in carbohydrate consumption, interacting together in the cell. Each enzyme is a small entity which performs a simple operation. Together they network for a nice and refined mechanism. In other words, selection and evolution moves to another level in real biology: not only the evolution of single components (enzymes) but also evolution of their mutual interaction. In our toy program, we don't allow interaction of "subroutines", nor of bacteria. The very fact that we are made of a system of interacting cells and not a huge unicellular bacterium is a hint that our case is very limited in possibilities.

Fourth point: there no chance for so called "lateral transfer" among bacteria. In biological systems, the DNA can be exchanged among bacteria, as it's universal and works in any case. Suppose that a very powerful enzymatic system would be obtained by the concerted presence of enzymes A,B,C and D. An organism happen to have enzymes A and B, but not C and D, because they are normally not evolved in its conditions. Another organism was able to evolve C and D to address its own environment. These two bacteria can come in contact, and exchange their genetic material. Suddenly, both organisms have the whole set of A,B,C, and D. This would have not been possible without the universality of the genetic code. It's also not possible in our program.

There are of course many other points and issues to consider. I think I reached my goal to share a personal experiment, and I would like to close with interesting links toward more evoluted (no pun intended) software to simulate digital life forms

Thanks for reading.