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Research Article
Received 26 August 2014, Accepted 26 August 2014 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/sim.6307
The basic science and mathematics of
random mutation and natural selection
Alan Kleinman*†
The mutation and natural selection phenomenon can and often does cause the failure of antimicrobial, herbicidal,
pesticide and cancer treatments selection pressures. This phenomenon operates in a mathematically predictable
behavior, which when understood leads to approaches to reduce and prevent the failure of the use of these selec-
tion pressures. The mathematical behavior of mutation and selection is derived using the principles given by
probability theory. The derivation of the equations describing the mutation and selection phenomenon is carried
out in the context of an empirical example. Copyright © 2014 John Wiley & Sons, Ltd.
Keywords: random mutation; natural selection; mathematics; probability theory
1. The mathematical model of mutation and natural selection
The mutation and natural selection phenomenon consists of two components. The mutation compo-
nent is a random stochastic event that occurs in the replication of a genome. Natural selection, on
the other hand, can be either random or non-random. Natural selection is random when there are a
variety of causes of death or impaired reproduction of members of a population. For example, some
members of the population are killed by accident, others by dehydration, others by starvation, others
by predation, and so on. This type of selection is characteristic in genetic drift and is not signi-
cant to the evolution of drug resistance, herbicide resistance, pesticide resistance and less than durable
cancer treatments and therefore will not be discussed in this paper. On the other hand, non-random
selection such as the use of antimicrobial agents, herbicides, pesticides and cancer treatments, which
cause the death or impaired reproduction across entire populations in a non-random manner, will be
described here.
In particular, the derivation of the equations, which describe mutation and non-random selection (from
here on will be simply termed ‘mutation and selection’), will be carried out in the context of a well-
measured empirical example of mutation and selection. This example describes the mutations required for
a bacterial population to become resistant to an antibiotic. While this example is of particular importance
to the use of selection pressures in the practice of treatment of infectious diseases, the principle is more
general and can be applied to the evolution of herbicide-resistant weeds, pesticide-resistant insects and
failure of cancer treatments.
The mathematical principles used to derive the equations of mutation and selection are obtained from
the text, Advanced Engineering Mathematics [1] by Erwin Kreyszig. Governing axioms and principles
will be repeated in this paper in order to clarify the derivation of the governing equations.
We start the derivation of the equations in the next section with the empirical example of mutation and
selection, which will form the context for the mathematics.
2. The empirical example of mutation and selection
The empirical example that we will use to frame the derivation of the equations, which describe muta-
tion and selection, was published in Science and is titled Darwinian Evolution Can Follow Only Very
PO Box 1240, Coarsegold, CA 93614, U.S.A.
*Correspondence to: Alan Kleinman, PO Box 1240, Coarsegold, CA 93614, U.S.A.
†E-mail: kleinman@sti.net
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
Few Mutational Paths to Fitter Proteins [2] and was written by Daniel M. Weinreich, Nigel F. Delaney,
Mark A. DePristo and Daniel L. Hartl. Included here is the abstract:
Abstract: Darwinian Evolution Can Follow Only Very Few Mutational Paths to Fitter Proteins
Five point mutations in a particular 𝛽-lactamase allele jointly increase bacterial resistance to a clinically impor-
tant antibiotic by a factor of ∼100,000. In principle, evolution to this high-resistance 𝛽-lactamase might follow
any of the 120 mutational trajectories linking these alleles. However, we demonstrate that 102 trajectories are
inaccessible to Darwinian selection and that many of the remaining trajectories have negligible probabilities of
realization, because four of these ve mutations fail to increase drug resistance in some combinations. Pervasive
biophysical pleiotropy within the 𝛽-lactamase seems to be responsible, and because such pleiotropy appears
to be a general property of missense mutations, we conclude that much protein evolution will be similarly
constrained. This implies that the protein tape of life may be largely reproducible and even predictable.
What this empirical example demonstrates is that the sequence of mutations must occur in an order of
ever increasing tness in order for the evolutionary process to have a reasonable chance of occurring. In
addition, this example demonstrates that there is more than a single sequential order, which can occur.
In other words, not every member of the population must have the same sequence of mutations in order
to evolve resistance to the antibiotic selection pressure. The population of bacteria has subdivided into
subpopulations, each taking their own trajectory to achieve resistance to this particular selection pressure.
If we label one subpopulation ‘1’, that subpopulation must get mutation A1 followed by mutation B1,
in turn followed by mutation C1, then D1 and nally E1 in order to evolve resistance to the antibiotic
selection pressure. If we label another subpopulation ‘2’, that subpopulation must get a different set of
mutations, which we can label A2 followed by mutation B2, in turn followed by mutation C2, then D2
and nally E2 in order to evolve resistance to the antibiotic selection pressure. Each of the subpopulations
that Weinreich and his co-authors describe has their own set of mutations, which lead to the evolution of
a high-resistance 𝛽-lactamase allele. Each of the subpopulations are evolving independently of the other
subpopulations. Once a particular subpopulation starts on an evolutionary trajectory, the replication of
members from that subpopulation to not contribute to trials for the next benecial mutation in a different
subpopulation on a different evolutionary trajectory.
In order to evolve a high-resistance 𝛽-lactamase allele, a bacterial subpopulation must accumulate ve
mutations in a particular sequence. We will now address how a subpopulation can accumulate these ve
mutations. We will drop the numeric and call these ve mutations A, B, C, D and E, and the ve mutations
must occur in that order. The rst mathematical question is to identify the trial and possible outcomes for
that trial in this stochastic process. We describe this mathematics in the following section.
3. The mathematics of the empirical example of mutation and selection
To derive the mathematical behavior of the empirical example of the mutation and selection phenomenon,
we rst dene some terms.
•nis the total population size.
•nAis the number of members in the subpopulation with mutation A.
•nGA is the number of generations that the members of the total population reproduce for the
probability that mutation A will occur.
•nGB is the number of generations that the members of the population with mutation A reproduce for
the probability that mutation B will occur.
•𝜇is the probability (frequency) that an error in replication will occur at a particular site in a single
member in one replication.
• P(BenecialA)is the probability that, of all the possible mutations that can occur at the particular
site, it will be the benecial mutation A.
• P(BenecialB)is the probability that, of all the possible mutations that can occur at the particular
site, it will be the benecial mutation B.
• P(A) is the probability that benecial mutation A will occur at a particular site.
• P(Ac) is the probability that benecial mutation A will not occur at a particular site.
• P(B) is the probability that benecial mutation B will occur at a particular site.
• P(Bc) is the probability that benecial mutation B will not occur at a particular site.
With these terms dened, we can determine the probability that mutation A will occur on some member
of the population. The rst step is to identify the trial (the random event) and the possible outcomes for
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
that trial. In the mutation and selection phenomenon, the mutation is the trial, and the possible outcomes
for that trial are the substitution of a base, Ad, Cy, Gu or Th, where Ad represents adenine, Cy cytosine,
Gu guanine and Th thymine, and we can denote the probabilities for each of these outcomes as P(Ad),
P(Cy), P(Gu) and P(Th), respectively. However, there are other possible outcomes from a mutation. For
example, a base can be inserted at the site. Let an ‘i’ preceding the base denote an insertion. Then,
P(iAd) means the probability that an Ad base will be inserted at the particular site and so on for the other
bases. If there is a deletion of a base at that site, then P(del) means the probability that a deletion of a
base will occur at that site. Additional terms can be added to include any other forms of mutation that
can occur.
The outcomes from a mutation are mutually exclusive events. That is, if a substitution of Ad occurs,
no other outcomes such as a substitution of a different base, insertion or deletion can occur and so on.
Then, to compute the outcomes for a mutation, we can use the addition rule of probabilities for mutually
exclusive events. This rule is repeated here from reference [1], p. 715.
If E1,…,Emare mutually exclusive events, then
P(E1∪E2∪…∪Em)=P(E1)+P(E2)+ …+P(Em)
Then, for our case, the mathematical expression for the possible outcomes for a mutation is
P(−∞ <X<+∞) = P(Ad)+P(Cy)+P(Gu)+P(Th)+P(iAd)+P(iCy)+P(iGu)
+P(iTh)+P(del)+…=1(1)
where ‘…’ represents any other mutation such as the probability of a double deletion or double insertion
of the base at that site and any other forms of mutation you might imagine could be included in the
sample space.
One could think of these outcomes as equivalent to an unfair die (with more than six sides) where a
roll of this die is more likely to give a substitution mutation rather than an insertion or deletion of a base.
Then, the probability for a benecial mutation A occurring on a single member in a single replication is
P(A) = P(BenecialA)𝜇(2)
where P(BenecialA)has a value between 0 and 1.
If the benecial mutation is the substitution of the Ad base, then P(BenecialA)=P(Ad); if the bene-
cial mutation is the substitution of a Gu base, then P(BenecialA)=P(Gu); if the deletion of the base
is the benecial mutation, then P(BenecialA)=P(del); and so on. With certainty, we know that 0 ⩽
P(BenecialA)⩽1.
Equation (2) is the probability that, in any given member of the population in a single replication, the
benecial mutation A will occur at the specic site to improve tness for that member. The P(BenecialA)
term takes into account that not all the possible mutations that might occur at a particular site will
be benecial.
Now, to compute the probability that mutation A will occur at that specic site in a population size
‘n’, we must use the complementary rule of probabilities. This theorem is given on p. 715 of reference
[1] and is repeated here.
The probability of an event E and its complement ECin a sample space S is related by the formula
P(E) = 1−P(EC)
Then,
P(Ac)=1−P(BenecialA)𝜇(3)
Equation (3) is the probability that, in any given member of the population in a single replication, the
benecial mutation A will not occur at the specic site to improve tness for that member.
Then, to compute the probability that a benecial mutation A will not occur in some member of a pop-
ulation size ‘n’ at a particular site in a single generation, we use the multiplication rule of probabilities.
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
Again from reference [1], p. 715 and repeated here, we obtain the following theorem for the
multiplication rule:
If A and B are events in a sample space S and P(A) ≠0 and P(B) ≠0, then
P(A∩B)=P(A)P(BA)=P(B)P(AB)
If the events A and B are such that P(A∩B)=P(A)P(B),then they are called independent events.
Equation (3) gives the probability that, in a single generation in a single member of the population,
the benecial mutation will not occur. To determine the probability that the benecial mutation will not
occur in a single generation in a population size ‘n’ raises the value in equation (3) to the ‘n’ power, by
using the multiplication rule of probabilities, which gives
P(Ac)=(1−P(BenecialA)𝜇)n(4)
Equation (4) gives the probability that, in a population size ‘n’, the benecial mutation A will not occur
in a single generation of ‘n’ replications.
Now, to compute the probability that the benecial mutation A will not occur in a population size
‘n’ in multiple generations nGA raises the result in equation (4) to the nGA power again, by using the
multiplication rule, which gives
P(Ac)=((1−P(BenecialA)μ)n)nGA =(1−P(BenecialA)𝜇)n∗nGA (5)
Equation (5) gives the probability that the mutation A will not occur in any member in population size
‘n’in‘nGA ’ generations. To compute the probability that mutation A will occur in some member in the
population size ‘n’in‘nGA’ generations, we again use the complementary rule and equation (5) to obtain
P(A) = 1−(1−P(BenecialA)𝜇)n∗nGA (6)
The next step of the calculation of this evolutionary process is to compute the probability that mutation
B will occur at the correct site in some member of the population that already has mutation A. The
computation just performed for mutation A is analogous except now that the population size for mutation
B is limited to those members that already have mutation A. We start this part of the computation by
computing the probability that our benecial mutation B will occur on some member of the population
that already has mutation A.
P(B) = P(BenecialB)𝜇(7)
where P(BenecialB)is dened in the same manner as was P(BenecialA).
Again, we recognize with certainty that 0 ⩽P(BenecialB)⩽1.
Equation (7) is the probability that, in any given member of the subpopulation ‘nA’ (members of the
population that already have mutation A) in a single replication, the benecial mutation B will occur at
the specic site to improve tness for that member.
Now, to compute the probability that mutation B will occur at the specic site in a population size ‘nA’,
we must again use the complementary rule of probabilities to compute the probability that mutation B
will not occur on some member of the population ‘nA’. Using equation (7) and the complementary rule,
we obtain
P(Bc)=1−P(BenecialB)𝜇(8)
Equation (8) gives the probability that, in a single generation in a single member of the population, the
benecial mutation B will not occur. To determine the probability that the benecial mutation will not
occur in a single generation in a population size ‘nA’ raises the value in equation (8) to the ‘nA’ power,
by using the multiplication rule, which gives
P(Bc)=(1−P(BenecialB)𝜇)nA(9)
Equation (9) gives the probability that, in a population size ‘nA’, the benecial mutation B will not occur
in a single generation.
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
Now, to compute the probability that the benecial mutation B will not occur in a population size
‘nA’ in multiple generations nGB raises the result in equation (9) to the nGB power again, by using the
multiplication rule, which gives
P(Bc)=((1−P(BenecialB)𝜇)nA)nGB =(1−P(BenecialB)𝜇)nA∗nGB (10)
Equation (10) gives the probability that the mutation B will not occur in any member in population size
nAin nGB generations. Again, using the complementary rule and equation (10), we obtain the probability
that mutation B will occur in at least one member of a population size nAin nGB generations:
P(B) = 1−(1−P(BenecialB)𝜇)nA∗nGB (11)
Equation (11) gives the probability that the benecial mutation B will occur on some member of the
subpopulation nAin nGB generations. The next step of the calculation of this evolutionary process is to
compute the joint probability that mutations A and B will occur at the correct sites in some member
of the population. Recognizing that mutations A and B are independent events, we compute the joint
probability that mutation B will occur on some member with mutation A using the multiplication rule
using equations (6) and (11):
P(A)P(B) = 1−(1−P(BenecialA)𝜇)n∗nGA 1−(1−P(BenecialB)𝜇)nA∗nGB (12)
Equation (12) gives the joint probability of mutation B occurring on a member of the population, which
already has mutation A as a function of population sizes and generations.
This computational scheme can be continued for mutations C, D and E, and we would obtain the
following joint probability equation:
P(A)P(B)P(C)P(D)P(E) = 1−(1−P(BenecialA)𝜇)n∗nGA 1−(1−P(BenecialB)𝜇)nA∗nGB 1−(1−
P(BenecialC)𝜇nAB∗nGC 1−(1−P(BenecialD)𝜇)nABC∗nGD 1−1−P(BenecialE)𝜇nABCD ∗nGE (13)
where nAB is the subpopulation size with mutations A and B; nABC is the subpopulation size with muta-
tions A, B and C; nABCD is the subpopulation size with mutations A, B, C and D; nGC is the number of
generations for mutation C to occur; nGD is the number of generations for mutation D to occur; nGE is the
number of generations for mutation E to occur; and P(BenecialC), P(BenecialD)and P(BenecialE)are
the corresponding probabilities for the benecial mutations C, D and E, which will occur at a particular
site when a mutation does occur.
Equation (13) gives the probability of accumulation of the ve mutations, A, B, C, D and E, as a
function of subpopulation size, number of generations and mutation rate for the evolutionary process to
occur. This mathematical behavior of mutation and selection gives rise to specic requirements for an
evolutionary process to have a reasonable probability of occurring. These requirements are described in
the discussion section, which follows.
4. Discussion of the mathematics of mutation and selection
Equation (13) gives rise to specic requirements for an evolutionary process by mutation and selection
to have a reasonable probability to occur. Each step in the evolutionary process occurs in a cyclical
manner. Mutation A occurs somewhere in the population and forms a new subpopulation of members
with that mutation A. However, until that subpopulation size increases over generations, the probability
that mutation B will fall on some member with mutation A will be small. There are simply not enough
trials occurring until the number of members in that subpopulation increases over the generations. The
number of trials for the particular mutation increases as the population size increases and the number
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
of generations that the subpopulation is able to replicate. Each of the components of equation (13) is of
the form:
P(X) = 1−(1−P(Benecial)𝜇)n∗nG(14)
P(Benecial)𝜇will always be a small number making the value of (1 −P(Benecial)𝜇) close to but
slightly less than 1, and for small values of nand nG,(1−P(Benecial)𝜇)n∗nGwill be close to 1 and the
probability for that component will be close to 0. As nand nGincrease, (1−P(Benecial)𝜇)n∗nGwill
decrease approaching 0, and the probability for that component will approach 1. The following graph
illustrates this where the probability of a mutation occurring is plotted as a function of number of members
and number of generations for various values of P(Benecial)𝜇(Figure 1).
When P(benecial)𝜇is less than 1e-7, there must be at least 100,000 replications of members who
would benet from the particular mutation before the probabilities of the benecial mutation occur-
ring starts to increase signicantly. Even when there are 1,000,000 replications, the probability of that
benecial mutation occurring is still only about 0.1.
This cycle of benecial mutation, followed by amplication of the benecial mutation, must repeat
itself over and over in order for the evolutionary process to have a reasonable probability to occur. Muta-
tion B will not have a reasonable probability of occurring on a member with mutation A until the number
of members with mutation A increases and/or the number of generations that members with mutation
A can replicate becomes large. Only when the number of members with mutation A and the number of
generations that members with mutation A can replicate reach a sufcient amount, there will be a rea-
sonable probability that mutation B will occur on some member with mutation A. And mutation C will
not have a reasonable probability of occurring on a member of the subpopulation with mutations A and
B until those members with mutation A and B can increase in number sufciently and/or replicate for a
sufcient number of generations for the mutation C event to occur.
This cyclical process of benecial mutation, followed by amplication of benecial mutation, is differ-
ent than the evolutionary process discussed by Haldane in his classic paper The Cost of Natural Selection
[3]. In this paper, Haldane proposes that ‘The principle unit process in evolution is the substitution of one
gene for another at the same locus’. The Weinreich example from reference [2] demonstrates that evolu-
tionary processes can lead to multiple different alleles. And the evolutionary process does not consist of
a substitution of one allele for another but is an amplication process where a benecial mutation (giving
a more benecial allele) must amplify (the members with that allele must increase in number) for there
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 1.0E+10 1.0E+12
P(X)
n*nG (number of replications)
P(X)=1-(1-P(beneficial)µ)^n*nG
10^-3
10^-4
10^-5
10^-6
10^-7
10^-8
10^-9
10^-10
Figure 1. Graph of P(X) as a function of n∗nG. [Correction added on 30 September 2014, after rst online
publication: gure 1 image has been corrected].
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014
A. KLEINMAN
to be a reasonable probability that another benecial mutation occurs on a member with the previous
benecial mutation.
Any disruption of this benecial mutation/amplication of benecial mutation cycle will stie the
evolutionary process. Introducing a second selection pressure, which targets a different genetic locus than
the rst selection pressure along with the rst selection pressure, forces the population to attempt to take
two evolutionary trajectories simultaneously and will disrupt the evolutionary cycle. It is this principle,
which has led to the success of combination therapy for the treatment of HIV [4]. Likewise, if Weinreich
and his co-authors had used two antibiotics where the second drug targets a different genetic locus instead
of his single beta-lactam drug, the evolutionary process that these authors measured in their experiment
would have been disrupted. Even if a benecial mutation for the rst drug were to occur, the second
drug would disrupt the amplication process for that benecial mutation for the rst drug. Likewise,
amplication of a benecial mutation for the second drug would be disrupted by the selection pressure of
the rst drug. This is a consequence of the basic science and mathematics of the mutation and selection
phenomenon and the multiplication rule of probabilities.
5. Glossary
Selection or selection pressure. Stressors on a population that kills or impairs the reproduction of some
or all members of a population.
Mutation. An error in the replication of a genetic sequence (DNA/RNA). This error can be a substitution
of a base, insertion of a base or bases, deletion of a base or bases or any other errors in replication at
a particular location in the genetic sequence. Most of these errors are detrimental; that is, they reduce
the tness to reproduce of that member, but occasionally, one of the kinds of mutations will improve the
tness to reproduce.
Mutation rate. The frequency or probability that an error will occur at a particular site in the genetic
sequence during a single replication.
References
1. Kreyszig E. Advanced Engineering Mathematics 3rd ed. John Wiley and Sons Inc.: New York, London, Sydney, Toronto,
1972.
2. Daniel MW. Darwinian evolution can follow only very few mutational paths to tter proteins. Science 7 2006; 312(5770):
111–114.
3. Haldane JBS. The Cost of Natural Selection. In Journal of Genetics, Vol. 55. Blackwell Publishing (Wiley-Blackwell):
New York, London, Sydney, Toronto, 1957; 511–524, (Available from: http://www.blackwellpublishing.com/ridley/
classictexts/haldane2.pdf) [Accessed on 12 September 2014].
4. David H. FRONTLINE Interview. (Available from: http://www.pbs.org/wgbh/pages/frontline/aids/interviews/ho.html)
[Accessed on 12 September 2014].
Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014