#### Abstract

We propose a molecular-level control system view of the gene mutations in DNA replication from the finite field concept. By treating DNA sequences as state variables, chemical mutagens and radiation as control inputs, one cell cycle as a step increment, and the measurements of the resulting DNA sequence as outputs, we derive system equations for both deterministic and stochastic discrete-time, finite-state systems of different scales. Defining the cost function as a summation of the costs of applying mutagens and the off-trajectory penalty, we solve the deterministic and stochastic optimal control problems by dynamic programming algorithm. In addition, given that the system is completely controllable, we find that the global optimum of both base-to-base and codon-to-codon deterministic mutations can always be achieved within a finite number of steps.

#### 1. Introduction

Systems biology is an emerging academic field aiming at system-level understanding of biological systems. The early development of systems biology started in the late 1940s [1]. Recent progress in molecular biology has enabled us to gain information on the interactions among the underlying molecules from comprehensive experimental data sets. In general, a system-level understanding of a biological system can be derived from insight into four key properties: (1) the system’s structure, (2) the system dynamics, (3) the control method, and (4) the design method [2]. Equivalently, identifying related components and their interactions, gathering qualitative and quantitative information about the system’s evolution under different circumstances, achieving the desired outputs by controlling the input with appropriate definitions of inputs and outputs of the system, and reconstructing analogous systems by eliminating the undesired properties are four essential steps in systems biology done by collaboration among engineers, biologists, and doctors. Figure 1 shows a typical method of system construction and verification commonly applied currently. Control engineers construct models, run simulations, and predict the system behaviors. Biologists design and carry out the experiments and measure the output data. Control engineers revise and verify the models by comparing the predictions and experimental results.

Systems biology is a cross-cutting research area connecting control engineering, biology, and medical science, as shown in Figure 2. It provides a systematic view of the biological system and related medical interventions. It aims at understanding the bare function and integration function of the cell to reconstruct the biological systems with desired features. Control and automation play critical roles in this novel field not only by providing new technology and equipment for biologists to design and perform meticulous experiments, to take high-throughput measurements, and to analyze experimental data efficiently, but also by offering doctors new medical applications and improving the precision of medical manipulations. The wide range of aspects which control and automation have been applied to include, but are not limited to, gene regulation [3, 4], drug delivery [2, 5], and neuron networks [6, 7]. The equipment provided by control engineers includes, but is not limited to, nanodevices, biochips, cuvettes for electroporation, and gene guns. Biologists perform various biological experiments, such as protein synthesis and virus DNA modifications, to gather measurements for model revisions and verifications, to conclude theoretical and practical results from evidence, and to help medical practice. Doctors use both theoretical and practical results from biologists to perform tissue engineering, such as organ transplants and artificial tissue construction.

According to their scales, biological systems can be divided into three levels: the molecular level (nm), cellular level (*μ*m), and tissue level (cm), analogous to the part, individual, and group, respectively. Molecular-level research focuses on how, when, where, and to what extent [10] a gene is expressed. The essential goal is to sketch a complete blueprint of genes by identifying the control sequences of coding DNA segments and their interactions. Cellular level research, in general, treats one cell as a plant in classical control theory and investigates the reactions of the cell to the changing environment, for instance, concentration changes of related chemicals. State-of-the-art medical therapies are primarily based on experimental results at the cellular level. Tissue-level research mainly concerns tissue reconstruction, artificial tissue substitutes, or tissue function recovery. The cell differentiation process is an important topic at the tissue level. Typical biological systems are collaboratively controlled at all three levels. Most current research work focuses on either cellular level or tissue level systems. Not much work has been done at the molecular-level. In contrast, understanding biological systems at the molecular level is crucial, since species have the same basic inheritance, DNA macromolecules, and follow a common rule in gene expression, the central dogma in molecular biology. Molecular level understanding of biological systems provides instrumental information about radical causes of many diseases and the genetic evidence of evolution. It also helps biologists to gain a better understanding of molecular level interactions, draw a complete blueprint of gene networks, improve existing means, create novel means to cure genetic diseases, and to elaborate on the theory of evolution. Recent technology in gene sequencing makes it possible to conquer the difficulty in measurement at the molecular level and to identify the nucleotide sequences of a particular DNA segment. Targeted sequencing is the most promising step toward maximizing the efficiency of the next-generation sequencing technology using polymerase chain reaction. The availability of DNA microarray makes it possible to accomplish tens of thousands of genetic tests for picomoles (10^{−12}) of a specific DNA sequence.

Researchers have applied various methods to model, simulate, and control the gene regulation processes. Early attempts to model and simulate gene regulatory systems are summarized in [10], including direct graphs, Bayesian networks, Boolean networks, ordinary and partial different equations, qualitative differential equations, quantative differential equations, stochastic equations, and role-base formalisms [10]. Other approaches include Petri nets [11], transformational grammars [12, 13], and process algebra [14]. Three important modeling methods in recent work are gene regulatory units viewed under compound control [4, 15–18], logic network models [19, 20], and base-to-base molecular-level formulation [21, 22]. The first modeling method quantitatively describes chemical concentration variations corresponding to external environmental changes at the cellular level. The second qualitatively illustrates the interactions among operons in gene regulatory units. The last modeling converts DNA segments to discrete vectors. In our paper, we adapt the base-to-base molecular level formulation to express state variables.

Current obstacles in systems biology are obvious. The structure and dynamics of biological systems are sometimes unclear. Most existing models are constructed by data-driven or hypothesis-driven methods, with only partial information available. Due to the complexity of the systems and incomplete information, the mathematical models are usually formulated by modifying empirical equations or proposing heuristic equations. The parameters of proposed models are obtained by estimation methods. Although those models can disclose significant details of the system’s structure and dynamics, the inconsistency between theoretical and experimental results creates difficulties for control engineers to verify the models, develop optimal controls, and reconstruct systems with desired properties.

In this paper, we use a novel approach to build up abstract mathematical models at the molecular level in Section 2.2, based directly on biological theory. With reasonable assumptions, we can avoid the conventional obstacles mentioned above. Different from existing methods, focusing on a gene or changes in chemical concentration, we emphasize the base change in the nucleotide bases. The cost function, a summation of costs for applying mutagens and the off-trajectory penalty, together with the system equations, formulates the optimal control problem. The optimal control is then solved by dynamic programming algorithm in Section 2.3. Section 3 shows simulation results of the optimal control problem at different scales and is followed by several important propositions.

#### 2. System Equations and Generalized Optimal Control Problem Formulation

The central dogma of molecular biology, first elaborated in [23] and restated in [24], illustrates the detailed residue-by-residue transfer of genetic sequential information. Nowadays, it is widely recognized as the backbone of molecular biology. It describes the genetic information flow among three kinds of biopolymers: DNA, RNA, and protein. In most living organisms, genetic information transfers from DNA to RNA, and then to protein. This process is usually irreversible, thus protein always acts as the sink of information flow. A codon consists of three consecutive nucleotide bases, corresponding to one amino acid according to the genetic codes. Since there are only 20 kinds of amino acids and 64 combinations of codons, there exists redundancy in genetic codes.

We are particularly interested in mutations that happen during the process of DNA replication, as DNA serves as long-term genetic information storage and is the basis of genetic inheritance, the accuracy of which is particularly important to ensure the correct expression of genes. DNA molecules consist of four kinds of nucleotide acids, *adenine* (), *thymine* (), *guanine* (), and *cytosine* (), and a backbone made of sugars and phosphate. In 1953, James D. Watson and Francis Crick found the double helix structure of DNA and the rule of basepairing, known as Watson-Crick basepairing [25, 26]. always pairs with , always pairs with , and vice versa. In nature, replication errors occur at a very low rate, one error for every 10^{7} nucleotides added [27]. The redundancy of information caused by the double-helix structure ensures the accuracy of DNA replication. Some DNA self-repair mechanisms, listed in [28], such as proofreading, also help to eliminate errors during the replication process.

Gene mutations are changes in the nucleotide sequence of DNA or RNA. Usually, we focus only on mutations occurring in coding DNA sequences and RNA. Mutations are caused by various reasons. Induced mutations are caused by either chemical mutagens or radiation. In general, radiation induces higher randomness than chemical mutagens. Point mutation is the simplest form of mutation, involving only one base. Point mutations can be further divided into transitions ( or ) and transversions (). Transversions are theoretically expected to be twice as frequent as transitions, but transitions may be favored over transversions in coding DNA because they usually result in a more conserved polypeptide sequence [29].

In this section, we first give the problem statement in Section 2.1, and then we construct system equations for both deterministic and stochastic mutations in Section 2.2. At last, in Section 2.3, we formulate the optimal control problem and apply dynamic programming algorithm to solve it.

##### 2.1. Problem Statement

Figure 3 shows the system diagram of restoring an abnormal DNA segment back to a normal sequence by applying mutagens during the process of DNA replication. After we obtain a patient’s genome, we compare the coding DNA segments with normal DNA segments in our database to figure out the possible range of mutated segments. Due to the redundancy in genetic codes, as long as any two DNA segments can be transcribed and then translated to the same amino acid sequence, the distance reference between them is considered to be zero. Therefore, instead of having a predetermined final state or a neighborhood of a final state, our final state lies in a set where the distance reference between any sequence in this set and the desired sequence is zero. We name this set the final desired set. The prescription is then determined by comparing the current measurement and every sequence in the desired set. Both internal noises and external disturbance can be eliminated by the measurement. We treat DNA sequences as state variables, the ON/OFF controls of all available mutagens at every spot on the given DNA segment as inputs, the measurements as the outputs, and one cell cycle as the step increment in our system equations.

The objective function is defined as a summation of the costs (including risks) of applying mutagens and the off-trajectory penalty. The optimal control sequences are computed beforehand to let doctors make treatment plans according to the patient’s condition. In general, the optimal control sequence and the corresponding optimal trajectory are not unique because the bases mutate independently in most cases and the order of mutating different bases does not matter if the number of medical treatment sessions is not under a tight restriction. Additional measurements are taken before and after each treatment, if necessary. In deterministic cases, the purpose of taking additional measurements is to check the current sequence and to eliminate both internal and external disturbances. The treatment plan is adjusted if the measurement is not the same as expected. In stochastic settings, the measurements are taken to conquer the randomness caused by both mutagens and other noises. The treatment is then updated accordingly.

To sum up, our system is a discrete-time dynamic system with finite state space and output space, and a set of ON/OFF switches as controls. Our goal is to optimally drive this system from a given initial state to a desired final set at the lowest cost.

##### 2.2. System Equations Formulation

We mainly focus on applying chemical mutagens and radiation to restore the original amino acid sequence during the process of DNA replication. Other factors that may affect the gene mutation, including temperature and electroporation, are not within our consideration. In addition, we assume that chemical mutagens or radiation target one and only one nucleotide base at any preset site, despite the technical limitation, and the results of applying chemical mutagens and radiation are independent. For simplicity, we normalize the dose of mutagens to transfer one nucleotide base to another in one step to 1. In most cases, nucleotide bases mutate independently, therefore there is no chain effect caused by mutagens. To avoid reactions among different mutagens, we require that at most one chemical mutagen and one radiation be applied in each cell cycle. While constructing a generalized model, since the order of applying chemical mutagens and radiation does not affect the results, without loss of generality, we require they be applied in the order shown in Figure 4. That is, chemical mutagens are always applied before the duplication process starts, radiation is always applied in the middle of the cell cycle, and the measurements are taken before every replication starts. Lastly, we assume that the measurements are always correct, and DNA replication error, background mutation rate, and other random noise can be eliminated from measurements by considering them as spontaneous mutation.

###### 2.2.1. Base-to-Base Deterministic Mutations

Denote the targeted DNA segment with nucleotide bases at th step by a column vector , as shown in Figure 5. is the th element of . Let be the transfer matrix from to , for all , , without mutation. Then, the perfect DNA replication process can be expressed as

Proposition 1. *.*

*Proof. *As no mutation occurs, is completely complementary to by Watson-Crick base pairing rule, and is completely complementary to . Therefore, is exactly the same as . thus,
Since every base mutates independently, every element of only depends on the corresponding element of , thus is diagonal. In addition, , we conclude .

Based on Proposition 1, we assign values to nucleotide bases set , where is an artificial nonsense base. Define an equivalence relationship between and , that is, , with

Proposition 2. * is a field under proper definitions of addition and multiplication. *

*Proof. *Defining the addition table and multiplication table as in Tables 1 and 2, we check if the set satisfies the definition of field.

*Closed under Addition and Multiplication*

Satisfied obviously from Tables 1 and 2.

*Associativity of Addition and Multiplication*

Implicitly satisfied by integer addition and multiplication.

*Commutativity of Addition and Multiplication*

Satisfied as Tables 1 and 2 are symmetric according to the diagonal.

*Additive and Multiplicative Identity*

Additive identity is 0, and multiplicative identity is 1.

*Additive and Multiplicative Inverses*

Additive inverses pair: , , .

Multiplicative inverses pair: , , .

*Distributivity of Multiplication over Addition*

Implicitly satisfied by integer addition and multiplication.

We conclude is a field under addition and multiplication defined by Tables 1 and 2.

From now on, we use to denote the field . And is the state vector representing a DNA segment with nucleotide bases, where is the set of -valued vectors of dimension .

We start with the simplest form of mutations, point mutation. Suppose there is a point mutation, we write it mathematically as where , and reduces to −1 as only one base is involved. The corresponding values of and , obtained by reverse engineering with all possible pairs of and , are listed in Table 3.

Here, represents the mutation from four normal nucleotide bases, and corresponds to mutation from nonsense base, that is, only if .

Rewriting (4) by collecting all values of and in Table 3, we getwhere , , representing the on/off controls, , , and . In (5a), and are constants for all and . and , the inputs of the system, are the on/off controls for chemical mutagens or radiation. Clearly, only if . Equation (5b) is a simplified version of (5a) as we put into vector form . and serve as vector basis for base-to-base deterministic model. and are now multi-input controls; each of them contains 5 on/off controls, corresponding to all possible transfer patterns. For a particular , at most one of s and s can be 1, as stated in Proposition 3. This is consistent with the fact that every state can be transferred to only one of the five states in the state space with corresponding mutagens available.

Proposition 3. *It is always transfer when mutation occurs, that is, one nucleotide base can only transfer to another one, therefore*(i)*if and , or , then ,*(ii)*if , then and is either 0 or a unit row vector,*(iii)*if , then and is either 0 or a unit row vector,*(iv)* is either 0 or a unit row vector, for all .*

Now, suppose for some reason we need to take an addition, measurement in the middle of the cell cycle, after the completion of the duplication and before the start of the th. We name this kind of measurement an intermediate state, and denote by it . Then, we have where the values of and , listed in Table 4, are obtained in the same way as getting and in Table 3.

Comparing Tables 3 and 4, we find the collection of and , and , form the same set, respectively. Thus, we continue using and when rewriting (6) in the form of (5a) and (5b), that is, where are the counterparts of , respectively, and are the same as in (5b).

Similar to Proposition 3, we get Proposition 4.

Proposition 4. * and in (7) need to satisfy the following conditions.*(i)*If and , or , then .*(ii)*If , then and is either 0 or a unit row vector.*(iii)*If , then and is either 0 or a unit row vector.*(iv)* is either 0 or a unit row vector, for all .*

Now take, both chemical mutagens and radiative rays under our consideration and apply them in the order as shown in Figure 4. Then, we can express our system equation aswhere and are the inputs of the system and is the measurement. Obviously, (8a) is modified from (5b) and (8b) from (7).

The two-step mutation and the intermediate state avoid the case is changed to different bases by radiation and chemical mutagens simultaneously, which causes confusion. Substituting (8a) and (8b), we getObviously, Proposition 3 still holds for and , and Proposition 4 holds for and for (9a).

For point mutations, we have 20 on/off controls in total for every step , 10 for chemical mutagens as described before, and the rest for radiation.

###### 2.2.2. Gene-to-Gene Deterministic Mutations

In general, mutations involve multiple bases. Therefore, large-scale deterministic model is necessary. Now, we show how to extend our model to large-scale systems.

Suppose we have a coding DNA segment with length , then . Since a coding DNA segment usually contains integer number of codons, which is made of three consecutive bases, is a multiple of 3. Let denote the component of . This notation is consistent with the one in Section 2.2.1. Initiated by the base-to-base deterministic model from Section 2.2.1, we write our system equation for large-scale system as where are on/off controls of the element, are square matrices corresponding to the mutations between normal bases or from normal bases by chemicals and radiation, respectively, are -dimensional column vectors representing mutations from nonsense bases by chemicals and radiation, respectively, and , .

and are diagonal matrices since each base mutates independently. The values in the first four rows of Tables 3 and 4 correspond to the diagonal elements of and , respectively. The last rows of Tables 3 and 4 are assigned to and , -dimensional vectors, at nonsense base’s spots for .

Define , a collection of matrices, where is the same as in (5a) and (5b), is the unit column vector of length with th component equal to 1 and all other components equal to 0, and is the square matrix with only the th element on the diagonal equals to 1, and 0 otherwise. Then, can be written as linear combinations of all elements from , with the coefficient of each element either 0 or 1 corresponding to the on/off control and , respectively.

Similarly, define , where is the same as (5a) and (5b). can be written as linear combinations of all components from , with coefficient of every component either 0 or 1 corresponding to the on/off control and , respectively.

Therefore, instead of using step-varying , we find matrix basis for those four square matrices to make the controls to be the only variables depending on , as we did for single-base cases. Then, we can, write (10) aswhere .

As shown in (11a), (11b), and (11c), multisites mutations contain controls in total for every step , where is the number of nucleotide bases on the targeted gene. Similar to point mutations, every single site has 20 controls in each step, 10 for chemical mutagens and 10 for radiation.

We can view as binary matrices of dimension , and , , are the corresponding element of row and column. Use , binary row vectors of dimension 5, to denote the row of , respectively. Again, .

Combining (11a) and (11b), and writing control variables in vector forms, we get

Proposition 5. *For large-scale deterministic system, satisfy conditions below.*(i)*If , then .*(ii)*If , or , then .*(iii)*For all , is either 0 or a row unit vector and .*(iv)*For all , is either 0 a row unit vector and .*(v)*For all , is either 0 or a row unit vector and .*(vi)*For all , is either 0 or a row unit vector and .*(vii)*For all , , , is either 0 or a unit row vector and is either 0 or a unit row vector.*

The mathematical model (12) is quite flexible and can be easily extended to many cases, such as transcription process, multiple spot mutations within one-step or broken DNA strands.

Take broken DNA strands as an example. DNA strand breaks due to various reasons. Our system equation can represent this phenomenon by dividing the whole system into small subsystems. Significant brokage of DNA strands is simply eliminated by cell mechanism to ensure the accuracy to DNA replication. Equation (13) shows the case of one single DNA strand breaking into two segments by chemical mutagens

###### 2.2.3. Gene-to-Gene Stochastic Mutations

In reality, mutagens, no matter chemical or radiative, always cause randomness in mutation. Therefore, we need to derive the model for gene-to-gene stochastic mutations.

Introduce new random variables, , associated with probability , , for all , , , respectively, where is the step index, are indices for chemical mutagens inducing mutation from normal bases and from , respectively, are indices for radiation inducing mutation from normal bases and from , respectively, is the index of DNA segment, and the value of corresponds to the transfer pattern, which can be found in Tables 3 and 4. Note different mutagens have different probability assignments, the probability assignments are only related to the type of mutagens, and the probability associated with every kind of mutagens sums up to 1, that is, The controls are , with the fact that 1 representing mutagen with corresponding index is applied at spot of DNA segment at generation, and 0 representing mutagen with corresponding index is not applied at spot at step, similar to Sections 2.2.1 and 2.2.2. The mutagen indices can be omitted in deterministic mutations since given the current state and control, the next state is unique. However, they are necessary for stochastic mutations, because there exist multiple possible states for the next stage given the control. In other words, the next state is determined by random variables , given the values of , and .

Suppose we have kinds of chemical mutagens available, with kinds to induce mutations from normal bases and kinds to induce mutations from . And we have kinds of radiation available, with kinds to induce mutations from normal bases and kinds to induce mutations from . Therefore, we have total controls for each spot at each step . We can write our system equation asAgain, we define the elements at row and column of binary matrices , respectively. , , and binary row vectors of dimension 5 denote the row of , respectively. Then, we can simplify (15a), (15b), and (15c) and combine (15a) and (15b) as

Proposition 6. *For large-scale stochastic system, , follow the rules below.*(i)*If , then .*(ii)*If and , then .*(iii)*If , and , then .*(iv)*For all , , ,, if , then is a unit row vector.*(v)*For all , , , , if , then is a unit row vector.*(vi)*For all , , , , if , then is a unit row vector.*(vii)*For all , , , , if , then is a unit row vector.*(viii)*For all , and , for all .*(ix)*For all , or 1 and , for all , .*(x)*For all , or 1 and , for all, .*(xi)*For all , or 1 and , for all , .*(xii)*For all , , , or 1 and or 1.*

We close this section with the definition of controllability to the system equations proposed above. DNA replication systems with system equations proposed as (9a), (9b), (12), and (16) are *completely controllable* if and only if for all , at least one path from to and at least one path from to by applying proper mutagens in the correct order, with finite.

##### 2.3. Generalized Optimal Control Problem Formulation

We first define our objective function that can be adapted to all kinds of systems proposed in Section 2.2 with minor changes. Mathematically, in systems where the controllable parameters of interest are discrete, the objective function is usually a weighted sum representing the number of times that a piece of equipment is turned “on” or “off” or the number of resources needed to execute certain tasks in the frequent cases [30]. In our case, this summation is the total number of times that different mutagens are applied weighted by the corresponding cost (including the risk). Another key factor of objective function is the off-trajectory penalty. Designing a trajectory beforehand is necessary to avoid other hidden risks. If the measurement indicates that the current state is off the predefined trajectory, we include a distance reference between current state and desired state as penalty and change the treatment plan accordingly.

Therefore, our objective function can be expressed as with , , () given. The physical meaning of , , , , , , , is the same as in Section 2.2.3. , for all, , , , , are the corresponding cost of applying chemical mutagens and radiative rays indexed , respectively. denotes the desired set at th stage, generated by the DNA sequences representing the same amino acid sequence as, the desired state at th step. And is the distance reference of the current state to the desired set at step. The final penalty, the distance reference from the final state to the desired set at , is included in the last term.

In general, , for all , , , , , because physically is a set of nonsense bases and more details are necessary to convert an back to normal bases, for instance, the cost to identify the exact element in the set . Our goal is to drive our system optimally from initial state to the desired final set by applying a sequence of mutagens indexed with , at problematic positions , and in a correct order .

In (17), the first four terms inside the expectation do not depend on random variables , , , and , for all as the treatment plan is computed based on the initial state . Given , the updated treatment plan is computed accordingly but still not related to random variables. The last term inside expectation, , is the only term in summation that depends on the distribution of the random variables.

The constraint of the optimal control problem, in general, is the system equation. We choose multidimensional stochastic system equation as the generalized constraints as it can be degenerated to one-dimensional and multidimensional deterministic cases by proper modifications.

Therefore, we can rewrite our objective function and formulate our optimal control problem as subject to We need to choose a proper distance reference to quantitatively describe the relationship between DNA segments of same length. We first define the distance reference between codons, and the distance reference between DNA segments is a weighted sum of distance reference between every pair of codons.

The distance reference between codons, , needs to fulfill the biological requirements as below.(1)Nonnegativity: the distance reference between any two codons is either positive or zero. Mathematically, , .(2)The distance reference between two codons corresponding to the same amino acid is zero.(3)Symmetry: the distance reference from codon to codon equals the distance reference from codon to codon , that is, .(4)The distance reference between two codons corresponding to different amino acids should reveal the chemical and physical differences between two amino acids.(5)The distance reference from stop codons to all other codons is much larger than those between other codons as early termination of amino acid sequences is more harmful than other forms of mutations.

All the existing metric defined on the finite field cannot achieve all the requirements above. The second requirement violates the identity of indiscernible, that is, if and only if . The redundancy in genetic codes implies if those two amino acids, and , are translated into the same amino acids. In addition, the triangular inequality is not necessarily true, according to the underlying physical meanings. We take the assumption that the stop codons are of the same distance reference from and to all other codons.

Important physical and chemical properties are listed in Table 5. We ignore codons containing since their chemical and physical properties cannot be found in literature.

From Table 5, we can see all codons are divided into different sets with each set corresponding to one amino acid. The size and the elements in one codon set vary from one amino acid to another. This implies that the costs of driving one codon to the desired final set generated by the desired final state might be different from the costs of driving the complementary codon to the desired final set generated by the complementary of desired final state. More discussions about this issue are presented in Section 3.

The distance reference between any two codons can be defined by a weighted sum of the differences between physical and chemical properties or other reasonable functions. And the distance reference between two DNA sequences is defined as the sum of distance reference between the corresponding pair of codons. The biological statics plays a crucial rule to define this distance function in practical.

An example of the distance function can be expressed as where are two amino acids.

is then assigned to with corresponding to amino acids , respectively.

Since the generalized optimal control problem in (18) and (19) is a multistage problem that can be broken down into simpler steps at different time points. Therefore, we can solve it by dynamic programming.

For dynamic programming, the optimal control policy is constructed backward. And Bellman’s principle of optimality states that the optimal policy for to is also the optimal policy for the tail problem, from to .

The tail problem is defined as The iterative update equation to find optimal policy can be expressed by (22), according to the dynamic programming algorithm in [31].

#### 3. Results and Discussion

In the following examples, we consider applying chemical mutagens only because the randomness of applying radiation is much larger and more difficult to control. We also omit the mutations between a normal base and because of high-cost and the unavailable chemical and physical properties for codons containing .

The distance reference between codons used in Sections 3.2 and 3.3 is computed by (20) with , , , , and . We only keep the final penalty but omit the off-trajectory penalty along the trajectory.

##### 3.1. Base-to-Base, Deterministic Optimal Control Problem

We define the distance reference between bases as with , where denotes the set excluding the element 0.

Our optimal control problem for point mutations is subject to with given, .

Suppose that there are 12 kinds of mutagens (), each corresponding to a specific transfer pattern as in Table 6, all available controls and the respective costs can be immediately listed as in Tables 7 and 8.

The elements along the antidiagonal of Table 7, , and , are artificially added, because the complementary transfers naturally happen and no mutagen is necessary. Thus, the costs along the antidiagonal of Table 8 are all zero, that is, . The equivalence relationship between subscription in two nucleotide bases and subscription in integer is defined by Table 6.

Under the above assumptions, we can write update equation for optimal policy explicitly as

Proposition 7. *For the same , , for all , , for all , where denotes the complementary base of . If, in addition, the system is completely controllable, , s.t. is the global minimum and for all , if 1 (mod 2), and if (mod 2). In our example, .*

*Proof. *This first part is due to the zero cost for the transfers between complementary bases in the consecutive steps.

For any , the relationship between minimal costs in consecutive steps is shown in (26). Since , is one of the four elements in the set from which the is picked. Moreover, . Therefore, is one of the four elements in the set. Since is the minimum picking for a set containing , we conclude that .

The value in our example is proved by brute force method, that is, is a guaranteed global minimum. For completely controllable systems, this always exists. The existence of implies that for without limitation in the number of steps, we can reach the global optimal in steps, 6 steps in our example.

Suppose that , is the global minimum, thus . However, according to the first part of the proposition. Therefore, for the same . Therefore, is also a global minimum.

By backward induction, suppose for , the statement is true, that is, is the global optimal either from or to . Obviously, for , the statement is still true. Therefore, , for all , .

In the proof of global minimum that can be reached in the finite step in Proposition 7, we also discover Proposition 8. Here, denotes the optimal cost from to .

Proposition 8. *Given two single base mutation optimal control problems, with the same fixed , with and desired final states complementary to each other. If is the global minimum, then is also the global minimum, that is, the global minimum of both systems is reach at the same stage . Moreover, for all , ,
*

Physically, Proposition 8 states that the optimal can be achieve at the same step from a pair of complementary bases to another pair of complementary bases at the same cost. However, this fact is true only for base-to-base deterministic mutations, because the distance reference is well defined by (23).

Now, we show an example with simulation results.

The costs of applying different mutagens are listed in Table 9. It is a numerical assignment to Table 8. Since we apply mutagens before the replication starts, actually transfer to and then to by replication. For simplicity, we just use the th and th step states as subscripts to represent the corresponding control and cost. The costs of transitions are lower than the costs of transversions. Therefore, is smaller than other mutagens, except artificial ones.

If we use to denote the costs of mutagens as listed in Table 9, then

Running the dynamic programming for every pair of , . Here, we sightly modify our notation. We use to denote the optimal cost from to . Then,

The path to reach the optimal cost is denoted by

where is the th state from to , that is, .

The simulation results are shown as below, including optimal costs for all possible transfer pairs , , , graphical representation in Figure 6, and optimal path , .