Optimal Control of Gene Mutation in DNA Replication

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.

Figure 1 

Typical analysis of biological systems

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.

Figure 2 

Systems biology is a cross-cutting research area connecting control engineering, biology, and medical science. Sources: protein synthesis http://www.anticancer.de/, liposome [8], corneal transplant http://www.avclinic.com/, microarray hybridization [9], cuvettes for electroportation http://www.en.wikipedia.org/, biochip http://www.clemson.edu/, nano robot http://www.molecularlab.it/.

According to their scales, biological systems can be divided into three levels: the molecular level (nm), cellular level (μm), 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 (𝐴), thymine (𝑇), guanine (𝐺), and cytosine (𝐶), 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]. 𝐴 always pairs with 𝑇, 𝐺 always pairs with 𝐶, and vice versa. In nature, replication errors occur at a very low rate, one error for every 10⁷ 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 (𝐴↔𝐺 or 𝐶↔𝑇) and transversions (𝐴/𝐺↔𝐶/𝑇). 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.

Figure 3 

System diagram of restoring an abnormal DNA segment back to a normal sequence by applying mutagens during the process of DNA replication.

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.

Figure 4 

The order of taking measurements and applying chemical mutagens and radiation in a cell cycle.

2.2.1. Base-to-Base Deterministic Mutations

Denote the targeted DNA segment with 𝑛 nucleotide bases at 𝑘th step by a column vector 𝑥_𝑘, as shown in Figure 5. 𝑥^𝑖_𝑘 is the 𝑖th element of 𝑥_𝑘. Let 𝑃 be the transfer matrix from 𝑥_𝑘 to 𝑥_𝑘+1, for all 𝑘, 𝑘∈ℤ⁺∪{0}, without mutation. Then, the perfect DNA replication process can be expressed as𝑥_𝑘+1=𝑃𝑥_𝑘.(1)

Figure 5 

Biological information flow in central dogma of molecular biology.

Proposition 1. 𝑃=−𝐼.

Proof. As no mutation occurs, 𝑥_𝑘+1 is completely complementary to 𝑥_𝑘 by Watson-Crick base pairing rule, and 𝑥_𝑘+2 is completely complementary to 𝑥_𝑘+1. Therefore, 𝑥_𝑘+2 is exactly the same as 𝑥_𝑘. thus, 𝑥_𝑘+2=𝑃𝑥_𝑘+1=𝑃²𝑥_𝑘⟹𝑃²=𝐼.(2) Since every base mutates independently, every element of 𝑥^𝑖_𝑘+1 only depends on the corresponding element of 𝑥^𝑖_𝑘, thus 𝑃 is diagonal. In addition, 𝑥_𝑘+1≠𝑥_𝑘, we conclude 𝑃=−𝐼.

Based on Proposition 1, we assign values to nucleotide bases set {𝐴,𝐺,𝐶,𝑇,𝑂}, where 𝑂 is an artificial nonsense base. Define an equivalence relationship between {𝐴,𝐺,𝐶,𝑇,𝑂} and {1,2,−2,−1,0}, that is, {𝐴,𝐺,𝐶,𝑇,𝑂}⇔{1,2,−2,−1,0}, with𝑥^𝑖_𝑘=⎧⎪⎪⎪⎨⎪⎪⎪⎩1,if𝐴,2,if𝐺,−2,if𝐶,−1,if𝑇,0,if𝑂.(3)

Proposition 2. {1,2,−2,−1,0} 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 {1,2,−2,−1,0} satisfies the definition of field.

Table 1 

Addition table for {1,2,−2,−1,0}.

Table 2 

Multiplication table for {1,2,−2,−1,0}.

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: 1↔−1, 2↔−2, 0↔0.
Multiplicative inverses pair: 1↔1, 2↔−2, −1↔−1.

Distributivity of Multiplication over Addition
Implicitly satisfied by integer addition and multiplication.

We conclude {0,1,2,−2,−1} is a field under addition and multiplication defined by Tables 1 and 2.

From now on, we use ℱ to denote the field {0,1,2,−2,−1}. And 𝑥_𝑘∈ℱ^𝑛 is the state vector representing a DNA segment with 𝑛 nucleotide bases, where ℱ^𝑛 is the set of ℱ-valued vectors of dimension 𝑛.

We start with the simplest form of mutations, point mutation. Suppose there is a point mutation, we write it mathematically as𝑥_𝑘+1=(−𝐼+Δ𝑠)𝑥_𝑘+Δ𝑤,(4) where 𝑥_𝑘+1,𝑥_𝑘∈ℱ, and −𝐼 reduces to −1 as only one base is involved. The corresponding values of Δ𝑠 and Δ𝑤, obtained by reverse engineering with all possible pairs of 𝑥_𝑘 and 𝑥_𝑘+1, are listed in Table 3.

Table 3 

Possible values of Δ𝑠 and Δ𝑤. The corresponding values of Δ𝑠 and Δ𝑤 are obtained by substituting the value of 𝑥_𝑘 and 𝑥_𝑘+1 into (4), with Δ𝑤≠0 only if 𝑥_𝑘=0.

Here, Δ𝑠 represents the mutation from four normal nucleotide bases, and Δ𝑤 corresponds to mutation from nonsense base, that is, Δ𝑤≠0 only if 𝑥_𝑘=0.

Rewriting (4) by collecting all values of Δ𝑠 and Δ𝑤 in Table 3, we get𝑥_𝑘+1=⎛⎜⎝−𝐼+⁴_𝑗=0𝑢^𝑗_𝑘𝑠_𝑗⎞⎟⎠𝑥_𝑘+⁴_𝑗=0𝑐^𝑗_𝑘𝑤_𝑗(5a)=−𝐼+𝑢_𝑘𝑠𝑥_𝑘+𝑐_𝑘𝑤,(5b)where {𝑠₀,𝑠₁,𝑠₂,𝑠₃,𝑠₄}={𝑤₀,𝑤₁,𝑤₂,𝑤₃,𝑤₄}={0,1,2,−2,−1}, 𝑢^𝑗_𝑘,𝑐^𝑗_𝑘∈{0,1}, representing the on/off controls, 𝑢_𝑘=[𝑢⁰_𝑘𝑢¹_𝑘𝑢²_𝑘𝑢³_𝑘𝑢⁴_𝑘], 𝑐_𝑘=[𝑐⁰_𝑘𝑐¹_𝑘𝑐²_𝑘𝑐³_𝑘𝑐⁴_𝑘], and 𝑠=𝑤=[012−2−1]^𝑇. In (5a), 𝑠_𝑗 and 𝑤_𝑗 are constants for all 𝑘 and 𝑗. 𝑢^𝑗_𝑘 and 𝑐^𝑗_𝑘, the inputs of the system, are the on/off controls for chemical mutagens or radiation. Clearly, ∑⁴_𝑗=0𝑐^𝑗_𝑘=1 only if 𝑥_𝑘=0. Equation (5b) is a simplified version of (5a) as we put 𝑢^𝑗_𝑘,𝑠_𝑗,𝑐^𝑗_𝑘,𝑤_𝑗 into vector form 𝑢_𝑘,𝑠,𝑐_𝑘,𝑤. 𝑠 and 𝑤 serve as vector basis for base-to-base deterministic model. 𝑢_𝑘 and 𝑐_𝑘 are now multi-input controls; each of them contains 5 on/off controls, corresponding to all possible transfer patterns. For a particular 𝑘, at most one of 𝑢^𝑗_𝑘s and 𝑐^𝑗_𝑘s 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 ℱ with corresponding mutagens available.

Proposition 3. It is always 1−1 transfer when mutation occurs, that is, one nucleotide base can only transfer to another one, therefore(i)if 𝑥_𝑘=0 and 𝑐_𝑘=0, or 𝑐_𝑘=[10000], then 𝑥_𝑘+1=0,(ii)if 𝑥_𝑘≠0, then 𝑐_𝑘=0 and 𝑢_𝑘 is either 0 or a unit row vector,(iii)if 𝑥_𝑘=0, then 𝑢_𝑘=0 and 𝑐_𝑘 is either 0 or a unit row vector,(iv)𝑢_𝑘+𝑐_𝑘 is either 0 or a unit row vector, for all 𝑘∈ℤ⁺∪{0}.

Now, suppose for some reason we need to take an addition, measurement in the middle of the cell cycle, after the completion of the 𝑘th duplication and before the start of the (𝑘+1)th. We name this kind of measurement an intermediate state, and denote by it 𝑥^_𝑘. Then, we have𝑥_𝑘+1=𝐼+Δ𝑠^𝑥^_𝑘+Δ𝑤^,(6) where the values of Δ𝑠^ and Δ𝑤^, listed in Table 4, are obtained in the same way as getting Δ𝑠 and Δ𝑤 in Table 3.

Table 4 

Possible values of Δ𝑠^ and Δ𝑤^. The corresponding values of Δ𝑠^ and Δ𝑤^ are obtained by substituting the value of 𝑥^_𝑘 and 𝑥_𝑘+1 into (4), with Δ𝑤^≠0 only if 𝑥^_𝑘=0.

Comparing Tables 3 and 4, we find the collection of Δ𝑠 and Δ𝑠^, Δ𝑤 and Δ𝑤^, form the same set, respectively. Thus, we continue using 𝑠 and 𝑤 when rewriting (6) in the form of (5a) and (5b), that is, 𝑥_𝑘+1=𝐼+𝑣_𝑘𝑠𝑥^_𝑘+𝑐^_𝑘𝑤,(7) where 𝑣_𝑘,𝑐^_𝑘 are the counterparts of 𝑢_𝑘,𝑐_𝑘, respectively, and 𝑠,𝑤 are the same as in (5b).

Similar to Proposition 3, we get Proposition 4.

Proposition 4. 𝑣_𝑘 and 𝑐^_𝑘 in (7) need to satisfy the following conditions.(i)If 𝑥_𝑘=0 and 𝑐^_𝑘=0, or 𝑐^_𝑘=[10000], then 𝑥_𝑘+1=0.(ii)If 𝑥_𝑘≠0, then 𝑐^_𝑘=0 and 𝑣_𝑘 is either 0 or a unit row vector.(iii)If 𝑥_𝑘=0, then 𝑣_𝑘=0 and 𝑐_𝑘 is either 0 or a unit row vector.(iv)𝑣_𝑘+𝑐^_𝑘 is either 0 or a unit row vector, for all 𝑘∈ℤ⁺∪{0}.

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 as𝑥^_𝑘=⎛⎜⎜⎜⎝−𝐼+𝑢_𝑘𝑠_{mutationscausedbychemicalmutagensfromnormalbases}⎞⎟⎟⎟⎠𝑥_𝑘+𝑐_𝑘𝑤_{mutationscausedbychemicalmutagensfrom𝑂},(8a)𝑥_𝑘+1=⎛⎜⎜⎜⎝𝐼+𝑣_𝑘𝑠_{mutationscausedbyradiativeraysfromnormalbases}⎞⎟⎟⎟⎠𝑥^_𝑘+𝑐^_𝑘𝑤_{mutationscausedbyradiativeraysfrom𝑂},(8b)𝑦_𝑘=𝑥_𝑘,(8c)where 𝑢_𝑘 and 𝑣_𝑘 are the inputs of the system and 𝑦_𝑘 is the measurement. Obviously, (8a) is modified from (5b) and (8b) from (7).

The two-step mutation and the intermediate state 𝑥^_𝑘avoid the case 𝑥_𝑘 is changed to different bases by radiation and chemical mutagens simultaneously, which causes confusion. Substituting (8a) and (8b), we get𝑥_𝑘+1=𝐼+𝑣_𝑘𝑠−𝐼+𝑢_𝑘𝑠𝑥_𝑘+𝐼+𝑣_𝑘𝑠𝑐_𝑘𝑤+𝑐^_𝑘𝑤,(9a)𝑦_𝑘=𝑥_𝑘.(9b)Obviously, Proposition 3 still holds for 𝑢_𝑘 and 𝑐_𝑘, and Proposition 4 holds for 𝑣_𝑘 and 𝑐^_𝑘 for (9a).

For point mutations, we have 20 on/off controls in total for every step 𝑘, 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 𝑛, then 𝑥_𝑘∈ℱ^𝑛. Since a coding DNA segment usually contains integer number of codons, which is made of three consecutive bases, 𝑛 is a multiple of 3. Let 𝑥^𝑖_𝑘 denote the 𝑖th component of 𝑥_𝑘. 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 𝑥^_𝑘=⎛⎜⎜⎜⎜⎜⎜⎝−𝐼+^𝑛_𝑖=1𝑢^𝑖_𝑘𝑆^𝑖_{𝑘mutationscausedbychemicalmutagensfromnormalbases}⎞⎟⎟⎟⎟⎟⎟⎠𝑥_𝑘+_{𝑖∈𝒪𝑘}𝑐^𝑖_𝑘𝑊^𝑖_{𝑘mutationscausedbychemicalmutagensfrom𝑂},𝑥_𝑘+1=⎛⎜⎜⎜⎜⎜⎝𝐼+^𝑛_𝑖=1𝑣^𝑖_𝑘𝑄^𝑖_{𝑘mutationscausedbyradiativeraysfromnormalbases}⎞⎟⎟⎟⎟⎟⎠𝑥^_𝑘+_{𝑖∈𝒪𝑘}𝑏^𝑖_𝑘𝑅^𝑖_{𝑘mutationscausedbyradiativeraysfrom𝑂},𝑦_𝑘=𝑥_𝑘,(10) where 𝑢^𝑖_𝑘,𝑣^𝑖_𝑘,𝑐^𝑖_𝑘,𝑏^𝑖_𝑘 are on/off controls of the 𝑖th element,𝑆^𝑖_𝑘,𝑄^𝑖_𝑘 are 𝑛×𝑛 square matrices corresponding to the mutations between normal bases or from normal bases by chemicals and radiation, respectively, 𝑊^𝑖_𝑘,𝑅^𝑖_𝑘 are 𝑛-dimensional column vectors representing mutations from nonsense bases by chemicals and radiation, respectively, and 𝒪_𝑘={𝑖∶𝑥^𝑖_𝑘=0,1≤𝑖≤𝑛}, 𝒪^_𝑘={𝑖∶𝑥^𝑖_𝑘=0,1≤𝑖≤𝑛}.

𝑆^𝑖_𝑘 and 𝑄^𝑖_𝑘 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 𝑆^𝑖_𝑘 and 𝑄^𝑖_𝑘, respectively. The last rows of Tables 3 and 4 are assigned to 𝑊^𝑖_𝑘 and 𝑅^𝑖_𝑘, 𝑛-dimensional vectors, at nonsense base’s spots for 𝑥_𝑘.

Define 𝒮={𝑠_𝑗𝑒_𝑖𝑒^𝑇_𝑖,∀𝑖,𝑗,0≤𝑗≤4,1≤𝑖≤𝑛}, a collection of 𝑛×𝑛 matrices, where 𝑠_𝑗 is the same as in (5a) and (5b), 𝑒_𝑖 is the unit column vector of length 𝑛 with 𝑖th component equal to 1 and all other components equal to 0, and 𝑒_𝑖𝑒^𝑇_𝑖 is the square matrix with only the 𝑖th element on the diagonal equals to 1, and 0 otherwise. Then, 𝑆^𝑖_𝑘,𝑄^𝑖_𝑘 can be written as linear combinations of all elements from 𝒮, with the coefficient of each element either 0 or 1 corresponding to the on/off control 𝑢^(𝑖,𝑗)_𝑘 and 𝑣^(𝑖,𝑗)_𝑘, respectively.

Similarly, define 𝒲={𝑤_𝑗𝑒_𝑖,∀𝑖,𝑗,0≤𝑗≤4,1≤𝑖≤𝑛}, where 𝑤_𝑗 is the same as (5a) and (5b). 𝑊^𝑖_𝑘,𝑅^𝑖_𝑘 can be written as linear combinations of all components from 𝒲, with coefficient of every component either 0 or 1 corresponding to the on/off control 𝑐^(𝑖,𝑗)_𝑘 and 𝑐^{(𝑖,𝑗)}_𝑘, respectively.

Therefore, instead of using step-varying 𝑆^𝑖_𝑘,𝑆^𝑖_𝑘,𝑊^𝑖_𝑘,𝑅^𝑖_𝑘, we find matrix basis for those four square matrices to make the controls to be the only variables depending on 𝑘, as we did for single-base cases. Then, we can, write (10) as𝑥^_𝑘=⎛⎜⎝−𝐼+^𝑛_𝑖=1⁴_𝑗=0𝑢^(𝑖,𝑗)_𝑘𝑠_𝑗𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥_𝑘+_{𝑖∈𝒪𝑘}⁴_𝑗=0𝑐^(𝑖,𝑗)_𝑘𝑤_𝑗𝑒_𝑖,(11a)𝑥_𝑘+1=⎛⎜⎝𝐼+^𝑛_𝑖=1⁴_𝑗=0𝑣^(𝑖,𝑗)_𝑘𝑠_𝑗𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥^_𝑘+_{𝑖∈𝒪𝑘}⁴_𝑗=0𝑐^{(𝑖,𝑗)}_𝑘𝑤_𝑗𝑒_𝑖,(11b)𝑦_𝑘=𝑥_𝑘,(11c)where 𝑢^(𝑖,𝑗)_𝑘,𝑣^(𝑖,𝑗)_𝑘,𝑐^(𝑖,𝑗)_𝑘,𝑐^{(𝑖,𝑗)}_𝑘∈{0,1}.

As shown in (11a), (11b), and (11c), multisites mutations contain 20𝑛 controls in total for every step 𝑘, where 𝑛 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 𝑢_𝑘,𝑣_𝑘,𝑐_𝑘,𝑐^_𝑘 as binary matrices of dimension 𝑛×5, and 𝑢^(𝑖,𝑗)_𝑘, 𝑣^(𝑖,𝑗)_𝑘𝑐^(𝑖,𝑗)_𝑘, 𝑐^{(𝑖,𝑗)}_𝑘are the corresponding element of 𝑖th row and 𝑗th column. Use 𝑢^𝑖_𝑘,𝑣^𝑖_𝑘,𝑐^𝑖_𝑘,𝑏^𝑖_𝑘, binary row vectors of dimension 5, to denote the 𝑖th row of 𝑢_𝑘,𝑣_𝑘,𝑐_𝑘,𝑐^_𝑘, respectively. Again, 𝑠=𝑤=[012−2−1]^𝑇.

Combining (11a) and (11b), and writing control variables in vector forms, we get𝑥_𝑘+1=⎛⎜⎝𝐼+^𝑛_𝑖=1𝑣^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠⎛⎜⎝−𝐼+^𝑛_𝑖=1𝑢^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥_𝑘+⎛⎜⎝𝐼+^𝑛_𝑖=1𝑣^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠_𝑖_∈𝒪𝑘𝑐^𝑖_𝑘𝑤𝑒_𝑖+_{𝑖∈𝒪𝑘}𝑏^𝑖_𝑘𝑤𝑒_𝑖,𝑦_𝑘=𝑥_𝑘.(12)

Proposition 5. For large-scale deterministic system, 𝑢_𝑘,𝑣_𝑘,𝑐_𝑘,𝑐^_𝑘 satisfy conditions below.(i)If 𝑒^𝑇_𝑖𝑥_𝑘=0, then 𝑖∈𝒪_𝑘.(ii)If 𝑒^𝑇_𝑖𝑥_𝑘=0, 𝑐ⁱ_𝑘=0 or 𝑐^𝑖_𝑘=[10000], then 𝑖∈𝒪^_𝑘.(iii)For all 𝑖∉𝒪_𝑘, 𝑢^𝑖_𝑘 is either 0 or a row unit vector and 𝑐^𝑖_𝑘=0.(iv)For all 𝑖∈𝒪_𝑘, 𝑐^𝑖_𝑘 is either 0 a row unit vector and 𝑢^𝑖_𝑘=0.(v)For all 𝑖∉𝒪^_𝑘, 𝑣^𝑖_𝑘 is either 0 or a row unit vector and 𝑏^𝑖_𝑘=0.(vi)For all 𝑖∈𝒪^_𝑘, 𝑏^𝑖_𝑘 is either 0 or a row unit vector and 𝑣^𝑖_𝑘=0.(vii)For all 𝑖,𝑘, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, 𝑢^𝑖_𝑘+𝑐^𝑖_𝑘 is either 0 or a unit row vector and 𝑣^𝑖_𝑘+𝑏^𝑖_𝑘 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 ⎛⎜⎝𝑥^_𝑘(1)𝑥^_𝑘(2)⎞⎟⎠=⎛⎜⎜⎜⎜⎜⎝−𝐼_𝑚+∑^𝑚_𝑖=1𝑢^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖00−𝐼_𝑛−𝑚+∑^𝑛_𝑖=𝑚+1𝑢^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎟⎟⎟⎟⎠×𝑥_𝑘(1)𝑥_𝑘(2)+⎛⎜⎜⎝_{𝑖∈𝒪𝑘}∑_{,1≤𝑖≤𝑚}𝑐^𝑖_𝑘𝑤𝑒_{𝑖𝑖∈𝒪k,(𝑚}∑_{+1)≤𝑖≤𝑛}𝑐^𝑖_𝑘𝑤𝑒_𝑖⎞⎟⎟⎠,𝑥_𝑘+1(1)=⎛⎜⎝𝐼_𝑚+^𝑚_𝑖=1𝑣^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥^_𝑘(1)+_{𝑖∈𝒪𝑘,1≤𝑖≤𝑚}𝑏^𝑖_𝑘𝑤𝑒_𝑖,𝑥_𝑘+1(2)=⎛⎜⎝𝐼_𝑛−𝑚+^𝑛_𝑖=𝑚+1𝑣^𝑖_𝑘𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥^_𝑘(2)+_{𝑖∈𝒪𝑘,(𝑚+1)≤𝑖≤𝑛}𝑏^𝑖_𝑘𝑤𝑒_𝑖.(13)

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, ℎ^(𝑖,𝑗)_𝑘,𝑙1,𝑟^(𝑖,𝑗)_𝑘,𝑙2,ℎ^{(𝑖,𝑗)}_𝑘,𝑙3,𝑟^{(𝑖,𝑗)}_𝑘,𝑙4∈{0,1}, associated with probability 𝑝^(ℎ)_𝑙1,𝑗, 𝑝^(𝑟)_𝑙2,𝑗,𝑝^(ℎ)_𝑙3,𝑗,𝑝^(𝑟)_𝑙4,𝑗, for all 𝑖,𝑘, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, respectively, where 𝑘 is the step index, 𝑙₁,𝑙₂ are indices for chemical mutagens inducing mutation from normal bases and from 𝑂, respectively, 𝑙₃,𝑙₄ are indices for radiation inducing mutation from normal bases and from 𝑂, respectively, 𝑖 is the index of DNA segment, and the value of 𝑗 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, ⁴_𝑗=0𝑝^(ℎ)_𝑙1,𝑗=1,∀𝑙₁,1≤𝑙₁≤𝑙,₄_𝑗=0𝑝^(𝑟)_𝑙2,𝑗=1,∀𝑙₂,1≤𝑙₂≤𝑚,₄_𝑗=0𝑝^(ℎ)_𝑙3,𝑗=1,∀𝑙₃,1≤𝑙₃≤𝑙^,₄_𝑗=0𝑝^(𝑟)_𝑙4,𝑗=1,∀𝑙₄,1≤𝑙₄≤𝑚^.(14) The controls are 𝑢^𝑖_𝑘,𝑙1,𝑐^𝑖_𝑘,𝑙2,𝑣^𝑖_𝑘,𝑙3,𝑏^𝑖_𝑘,l4∈{0,1}, with the fact that 1 representing mutagen with corresponding index is applied at 𝑖th spot of DNA segment at 𝑘th generation, and 0 representing mutagen with corresponding index is not applied at spot 𝑖 at 𝑘th step, similar to Sections 2.2.1 and 2.2.2. The mutagen indices 𝑙₁,𝑙₂,𝑙₃,𝑙₄ 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 ℎ^(𝑖,𝑗)_𝑘,𝑙1,𝑟^(𝑖,𝑗)_𝑘,𝑙2,ℎ^{(𝑖,𝑗)}_𝑘,𝑙3,𝑟^{(𝑖,𝑗)}_𝑘,𝑙4, given the values of 𝑢^𝑖_𝑘,𝑙1,𝑐^𝑖_𝑘,𝑙2,𝑣^𝑖_𝑘,𝑙3,𝑏^𝑖_𝑘,𝑙4, and 𝑥_𝑘.

Suppose we have (𝑙+𝑚) kinds of chemical mutagens available, with 𝑙 kinds to induce mutations from normal bases and 𝑚 kinds to induce mutations from 𝑂. And we have (𝑙^+𝑚^) kinds of radiation available, with 𝑙^ kinds to induce mutations from normal bases and 𝑚^ kinds to induce mutations from 𝑂. Therefore, we have total (𝑙+𝑚+𝑙^+𝑚^) controls for each spot 𝑖 at each step 𝑘. We can write our system equation as𝑥^_𝑘=⎛⎜⎜⎜⎜⎜⎜⎝−𝐼+^𝑙_𝑙1=1^𝑛_𝑖=1𝑢^{𝑖𝑘,𝑙}₁⁴_𝑗=0ℎ^(𝑖,𝑗)_𝑘,𝑙1𝑠_𝑗𝑒_𝑖𝑒^𝑇_{𝑖mutationscausedbychemicalmutagensfromnormalbases}⎞⎟⎟⎟⎟⎟⎟⎠𝑥_𝑘+^𝑚_𝑙2=1_{𝑖∈𝒪𝑘}𝑐^{𝑖𝑘,𝑙}₂⁴_𝑗=0𝑟^(𝑖,𝑗)_𝑘,𝑙2𝑤_𝑗𝑒_{𝑖mutationscausedbychemicalmutagensfrom𝑂},(15a)𝑥_𝑘+1=⎛⎜⎜⎜⎜⎜⎜⎝𝐼+^𝑙_𝑙3=1^𝑛_𝑖=1𝑣^{𝑖𝑘,𝑙}₃⁴_𝑗=0ℎ^{(𝑖,𝑗)}_𝑘,𝑙3𝑠_𝑗𝑒_𝑖𝑒^𝑇_{𝑖mutationscausedbyradiativeraysfromnormalbases}⎞⎟⎟⎟⎟⎟⎟⎠𝑥^_𝑘+^𝑚_𝑙4=1_{𝑖∈𝒪𝑘}𝑏^{𝑖𝑘,𝑙}₄⁴_𝑗=0𝑟^{(𝑖,𝑗)}_𝑘,𝑙4𝑤_𝑗𝑒_{𝑖mutationscausedbyradiativeraysfrom𝑂},(15b)𝑦_𝑘=𝑥_𝑘.(15c)Again, we define ℎ^(𝑖,𝑗)_𝑘,𝑙1,𝑟^(𝑖,𝑗)_𝑘,𝑙2,ℎ^{(𝑖,𝑗)}_𝑘,𝑙3,𝑟^{(𝑖,𝑗)}_𝑘,𝑙4 the elements at 𝑖th row and 𝑗th column of 𝑛×5 binary matrices ℎ_𝑘,𝑙1,𝑟_𝑘,𝑙2,ℎ^_𝑘,𝑙3,𝑟^_𝑘,𝑙4, respectively. ℎ^𝑖_𝑘,𝑙1,𝑟^𝑖_𝑘,𝑙2,ℎ^𝑖_𝑘,𝑙3, 𝑟^𝑖_𝑘,𝑙4, and binary row vectors of dimension 5 denote the 𝑖th row of ℎ_𝑘,𝑙1,𝑟_𝑘,𝑙2,ℎ^_𝑘,𝑙3,𝑟^_𝑘,𝑙4, respectively. Then, we can simplify (15a), (15b), and (15c) and combine (15a) and (15b) as𝑥_𝑘+1=⎛⎜⎜⎝𝐼+^𝑙_𝑙3=1^𝑛_𝑖=1𝑣^𝑖_𝑘,𝑙3ℎ^𝑖_𝑘,𝑙3𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎟⎠×⎛⎜⎝−𝐼+^𝑙_𝑙1=1^𝑛_𝑖=1𝑢^𝑖_𝑘,𝑙1ℎ^𝑖_𝑘,𝑙1𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥_𝑘+⎛⎜⎜⎝𝐼+^𝑙_𝑙3=1^𝑛_𝑖=1𝑣^𝑖_𝑘,𝑙3ℎ^𝑖_𝑘,𝑙3𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎟⎠^𝑚_𝑙₂₌₁_{𝑖∈𝒪𝑘}𝑐^𝑖_𝑘,𝑙₂𝑟^𝑖_𝑘,𝑙2𝑤𝑒_𝑖+^𝑚_𝑙4=1_{𝑖∈𝒪𝑘}𝑏^𝑖_𝑘,𝑙4𝑟^𝑖_𝑘,𝑙4𝑤𝑒_𝑖,𝑦_𝑘=𝑥_𝑘.(16)

Proposition 6. For large-scale stochastic system, 𝑢^𝑖_𝑘,𝑙1,ℎ^𝑖_𝑘,𝑙1,𝑐^𝑖_𝑘,𝑙2,𝑟^𝑖_𝑘,𝑙2,𝑣^𝑖_𝑘,𝑙3,ℎ^𝑖_𝑘,𝑙3,𝑏^𝑖_𝑘,𝑙4,𝑟^𝑖_𝑘,𝑙4 follow the rules below.(i)If 𝑒^𝑇_𝑖𝑥_𝑘=0, then 𝑖∈𝒪_𝑘.(ii)If 𝑒^𝑇_𝑖𝑥_𝑘=0 and ∑^𝑚_𝑙2=1𝑐^𝑖_𝑘,𝑙2=0, then 𝑖∈𝒪^_𝑘.(iii)If 𝑒^𝑇_𝑖𝑥_𝑘=0, ∑^𝑚_𝑙2=1𝑐^𝑖_𝑘,𝑙2=1 and 𝑟^𝑖_𝑘,𝑙2=[10000], then 𝑖∈𝒪^_𝑘.(iv)For all 𝑖,𝑘,𝑙₁, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0},1≤𝑙₁≤𝑙, if 𝑢^𝑖_𝑘,𝑙1=1, then ℎ^𝑖_𝑘,𝑙1 is a unit row vector.(v)For all 𝑖,𝑘,𝑙₂, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, 1≤𝑙₂≤𝑚, if 𝑐^𝑖_𝑘,𝑙2=1, then 𝑟^𝑖_𝑘,𝑙2 is a unit row vector.(vi)For all 𝑖,𝑘,𝑙₃, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, 1≤𝑙₃≤𝑙^, if 𝑣^𝑖_𝑘,𝑙3=1, then ℎ^𝑖_𝑘,𝑙3 is a unit row vector.(vii)For all 𝑖,𝑘,𝑙₄, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, 1≤𝑙₄≤𝑚^, if 𝑏^𝑖_𝑘,𝑙4=1, then r^𝑖_𝑘,𝑙4 is a unit row vector.(viii)For all 𝑖∉𝒪_𝑘, ∑^𝑙_𝑙1=1𝑢^𝑖_𝑘,𝑙1=0or1 and 𝑐^𝑖_𝑘,𝑙2=0, for all 𝑙₂,1≤𝑙₂≤𝑚.(ix)For all 𝑖∈𝒪_𝑘, ∑^𝑚_𝑙2=1𝑐^𝑖_𝑘,𝑙2=0 or 1 and 𝑢^𝑖_𝑘,𝑙1=0, for all 𝑙₁, 1≤𝑙₁≤𝑙.(x)For all 𝑖∉𝒪^_𝑘, ∑^𝑙_𝑙3=1𝑣^𝑖_𝑘,𝑙3=0 or 1 and 𝑏^𝑖_𝑘,𝑙4=0, for all𝑙₄, 1≤𝑙₄≤𝑚^.(xi)For all 𝑖∈𝒪^_𝑘, ∑^𝑚_𝑙4=1𝑏^𝑖_𝑘,𝑙4=0 or 1 and 𝑣^𝑖_𝑘,𝑙3=0, for all 𝑙₃, 1≤𝑙₃≤𝑙^.(xii)For all 𝑖,𝑘, 1≤𝑖≤𝑛, 𝑘∈ℤ⁺∪{0}, ∑^𝑙_𝑙1=1𝑢^𝑖_𝑘,𝑙1+∑^𝑚_𝑙2=1𝑐^𝑖_𝑘,𝑙2=0 or 1 and ∑^𝑙_𝑙3=1𝑣^𝑖_𝑘,𝑙3+∑^𝑚_𝑙4=1𝑏^𝑖_𝑘,𝑙4=0 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 𝑥₀,𝑥_2𝑘1,𝑥_2𝑘2+1∈ℱ, 𝑘₁,𝑘₂∈ℤ⁺∪{0},∃ at least one path from 𝑥₀ to 𝑥_2𝑘1 and at least one path from 𝑥₀ to 𝑥_2𝑘2+1 by applying proper mutagens in the correct order, with 𝑘₁,𝑘₂ 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 𝐽₀𝑥₀=min_{𝑢,𝑐,𝑣,𝑐}𝔼_{ℎ,ℎ,𝑟,𝑟}⎡⎢⎢⎢⎢⎣^𝑁−1_𝑘=0^𝑙_𝑙1=1^𝑛_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑘,𝑙1+^𝑁−1_𝑘=0^𝑚_𝑙2=1^𝑛_𝑖=1𝛽_𝑙2𝑐^𝑖_{𝑘,𝑙2costofapplyingchemicalmutagens}+^𝑁−1_𝑘=0^𝑙_𝑙3=1^𝑛_𝑖=1𝛼^_𝑙3𝑣^𝑖_𝑘,𝑙3+^𝑁−1_𝑘=0^𝑚_𝑙4=1^𝑛_𝑖=1𝛽^_𝑙4𝑏^𝑖_{𝑘,𝑙4costofapplyingradiativerays}+^𝑁_𝑘=0𝑑𝑥_𝑘,𝑥^𝑑_𝑘_{tracingcost}⎤⎥⎥⎥⎥⎦,(17) with 𝑥₀,𝑥^𝑑_𝑘∈ℱ^𝑛, 1≤𝑘≤𝑁, 𝑛≡0 (mod3) given. The physical meaning of 𝑢^𝑖_𝑘,𝑙1, 𝑐^𝑖_𝑘,𝑙2, 𝑣^𝑖_𝑘,𝑙3, 𝑏^𝑖_𝑘,𝑙4, 𝑙₁, 𝑙₂, 𝑙₃, 𝑙₄ is the same as in Section 2.2.3. 𝛼_𝑙1,𝛽_𝑙2,𝛼^_𝑙3,𝛽^_𝑙4∈ℝ, for all𝑙₁,𝑙₂,𝑙₃,𝑙₄, 1≤𝑙₁≤𝑙, 1≤𝑙₂≤𝑚, 1≤𝑙₃≤𝑙^, 1≤𝑙₄≤𝑚^, are the corresponding cost of applying chemical mutagens and radiative rays indexed 𝑙₁,𝑙₂,𝑙₃,𝑙₄, respectively. {𝑥^𝑑_𝑘}∶ℱ^𝑛×ℱ^𝑛→ℝ⁺∪{0} denotes the desired set at 𝑘th stage, generated by the DNA sequences representing the same amino acid sequence as𝑥^𝑑_𝑘, the desired state at 𝑘th step. And 𝑑(𝑥_𝑘,{𝑥^𝑑_𝑘}) is the distance reference of the current state 𝑥_𝑘 to the desired set {𝑥^𝑑_𝑘} at 𝑘th step. The final penalty, the distance reference from the final state to the desired set at 𝑘=𝑁, is included in the last term.

In general, 𝛽_𝑙2,𝛽^_𝑙4≪𝛼_𝑙1,𝛼^_𝑙3, for all 𝑙₁,𝑙₂,𝑙₃,𝑙₄, 1≤𝑙₁≤𝑙, 1≤𝑙₂≤𝑚, 1≤𝑙₃≤𝑙^, 1≤𝑙₄≤𝑚^, because physically 𝑂 is a set of nonsense bases and more details are necessary to convert an 𝑂 back to normal bases, for instance, the cost to identify the exact element in the set 𝑂. Our goal is to drive our system optimally from initial state 𝑥₀ to the desired final set {𝑥^𝑑_𝑁} by applying a sequence of mutagens indexed with {𝑙₁,𝑙₂,𝑙₃,𝑙₄}, at problematic positions 𝑖, and in a correct order 𝑘.

In (17), the first four terms inside the expectation do not depend on random variables ℎ^𝑖_k,𝑙1, 𝑟^𝑖_𝑘,𝑙2, ℎ^𝑖_𝑘,𝑙3, and 𝑟^𝑖_𝑘,𝑙4, for all 𝑖,𝑘,𝑙₁,𝑙₂,𝑙₃,𝑙₄ as the treatment plan is computed based on the initial state 𝑥₀. Given 𝑦_𝑘, the updated treatment plan is computed accordingly but still not related to random variables. The last term inside expectation, ∑^𝑁_𝑘=0𝑑(𝑥_𝑘,{𝑥^𝑑_𝑘}), 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 as𝐽₀𝑥₀=min_{{𝑢,𝑐,𝑣,𝑐}0,1,…,𝑁−1}⎡⎢⎢⎣^𝑁−1_𝑘=0^𝑙_𝑙1=1^𝑛_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑘,𝑙1+^𝑁−1_𝑘=0^𝑚_𝑙2=1^𝑛_𝑖=1𝛽_𝑙2𝑐^𝑖_𝑘,𝑙2+^𝑁−1_𝑘=0^𝑙_𝑙3=1^𝑛_𝑖=1𝛼^_𝑙3𝑣^𝑖_𝑘,𝑙3+^𝑁−1_𝑘=0^𝑚_𝑙4=1^𝑛_𝑖=1𝛽^_𝑙4𝑏^𝑖_𝑘,𝑙4+^𝑁_𝑘=0𝔼_{{ℎ,𝑟,ℎ,𝑟}0,1,…,𝑁−1}𝑑𝑥_𝑘,𝑥^𝑑_𝑘⎤⎥⎦,(18) subject to𝑥_𝑘+1=⎛⎜⎜⎝𝐼+^𝑙_𝑙3=1^𝑛_𝑖=1𝑣^𝑖_𝑘,𝑙3ℎ^𝑖_𝑘,𝑙3𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎟⎠×⎛⎜⎝−𝐼+^𝑙_𝑙1=1^𝑛_𝑖=1𝑢^𝑖_𝑘,𝑙1ℎ^𝑖_𝑘,𝑙1𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥_𝑘+⎛⎜⎜⎝𝐼+^𝑙_𝑙3=1^𝑛_𝑖=1𝑣^𝑖_𝑘,𝑙3ℎ^𝑖_𝑘,𝑙3𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎟⎠^𝑚_𝑙₂₌₁_{𝑖∈𝒪𝑘}𝑐^𝑖_𝑘,𝑙2𝑟^𝑖_𝑘,𝑙2𝑤𝑒_𝑖+^𝑚_𝑙4=1_{𝑖∈𝒪𝑘}𝑏^𝑖_𝑘,𝑙4𝑟^𝑖_𝑘,𝑙4𝑤𝑒_𝑖,𝑦_𝑘=𝑥_𝑘.(19) 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, 𝑑(𝜑₁,𝜑₂),𝜑₁,𝜑₂∈ℱ³, needs to fulfill the biological requirements as below.(1)Nonnegativity: the distance reference between any two codons is either positive or zero. Mathematically, 𝑑∶ℱ³×ℱ³→ℝ⁺∪{0}, 𝑑(𝜑₁,𝜑₂)≥0.(2)The distance reference between two codons corresponding to the same amino acid is zero.(3)Symmetry: the distance reference from codon 𝜑₁ to codon 𝜑₂ equals the distance reference from codon 𝜑₂ to codon 𝜑₁, that is, 𝑑(𝜑₁,𝜑₂)=𝑑(𝜑₂,𝜑₁).(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, 𝑑(𝜑₁,𝜑₂)=0 if and only if 𝜑₁=𝜑₂. The redundancy in genetic codes implies 𝑑(𝜑₁,𝜑₂)=0 if those two amino acids, 𝜑₁ and 𝜑₂, 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 𝑂 since their chemical and physical properties cannot be found in literature.

Table 5 

Properties of amino acids.

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 as𝑑𝜉₁,𝜉₂=𝜁_polaritypolarity𝜉₁,𝜉₂+𝜁_PHPH𝜉₁,𝜉₂+𝜁_sizesize𝜉₁,𝜉₂,polarity𝜉₁,𝜉₂=⎧⎨⎩0if𝜉₁,𝜉₂arebothpolarornon-polar,1ifoneof𝜉₁,𝜉₂ispolar,andtheothernon-polar,PH𝜉₁,𝜉₂=||PHvalueof𝜉₁−PHvalueof𝜉₂||,size𝜉₁,𝜉₂=⎧⎪⎪⎨⎪⎪⎩0,if𝜉₁,𝜉₂arebothtiny,small,ornormal,𝜎₁,ifoneof𝜉₁,𝜉₂istiny,andtheothersmall,𝜎₂,ifoneof𝜉₁,𝜉₂istiny,andtheothernormal,𝜎₃,ifoneof𝜉₁,𝜉₂issmall,andtheothernormal,(20) where 𝜉₁,𝜉₂ are two amino acids.

𝑑(𝜉₁,𝜉₂) is then assigned to 𝑑(𝜑₁,𝜑₂) with 𝜑₁,𝜑₂ corresponding to amino acids 𝜉₁,𝜉₂, 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 𝑥₀ to {𝑥^𝑑_𝑁} is also the optimal policy for the tail problem, from 𝑥_𝑞 to {𝑥^𝑑_𝑁}.

The tail problem is defined as𝐽_𝑞𝑥_𝑞=min_{{𝑢,𝑐,𝑣,𝑐}𝑞,𝑞+1,…,𝑁−1}⎧⎨⎩^𝑁−1_𝑘=𝑞^𝑙_𝑙1=1^𝑛_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑘,𝑙1+^𝑁−1_𝑘=𝑞^𝑚_𝑙2=1^𝑛_𝑖=1𝛽_𝑙2𝑐^𝑖_𝑘,𝑙2+^𝑁−1_𝑘=𝑞^𝑙_𝑙3=1^𝑛_𝑖=1𝛼^_𝑙3𝑣^𝑖_𝑘,𝑙3+^𝑁−1_𝑘=𝑞^𝑚_𝑙4=1^𝑛_𝑖=1𝛽^_𝑙4𝑏^𝑖_𝑘,𝑙4+^𝑁_𝑘=𝑞𝔼_{{ℎ,𝑟,ℎ,𝑟}𝑞,𝑞+1,…,𝑁−1}𝑑𝑥_𝑘,𝑥^𝑑_𝑘⎫⎬⎭.(21) The iterative update equation to find optimal policy can be expressed by (22), according to the dynamic programming algorithm in [31]. 𝐽_𝑁𝑥_𝑁=𝑑𝑥_𝑁,𝑥^𝑑_𝑁,𝐽_𝑞𝑥_𝑞=min_{𝑢𝑞,𝑐𝑞,𝑣𝑞,𝑐𝑞}𝔼_{ℎ𝑞,𝑟𝑞,ℎ𝑞,𝑟𝑞}⎡⎢⎢⎣^𝑙_𝑙1=1^𝑛_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑞,𝑙1+^𝑚_𝑙2=1^𝑛_𝑖=1𝛽_𝑙2𝑐^𝑖_𝑞,𝑙2+^𝑙_𝑙3=1^𝑛_𝑖=1𝛼^_𝑙3𝑣^𝑖_𝑞,𝑙3+^𝑚_𝑙4=1^𝑛_𝑖=1𝛽^_𝑙4𝑐^𝑖_𝑞,𝑙4+𝑑𝑥_𝑞,𝑥^𝑑_𝑞+𝐽_𝑞+1𝑥_𝑞+1⎤⎥⎥⎦=min_{𝑢𝑞,𝑐𝑞,𝑣𝑞,𝑐𝑞}⎧⎪⎨⎪⎩^𝑙_𝑙1=1^𝑛_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑞,𝑙1+^𝑚_𝑙2=1^𝑛_𝑖=1𝛽_𝑙2𝑐^𝑖_𝑞,𝑙2+^𝑙_𝑙3=1^𝑛_𝑖=1𝛼^_𝑙3𝑣^𝑖_𝑞,𝑙3+^𝑚_𝑙4=1^𝑛_𝑖=1𝛽^_𝑙4𝑐^𝑖_𝑞,𝑙4+𝔼_{ℎ𝑞,𝑟𝑞,ℎ𝑞,𝑟𝑞}𝑑𝑥_𝑞,𝑥^𝑑_𝑞+𝐽_𝑞+1𝑥_𝑞+1⎫⎬⎭,𝑞=0,1,…,𝑁−1.(22)

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 𝑂 because of high-cost 𝛽_𝑙2 and the unavailable chemical and physical properties for codons containing 𝑂.

The distance reference between codons used in Sections 3.2 and 3.3 is computed by (20) with 𝜁_polarity=8, 𝜁_PH=3, 𝜁_size=1, 𝜎₁=2, 𝜎₂=5 and 𝜎₃=3. 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 as𝑑𝜓₁,𝜓₂=⎧⎨⎩0,if𝑥_𝑁=𝑥^𝑑_𝑁,∞,if𝑥_𝑁≠𝑥^𝑑_𝑁,(23) with 𝜓₁,𝜓₂∈ℱ_⧵{0}, where ℱ_⧵{0} denotes the set ℱ excluding the element 0.

Our optimal control problem for point mutations is𝐽₀𝑥₀=min_{𝑢𝑘,𝑙1,0≤𝑘≤𝑁,1≤𝑙1≤𝑙}⎧⎨⎩^𝑁−1_𝑘=0^𝑙_𝑙1=1𝛼_𝑙1𝑢_𝑘,𝑙1⎫⎬⎭,(24) subject to𝑥_𝑘+1=⎛⎜⎝−𝐼+^𝑙_𝑙1=1𝑢_𝑘,𝑙1𝑠⎞⎟⎠𝑥_𝑘,𝑥_𝑁=𝑥^𝑑_𝑁,(25) with 𝑥₀ given, 𝑥_𝑘∈ℱ_⧵{0}.

Suppose that there are 12 kinds of mutagens (𝑙₁=12), 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.

Table 6 

An example of chemical mutagens and their corresponding transfer patterns in deterministic mutations.

Table 7 

Controls corresponding to transfer between bases within one step. The leftmost column denotes the state 𝑘th step, and the upmost row denotes the (𝑘+1)th state.

Table 8 

Corresponding step cost of controls as shown in Table 7.

The elements along the antidiagonal of Table 7, 𝑢_𝐴𝑇,𝑢_𝐺𝐶,𝑢_𝐶𝐺, and 𝑢_𝑇𝐴, 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, 𝛼_𝐴𝑇=𝛼_𝐺𝐶=𝛼_𝐶𝐺=𝛼_𝑇𝐶=0. The equivalence relationship between subscription in two nucleotide bases and subscription in integer 𝑙₁(𝜓₁𝜓₂)∶{𝐴,𝑇,𝐺,𝐶}×{𝐴,𝑇,𝐺,𝐶}→{integersfrom1to12} is defined by Table 6.

Under the above assumptions, we can write update equation for optimal policy explicitly as𝐽_𝑞𝑥_𝑞=min_{𝑢𝑞,𝑙1}𝛼_𝑥𝑞𝜓+𝐽_𝑞+1(𝜓),∀𝜓∈{𝐴,𝑇,𝐺,𝐶}⟺ℱ_⧵{0}.(26)

Proposition 7. For the same 𝑥^𝑑_𝑁, 𝐽_𝑞(𝜓)≤𝐽_𝑞+1(𝜓), for all 𝑞, 0≤𝑞≤𝑁−1, for all 𝜓∈{𝐴,𝑇,𝐺,𝐶}, where 𝜓 denotes the complementary base of 𝜓. If, in addition, the system is completely controllable, ∃𝑀, s.t. 𝐽_𝑀(𝜓) is the global minimum and for all 𝑞≤𝑀, 𝐽_𝑞(𝜓)=𝐽_𝑀(𝜓) if 𝑀−𝑞≡ 1 (mod 2), and 𝐽_𝑞(𝜓)=𝐽_𝑀(𝜓) if 𝑀−𝑞≡0 (mod 2). In our example, 𝑀≥𝑁−6.

Proof. This first part is due to the zero cost for the transfers between complementary bases in the consecutive steps.
For any 0≤𝑞≤𝑁−1, the relationship between minimal costs in consecutive steps is shown in (26). Since 𝜓∈{𝐴,𝑇,𝐺,𝐶}, 𝛼_𝜓𝜓+𝐽_𝑞+1(𝜓) is one of the four elements in the set from which the 𝐽_𝑞(𝜓) is picked. Moreover, 𝛼_𝜓𝜓=0. Therefore, 𝐽_𝑞+1(𝜓) is one of the four elements in the set. Since 𝐽_𝑞(𝜓) is the minimum picking for a set containing 𝐽_𝑞+1(𝜓), we conclude that 𝐽_𝑞(𝜓)≤𝐽_𝑞+1(𝜓).
The 𝑀 value in our example is proved by brute force method, that is, 𝐽_𝑁−6(𝜓) is a guaranteed global minimum. For completely controllable systems, this 𝑀 always exists. The existence of 𝑀 implies that for without limitation in the number of steps, we can reach the global optimal in 𝑁−𝑀 steps, 6 steps in our example.
Suppose that 𝑞=𝑀, 𝐽_𝑀(𝜓) is the global minimum, thus 𝐽_𝑀−1(𝜓)≥𝐽_𝑀(𝜓). However,𝐽_𝑀−1(𝜓)≤𝐽_𝑀(𝜓) according to the first part of the proposition. Therefore, 𝐽_𝑀−1(𝜓)=𝐽_𝑀(𝜓) for the same 𝑥^𝑑_𝑁. Therefore, 𝐽_𝑀−1(𝜓) is also a global minimum.
By backward induction, suppose for 𝑞=𝑞₁, the statement is true, that is, 𝐽_𝑞1−1(𝜓)=𝐽_𝑞1(𝜓) is the global optimal either from 𝑥_𝑞1−1=𝜓 or 𝑥_𝑞1=𝜓 to 𝑥^𝑑_𝑁. Obviously, for 𝑞=𝑞₁−1, the statement is still true. Therefore, 𝐽_𝑞(𝜓)=𝐽_𝑞−2(𝜓)=𝐽_𝑞−1(𝜓),𝜓∈{𝐴,𝑇,𝐺,𝐶}, for all 𝑞, 2≤𝑞≤𝑀.

In the proof of global minimum that can be reached in the finite step in Proposition 7, we also discover Proposition 8. Here, 𝐽_𝑞(𝑥_𝑞,𝑥^𝑑_𝑁) denotes the optimal cost from 𝑥_𝑞 to 𝑥^𝑑_𝑁.

Proposition 8. Given two single base mutation optimal control problems, with the same fixed 𝑁, with and desired final states complementary to each other. If 𝐽_𝑀(𝜓,𝑥^𝑑_𝑁) is the global minimum, then 𝐽_𝑀(𝜓,𝑥^𝑑_𝑁) is also the global minimum, that is, the global minimum of both systems is reach at the same stage 𝑀. Moreover, for all 𝑞, 0≤𝑞≤𝑀, 𝐽_𝑞𝜓,𝑥^𝑑_𝑁=𝐽_𝑞𝜓,𝑥^𝑑_𝑁,𝜓,𝑥^𝑑_𝑁∈{𝐴,𝑇,𝐺,𝐶}.(27)

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, 𝑢_𝐴𝐴 actually transfer 𝐴 to 𝑇 and then to 𝐴 by replication. For simplicity, we just use the 𝑘th and (𝑘+1)th step states as subscripts to represent the corresponding control and cost. The costs of transitions are lower than the costs of transversions. Therefore, 𝛼_𝐴𝐶,𝛼_𝐶𝐴,𝛼_𝐺𝑇,𝛼_𝑇𝐺 is smaller than other mutagens, except artificial ones.

Table 9 

Sample step costs.

If we use 𝜒 to denote the costs of mutagens as listed in Table 9, then 𝜒=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.216.602.3306.158.9503.824.6109.177.2400.645.0910.28⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝛼_𝐴𝐴𝛼_𝐴𝐺𝛼_𝐴𝐶𝛼_𝐴𝑇𝛼_𝐺𝐴𝛼_𝐺𝐺𝛼_𝐺𝐶𝛼_𝐺𝑇𝛼_𝐶𝐴𝛼_𝐶𝐺𝛼_𝐶𝐶𝛼_𝐶𝑇𝛼_𝑇𝐴𝛼_𝑇𝐺𝛼_𝑇𝐶𝛼_𝑇𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⟺⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝛼₁𝛼₂𝛼₃0𝛼₄𝛼₅0𝛼₆𝛼₇0𝛼₈𝛼₉0𝛼₁₀𝛼₁₁𝛼₁₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(28)

Running the dynamic programming for every pair of (𝑥_𝑞,𝑥^𝑑_𝑁)∈{𝐴,𝑇,𝐺,𝐶}×{𝐴,𝑇,𝐺,𝐶}, 𝑁=9. Here, we sightly modify our notation. We use 𝐽_𝑞(𝑥_𝑞,𝑥^𝑑_𝑁) to denote the optimal cost from 𝑥_𝑞 to 𝑥^𝑑_𝑁. Then, 𝐽_𝑞=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐽_𝑞(𝐴,𝐴)𝐽_𝑞(𝐴,𝐺)𝐽_𝑞(𝐴,𝐶)𝐽_𝑞(𝐴,𝑇)𝐽_𝑞(𝐺,𝐴)𝐽_𝑞(𝐺,𝐺)𝐽_𝑞(𝐺,𝐶)𝐽_𝑞(𝐺,𝑇)𝐽_𝑞(𝐶,𝐴)𝐽_𝑞(𝐶,𝐺)𝐽_𝑞(𝐶,𝐶)𝐽_𝑞(𝐶,𝑇)𝐽_𝑞(𝑇,𝐴)𝐽_𝑞(𝑇,𝐺)𝐽_𝑞(𝐶,𝐶)𝐽_𝑞(𝑇,𝑇)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(29)

The path to reach the optimal cost is denoted by 𝑃_𝑞=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑃_𝑞(𝐴,𝐴)𝑃_𝑞(𝐴,𝐺)P_𝑞(𝐴,𝐶)𝑃_𝑞(𝐴,𝑇)𝑃_𝑞(𝐺,𝐴)𝑃_𝑞(𝐺,𝐺)𝑃_𝑞(𝐺,𝐶)𝑃_𝑞(𝐺,𝑇)𝑃_𝑞(𝐶,𝐴)𝑃_𝑞(𝐶,𝐺)𝑃_𝑞(𝐶,𝐶)𝑃_𝑞(𝐶,𝑇)𝑃_𝑞(𝑇,𝐴)𝑃_𝑞(𝑇,𝐺)𝑃_𝑞(𝑇,𝐶)𝑃_𝑞(𝑇,𝑇)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(30)

where 𝑃_𝑞(𝑥_𝑞,𝑥^𝑑_𝑁) is the (𝑞+1)th state from 𝑥_𝑞 to 𝑥^𝑑_𝑁, that is, 𝑥_𝑞+1=𝑃_𝑞(𝑥_𝑞,𝑥^𝑑_𝑁).

The simulation results are shown as below, including optimal costs for all possible transfer pairs (𝑥_𝑞,𝑥^𝑑_𝑁)∈{𝐴,𝑇,𝐺,𝐶}×{𝐴,𝑇,𝐺,𝐶}, 𝐽_𝑞, 0≤𝑞≤8, graphical representation in Figure 6, and optimal path 𝑃_𝑞, 0≤𝑞≤7. 𝐽₀=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.215.090.6406.156.7903.823.8206.796.1500.645.095.21⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣00.645.095.213.8206.796.156.156.7903.825.215.090.640⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.215.090.6406.156.7903.823.8206.796.1500.645.095.21⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₃=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣00.645.095.213.8206.796.156.156.7903.825.215.090.640⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₄=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.215.090.6406.156.7903.823.8206.796.1500.645.095.21⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₅=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣00.645.095.213.8206.796.156.156.7903.825.215.090.640⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₆=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.215.090.6406.156.7903.823.8207.886.1500.645.095.21⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₇=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣00.645.095.213.8208.486.156.157.8803.825.215.090.640⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐽₈=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣5.216.602.3306.158.9503.824.6109.177.2400.645.0910.28⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(31)𝑃₀=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴,𝑇𝑇𝑇𝑇𝐴,𝐶𝐴,𝐶𝐶𝐶,𝑇𝐺𝐺𝐺𝐺𝐴𝐴,𝐺𝐴,𝐶𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑇𝑇𝑇𝐴,𝑇𝐶,𝑇𝐶𝐴,𝐶𝐴,𝐶𝐺𝐺𝐺𝐺𝐴𝐴,𝐶𝐴,𝐺𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴,𝑇𝑇𝑇𝑇𝐴,𝐶𝐴,𝐶𝐶𝐶,𝑇𝐺𝐺𝐺𝐺𝐴𝐴,𝐺𝐴,𝐶𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₃=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑇𝑇𝑇𝐴,𝑇𝐶,𝑇𝐶𝐴,𝐶𝐴,𝐶𝐺𝐺𝐺𝐺𝐴𝐴,𝐶𝐴,𝐺𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₄=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴,𝑇𝑇𝑇𝑇𝐴,𝐶𝐴,𝐶𝐶𝐶,𝑇𝐺𝐺𝐺𝐺𝐴𝐴,𝐺𝐴,𝐶𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₅=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑇𝑇𝑇𝐴,𝑇𝐶,𝑇𝐶𝐴𝐴,𝐶𝐺𝐺𝐺𝐺𝐴𝐴,𝐶𝐴,𝐺𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₆=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴,𝑇𝑇𝑇𝑇𝐴,𝐶𝐴𝐶𝐶,𝑇𝐺𝐺𝑇𝐺𝐴𝐴,𝐺𝐴,𝐶𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑃₇=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑇𝑇𝑇𝐴𝑇𝐶𝐴𝐴𝐺𝑇𝐺𝐺𝐴𝐶𝐺𝐴⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(32)

Figure 6 

Graphical representation of 𝐽𝑞, 0≤𝑞≤8, 𝑁=9, in single-base deterministic mutation example. The 𝑥-axis and 𝑦-axis represent 𝑥𝑞 and 𝑥𝑑𝑁, respectively. 𝐽𝑞(𝑥𝑞,𝑥𝑑𝑁) are represented by 16 isolated points. Those discrete points are connected together to show the surface.

For simplicity, we use 1 to represent 𝐴, 2 to 𝐺, 3 to 𝐶, and 4 to 𝑇 in graphical interpretation. From Figure 6, we can see clearly that the optimal cost decreases as 𝑞 decreases in the first few steps, and then optimal cost remains at the global minimum. This phenomenon obeys Proposition 7. In this example, global optimal is reached at 𝑀=5 for all pairs of initial and final states as 𝐽₇≠𝐽₅=𝐽₃ and 𝐽₆≠𝐽₄=𝐽₂. So, the global minimum is achieved before we reach 𝑁−5=4 in this particular case. This also implies that with 𝑁 free we can reach desired final state in 4 steps from given initial state.

Observing closely to 𝐽_𝑞, 0≤𝑞≤5, 𝐽_𝑞−1 equals to 𝐽_𝑞 by exchanging the first and the last columns, and the second and the third columns, which is consistent with Proposition 8. Or we can exchange the first and the last rows, and the second and the third rows of 𝐽_𝑞 to obtain 𝐽_𝑞−1. 𝐽_𝑞1 and 𝐽_𝑞2 are the same for 𝑞₁,𝑞₂≤𝑀=5 for 𝑞₁−𝑞₂=0 (mod 2). This obeys Proposition 7.

The optimal trajectories are generated from 𝑃_𝑞(𝑥_𝑞,𝑥^𝑑_𝑁). For example, given 𝑥₂=𝑇, and the final state 𝑥^𝑑₉=𝐺, we want to generate the optimal trajectories.

𝑥₃=𝑃₂(𝑇,𝐺)=𝐴,𝐺.

If 𝑥₃=𝐴, 𝑥₄=𝑃₃(𝐴,𝐺)=𝑇; if 𝑥₃=𝐺, 𝑥₄=𝑃₃(𝐴,𝐺)=𝐶.

If 𝑥₄=𝑇, 𝑥₅=𝑃₄(𝑇,𝐺)=𝐴,𝐺; if 𝑥₄=𝐶, 𝑥₅=𝑃₄(𝐶,𝐺)=𝐺.

If 𝑥₅=𝐴, 𝑥₆=𝑃₅(𝐴,𝐺)=𝑇; if 𝑥₅=𝐺, 𝑥₆=𝑃₅(𝐺,𝐺)=𝐶.

If 𝑥₆=𝑇, 𝑥₇=𝑃₆(𝑇,𝐺)=𝐴,𝐺; if 𝑥₆=𝐶, 𝑥₇=𝑃₆(𝐶,𝐺)=𝐺.

If 𝑥₇=𝐴, 𝑥₈=𝑃₇(𝐴,𝐺)=𝑇; if 𝑥₇=𝐺, 𝑥₈=𝑃₇(𝐺,𝐺)=𝐶.

So, the optimal routes are𝑇→𝐴→𝑇→𝐴→𝑇→𝐴→𝑇→^𝑢𝑇𝐺_𝛼𝑇𝐺𝐺;𝑇→𝐴→𝑇→𝐴→𝑇→^𝑢𝑇𝐺_𝛼𝑇𝐺𝐺→𝐶→𝐺;𝑇→𝐴→𝑇→^𝑢𝑇𝐺_𝛼𝑇𝐺𝐺→𝐶→𝐺→𝐶→𝐺;𝑇→^𝑢𝑇𝐺_𝛼𝑇𝐺𝐺→𝐶→𝐺→𝐶→𝐺→𝐶→𝐺.

Consequently, the optimal cost is 𝐽₂(𝑇,𝐺)=0.64=𝛼_𝑇𝐺.

Optimal trajectories for other pairs of initial and final states can be obtained in the same manner.

It takes less than 1 second to generate optimal trajectories for all pairs of initial and final states with 𝑁=9 on a regular desktop. Since we have already proven by the brute force method that the global optimal can be achieved with 𝑀≤6, the computation time can be further reduced by taking 𝑁=6 with all the results necessary for this example.

3.2. Codon-to-Codon, Deterministic Optimal Control Problem

For codon-to-codon deterministic mutations, we formulate our optimal control problem as𝐽₀𝑥₀=min_{𝑢𝑖𝑘,𝑙1,0≤𝑘≤𝑁,1≤𝑙1≤𝑙,1≤𝑖≤3}⎧⎨⎩^𝑁−1_𝑘=0^𝑙_𝑙1=1³_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑘,𝑙1+𝑑𝑥_𝑁,𝑥^𝑑_𝑁⎫⎬⎭,(33) subject to𝑥_𝑘+1=⎛⎜⎝−𝐼+^𝑙_𝑙1=1³_𝑖=1𝑢^𝑖_𝑘,𝑙1𝑠𝑒_𝑖𝑒^𝑇_𝑖⎞⎟⎠𝑥_𝑘,(34) with 𝑥₀,𝑥^𝑑_𝑁∈ℱ³_⧵{0} given, 𝑥_𝑘∈ℱ³_⧵{0}, for all 𝑘, 0≤𝑘≤𝑁, and 𝑑(𝜑₁,𝜑₂),𝜑₁,𝜑₂∈ℱ³_⧵{0} as defined in Section 2.3.

If we take the same assumption on available mutagens as in Section 3.1, then we can write update equation for optimal control policy explicitly as 𝐽_𝑞𝑥_𝑞=min_{𝑢𝑖𝑞,𝑙1,1≤𝑙1≤𝑙,1≤𝑖≤3}×⎧⎪⎪⎨⎪⎪⎩𝛼_𝑥¹_𝑞𝜓1+𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝜓₁𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠,𝛼_𝑥²_𝑞𝜓2+𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝑥¹_𝑞𝜓₂𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠,𝛼_𝑥³_𝑞𝜓3+𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝑥¹_𝑞𝑥²_𝑞𝜓₃⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠,𝜓₁,𝜓₂,𝜓₃∈{𝐴,𝑇,𝐺,𝐶}⟺ℱ_⧵{0}⎫⎪⎪⎬⎪⎪⎭,(35) with 𝑥^𝑖_𝑞∈{𝐴,𝑇,𝐺,𝐶}⇔ℱ_⧵{0}, 1≤𝑖≤3 denotes the 𝑖th element of 𝑥_𝑞∈ℱ³_⧵{0}, and 𝑥^𝑖_𝑞 denotes the complementary base of 𝑥^𝑖_𝑞.

The optimal control sequences depend on the numerical values of 𝛼_𝑙1s and 𝑑(𝜑₁,𝜑₂), 𝜑₁,𝜑₂∈ℱ³_⧵{0}. Though we do not have real values of 𝛼_𝑙1s and 𝑑(𝜑₁,𝜑₂), we can always obtain simulation results to compare the result differences by assigning different numerical values to those parameters.

Therefore, we use three different assignments of 𝛼_𝑙1s and the same 𝑑(𝜑₁,𝜑₂) to generate our simulation results. Those three assignments of 𝛼_𝑙1s are 𝜒, 5𝜒, and 0.5𝜒, respectively, with 𝜒 the same as assigned in Section 3.1. In every particular example, it takes approximately 2 seconds on a regular desktop to generate the optimal path table for all pairs of initial and final states with 𝑁=19, and the dynamic programming algorithm ensures that the optimal control for tail problems is generated at the same time.

The graphical interpretation of three assignments are shown in Figures 7, 8, and 9, respectively. The 𝑥-axis and 𝑦-axis denote 𝑥_𝑞 and 𝑥^𝑑_𝑁, respectively. For a codon [𝜓₁𝜓₂𝜓₃]^𝑇,𝜓₁,𝜓₂, 𝜓₃∈{𝐴,𝑇,𝐺,𝐶}⇔ℱ_⧵{0}, its index is calculated by 4²𝜓₁−1+4𝜓₂−1+𝜓₃,(36)where 𝜓_𝑖=1 if 𝐴, 𝜓_𝑖=2 if 𝐺, 𝜓_𝑖=3 if 𝐶, and 𝜓_𝑖=4 if 𝑇, 1≤𝑖≤3, for the simplicity of graphical interpretation. A codon has 4³=64 combinations. Thus, there are 64² pairs of initial and final desired states, and there are 64×21 pairs of initial state and final desired set. The surface is generated by connecting 64×64 discrete points together. 𝐽_𝑞 is calculated following the same procedure as in base-to-base deterministic cases. The value of optimal cost can be read directly from graphical interpretation, and the optimal path can be generated from path matrix 𝑃_𝑞, similar to base-to-base deterministic case. Both 𝐽_𝑞 and 𝑃_𝑞,forall𝑞,0≤𝑞≤𝑁, are of 64×64 dimension.

Figure 7 

Graphical representation of 𝐽𝑞, 𝑞=0,1,2,3,15,16,17,18 for codon-to-codon deterministic mutations, with 𝛼𝑙1=𝜒, 𝑁=19.

Figure 8 

Graphical representation of 𝐽𝑞, 𝑞=0,1,2,3,15,16,17,18 for codon-to-codon deterministic mutations, with 𝛼𝑙1=5𝜒, 𝑁=19.

Figure 9 

Graphical representation of 𝐽𝑞, 𝑞=0,1,2,3,15,16,17,18 for codon-to-codon deterministic mutations, with 𝛼𝑙1=0.5𝜒, 𝑁=19.

From the graphical interpretation and Table 10, we find that the value of 𝑞 where the global minimum is reached at the first time decreases as 𝛼_𝑙1 decreases. And 𝐽₀ is more similar to 𝐽₁₈ with 𝛼=5𝜒 than with 𝛼=𝜒 or 𝛼=0.5𝜒. This implies that if 𝑑(𝜑₁,𝜑₂) are the deterministic term in our objective function, then the treatment plan is made to drive the final state as close to the desired set as possible; if the costs of applying mutagens is the deterministic term in the objective function, then the treatment plan tends to stay in the original state and applies less mutagens; if they are of equal weight, then the treatment plan deals with this tradeoff.

Table 10 

Simulation results with different 𝛼_𝑙1 assignments and the first 𝑞 where the global optimal is reached.

Moreover, no matter how the numerical values of final penalty and the costs of applying mutagens changes in our objective function as shown in (33), there is always a 𝑀, 𝑀≤𝑁−18, 𝐽_𝑀(𝑥_𝑀) is global minimum. Proposition 7 can be extended to codon-to-codon deterministic mutations as stated in Proposition 9.

Proposition 9. Given an optimal control problem with objective function in the form of (33), constraints in the form of (34), and all available chemical mutagens and their corresponding transfer pairs and costs as listed in Tables 6, 7, and 8, 𝐽_𝑞(𝜑)≤𝐽_𝑞+1(𝜑),𝜑∈ℱ³_⧵{0}. If, in addition, the system is completely controllable, ∃𝑀, s.t. 𝐽_𝑀(𝜑) is the global minimum and for all 𝑞≤𝑀, 𝐽_𝑞(𝜑)=𝐽_𝑀(𝜑) if 𝑀−𝑞≡1 (mod 2), and 𝐽_𝑞(𝜑)=𝐽_𝑀(𝜑) if 𝑀−𝑞≡0 (mod2). In our example, 𝑀≥𝑁−18.

Proof. We only prove that 𝑀≥𝑁−18 here since the rest is similar to the proof of Proposition 7.
The objective function in (33) can be written as the summation of three separate single-base mutation systems and the distance reference between final states and the final desired set, that is, 𝐽_𝑞𝑥_𝑞=min_{𝑛1,𝑛2,𝑛3≥02𝑁≤𝑛1+𝑛2+𝑛3≤3𝑁−1}⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐽_𝑛1(𝑥_𝑛−1)𝑥¹_𝑞,𝜓₁_{optimalcostsofbase-to-basedeterministicoptimalcontrolproblemformedbythe1stbase}+𝐽_𝑛2(𝑥_𝑛−2)𝑥²_𝑞,𝜓₂_{optimalcostsofbase-to-basedeterministicoptimalcontrolproblemformedby-the2ndbase}+𝐽_𝑛3(𝑥_𝑛−3)𝑥³_𝑞,𝜓₃_{optimalcostsofbase-to-basedeterministicoptimalcontrolproblemformedbythe3rdbase}+𝑑⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝜓₁𝜓₂𝜓₃⎤⎥⎥⎥⎥⎥⎦,𝑥^𝑑_𝑁⎞⎟⎟⎟⎟⎟⎠⎫⎪⎪⎪⎬⎪⎪⎪⎭,(37) where 𝑁−𝑞=(𝑁−𝑛₁)+(𝑁−𝑛₂)+(𝑁−𝑛₃).
According to Proposition 7, 𝐽_𝑁−6(𝑥_𝑁−6) is guaranteed to be the global optimal for single-base mutations. Therefore, optimal costs corresponding to three single-base mutation systems, 𝐽_𝑛1(𝑥_𝑛−1)(𝑥¹_𝑞,𝜓₁), 𝐽_𝑛2(𝑥_𝑛−2)(𝑥²_𝑞,𝜓₂), 𝐽_𝑛3(𝑥_𝑛−3)(𝑥³_𝑞,𝜓₃) are guaranteed to reach their own global optimal at 𝑛₁=𝑛₂=𝑛₃=𝑁−6 with all possible combinations of 𝜓₁,𝜓₂,𝜓₃∈{𝐴,𝑇,𝐺,𝐶}. Therefore, 𝑞=𝑁−18 is a guaranteed global optimal.

Indeed, Proposition 9 is a quite loose condition. In real examples above, the 𝑀 value where the first global optimal is reached at earlier stage as listed in Table 10.

Graphically, the indices of complementary codons 𝜑₁=[𝜓₁𝜓₂𝜓₃]^𝑇 and 𝜑₂=[𝜓₁𝜓₂𝜓₃]^𝑇 sum up to 65, that is, 16𝜓₁−1+4𝜓₂−1+𝜓₃+16𝜓₁−1+4𝜓₂−1+𝜓₃=16𝜓₁−1+4𝜓₂−1+𝜓₃+165−𝜓₁−1+45−𝜓₂−1+5−𝜓₃=65.(38)

Therefore, 𝐽_𝑞 and 𝐽_𝑞−1, 1≤𝑞≤𝑀 are symmetric about the plane 𝑥=32.5, 𝐽_𝑞−2 and 𝐽_𝑞, 2≤𝑞≤𝑀 are the same, as shown in Figures 7, 8, and 9.

However, Proposition 8 cannot be extended to codon-to-codon deterministic case due to the redundancy of genetic codes, that is, the set of codons translated to the same amino acid varies from one amino acid to another as shown in Table 5. The simulation results show that the costs, a pair of complementary codons, to two final desired set generated by a pair of complementary final desired codons are different, that is, 𝐽_𝑞(𝑥_𝑞,{𝑥^𝑑_𝑁})≠𝐽_𝑞(𝑥_𝑞,{𝑥^𝑑_𝑁}), in general, for any 𝑞. Graphically, the optimal cost profile 𝐽_𝑞 is not symmetric about the plane 𝑦=32.5 for 𝐽_𝑞−1, 1≤𝑞≤𝑀. Therefore, the doctors need to pick the strand with lower cost to make the treatment plan. This also implies that in large-scale cases, for instance, a gene containing hundreds of nucleotide bases, the doctors should make the treatment plan based on the strand the total cost of which is lower than the other.

3.3. Codon-to-Codon, Stochastic Optimal Control Problem

The optimal control problem of codon-to-codon stochastic mutations can be written as𝐽₀𝑥₀=min_{𝑢𝑖𝑘,𝑙1,0≤𝑘≤𝑁−11≤𝑙1≤𝑙,1≤𝑖≤3}⎧⎪⎨⎪⎩^𝑁−1_𝑘=0^𝑙_𝑙1=1³_𝑖=1𝛼_𝑙1𝑢^𝑖_𝑘,𝑙1+𝔼_{ℎ𝑖𝑘,𝑙1,0≤𝑘≤𝑁−11≤𝑙1≤𝑙,1≤𝑖≤3}𝑑𝑥_𝑁,𝑥^𝑑_𝑁⎫⎪⎬⎪⎭,(39) subject to𝑥_𝑘+1=−𝐼𝑥_𝑘+^𝑙_𝑙1=1³_𝑖=1𝑢^𝑖_𝑘,𝑙1ℎ^𝑖_𝑘,𝑙1𝑠𝑒_𝑖𝑒^𝑇_𝑖𝑥_𝑘,(40) with 𝑥₀,𝑥^𝑑_𝑁∈ℱ³_⧵{0} given, 𝑥_𝑘∈ℱ³_⧵{0}.

The major difference between deterministic and stochastic systems is that we impose the random binary vector, ℎ^𝑖_𝑘,𝑙1, in our system equation (40). We denote the probability associated with ℎ^𝑖,𝑗_𝑘,𝑙1 to be 𝑝^(ℎ)_{𝑙1,𝜓1𝜓2} with 𝜓₁,𝜓₂∈{𝐴,𝑇,𝐺,𝐶}. The equivalence relationship between 𝑗 and 𝜓₁𝜓₂ can be found in Table 3.

Again, we assume that we have 𝑙₁=12 kinds of mutagens, each corresponding to one major transfer pattern, associated with probability assignments, as listed in Table 11.

Table 11 

12 kinds of mutagens, each corresponding to major transfer patterns, and the probabilities of different mutagens on different transfer patterns.

Then, we can write updated formula for optimal control policy explicitly as 𝐽_𝑞𝑥_𝑞=min_{𝑢𝑖𝑞,𝑙1,1≤𝑙1≤𝑙,1≤𝑖≤3}⎧⎪⎪⎨⎪⎪⎩𝛼_𝑥¹_𝑞𝜓1+𝔼_{ℎ1𝑞,𝑙1(𝑥1𝑞𝜓1)}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣⋅𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦,𝛼_𝑥²_𝑞𝜓2+𝔼_{ℎ2𝑞,𝑙1(𝑥2𝑞𝜓2)}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝑥¹_𝑞⋅𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦,𝛼_𝑥³_𝑞𝜓3+𝔼_{ℎ3𝑞,𝑙1(𝑥3𝑞𝜓3)}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝑥¹_𝑞𝑥²_𝑞⋅⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦,𝜓₁,𝜓₂,𝜓₃∈{𝐴,𝑇,𝐺,𝐶}⟺ℱ_⧵{0}⎫⎪⎪⎬⎪⎪⎭(41) where_{ℎ1𝑞,𝑙1}𝔼_{(𝑥1𝑞𝜓1)}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣⋅𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦=𝑝^(ℎ)_{𝑙1(𝑥1𝑞𝜓1),𝑥1𝑞𝐴}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝐴𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦+𝑝^(ℎ)_{𝑙1(𝑥1𝑞𝜓1),𝑥1𝑞𝐺}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝐺𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦+𝑝^(ℎ)_{𝑙1(𝑥1𝑞𝜓1),𝑥1𝑞𝐶}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝐶𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦+𝑝^(ℎ)_{𝑙1(𝑥1𝑞𝜓1),𝑥1𝑞𝑇}⎡⎢⎢⎢⎢⎢⎣𝐽_𝑞+1⎛⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎣𝑇𝑥²_𝑞𝑥³_𝑞⎤⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎥⎦,(42)

where 𝑥^𝑖_𝑞∈{𝐴,𝑇,𝐺,𝐶}⇔ℱ_⧵{0}, 0≤𝑞≤𝑁−1, 1≤𝑖≤3 denotes the 𝑖th element of 𝑥_𝑞∈ℱ³_⧵{0}, 𝑥^𝑖_𝑞 denotes the complementary base of 𝑥^𝑖_𝑞, and 𝑙₁∶𝜓₁𝜓₂∈{𝐴,𝑇,𝐺,𝐶}×{𝐴,𝑇,𝐺,𝐶}→{integersfrom1to12}, the mapping from major transfer pattern 𝜓₁→𝜓₂ to mutagen index, as shown in Table 11. The mathematical expression of 𝔼_{ℎ2𝑞,𝑙1(𝑥2𝑞𝜓2)}[𝐽_𝑞+1([𝑥¹_𝑞⋅𝑥³_𝑞]^𝑇)] and 𝔼_{ℎ3𝑞,𝑙1(𝑥3𝑞𝜓3)}[𝐽_𝑞+1(^{𝑥1𝑞𝑥2𝑞⋅}^𝑇)] is similar to 𝔼_{ℎ1𝑞,𝑙1(𝑥1𝑞𝜓1)}[𝐽_𝑞+1([⋅𝑥²_𝑞𝑥³_𝑞]^𝑇)] as shown above.

In order to run the simulation, we assign numerical values to probabilities in Table 11, as illustrated in Table 12.

Table 12 

Sample probabilities with respect to different mutagens and different transfer patterns.

As in Section 3.2, we use three different assignments for 𝛼_𝑙1s, 𝜒,5𝜒, and 0.5𝜒, respectively. The optimal cost profile 𝐽_𝑞 with selected 𝑞 values, for every pair of (𝑥_𝑞,𝑥^𝑑_𝑁), is graphically interpreted in Figures 10, 11, and 12, respectively, with 𝑁=29, with computation time of approximately 7 seconds on a regular desktop.

Figure 10 

Graphical representation of 𝐽𝑞(𝑥𝑞), 𝑞=0,1,2,3,15,19,24,28 for codon-to-codon stochastic mutations, with 𝛼𝑙1=𝜒, probability assignment as in Table 12, 𝑁=29.

Figure 11 

Graphical representation of 𝐽𝑞(𝑥𝑞), 𝑞=0,1,2,3,15,19,24,28 for codon-to-codon stochastic mutations, with 𝛼𝑙1=5𝜒, 𝑁=29.

Figure 12 

Graphical representation of 𝐽𝑞(𝑥𝑞), 𝑞=0,1,2,3,15,19,24,28 for codon-to-codon stochastic mutations, with 𝛼𝑙1=0.5𝜒, 𝑁=29.

The simulation results are similar to those in Section 3.2. The profile of 𝐽₀ is more similar to 𝐽₂₉ when 𝛼_𝑙1s are assigned 5𝜒 than 𝜒 or 0.5𝜒. This implies in codon-to-codon stochastic mutations; the optimal control sequence behaves as codon-to-codon deterministic cases, that is, the system tends to getting as close as possible to the final desired set if 𝛼_𝑙1s are much smaller than 𝑑(𝜑₁,𝜑₂), and the system tends to remain in the same state with minor mutations when 𝛼_𝑙1s are relatively larger than 𝑑(𝜑₁,𝜑₂),𝜑₁,𝜑₂∈ℱ³_⧵{0}. And 𝐽_𝑞(𝜑)≤𝐽_𝑞+1(𝜑), 𝜑∈ℱ³_⧵{0} is still valid in codon-to-codon stochastic case.

However, for stochastic cases, we cannot reach a global minimum because of the randomness caused by mutagens. Since, in usual cases, there exists no stationary global minimum, we need to define error tolerance 𝜖, that is, if |𝐽_𝑞(𝜓)−𝐽_𝑞−1(𝜓)|≤𝜖 with the same {𝑥^𝑑_𝑁}, then we can stop at 𝐽_𝑞(𝑥_𝑞). Otherwise, we need to proceed to calculate 𝐽_𝑞−2(𝜓). The value of 𝜖 is decided based on the doctors, experience. Obviously, the smaller 𝜖 is, the better treatment plan is. However, we can still observe Figures 10, 11, and 12 to conclude that 𝐽₀ and 𝐽₂ are almost of the same shape in all three different parameter assignments. Higher dimensional optimal control problems, gene-to-gene stochastic mutations, can be solved as a series of cascade codon-to-codon stochastic problems.

4. Conclusion

In this paper, we present a mathematical model to deal with mutations in the process of DNA replication in the view of control systems. Different from the existing models, our model is constructed directly from the basic biological theories, the central dogma in molecular biology, and the complementary base pair for DNA molecules with double helix structure. It precisely describes how the induced mutations affect the targeted DNA segments at molecular level. It provides instrumental information of molecule interactions in gene mutation for biologists and doctors to gain a better understanding of cellular and tissue level systems’ behavior. Our model is adaptive to point and multisite, deterministic and stochastic mutations. Though we emphasize that we target at induced mutations during the process of DNA replication in our work, this model can be extended to other biological processes at molecular level, such as transcription process and DNA brokage.

In our optimal control problem, the objective function includes two factors: the risk/cost of applying mutagens and the off-trajectory penalty. Under optimal control policy, the summation of those two factors are minimized, by dynamic programming, to propose a low-risk treatment plan. We define the distance reference following the chemical and physical properties of amino acids, representing the penalty. Our objective is to drive the system from given initial state to the final desired set generated by the final desired state at the lowest cost. We define the final desired set since redundancy in genetic codes gives us additional options of final desired state to further lower the cost. Dynamic programming algorithm ensures the optimality of the solution. We also discuss three different small-scale system, and show the simulation results of examples. The optimal control problems of base-to-base deterministic mutations and codon-to-codon deterministic mutations are of theoretical importance. As shown in the propositions, the global optimal can be reached within finite steps. If the step limit is larger than the number of steps that global optimal can achieve, then we have some flexibility in our treatment plan. In addition, there exist multiple optimal paths with the same total cost, given the initial state and the final desired set. The optimal control problem of codon-to-codon stochastic mutations is of practical importance, since codon is the basic component forming long DNA sequences. The step limit 𝑁 is decided by doctors according to patients’ conditions, and the treatment plan is made according to the initial state, the final desired set, and the step limit. Since the doctors constantly take measurement to see the result of treatments at current stage, the treatment plan is updated accordingly. Solving codon-to-codon stochastic optimal control problem is a key step to realize the optimal control to gene-to-gene stochastic mutations in the real world.

Our work contributes to several aspects in systems biology. The optimal control sequences generated by dynamic programming make it possible for biologists and doctors to mutate certain sections of genes on purpose in laboratory, at a relative low cost and low risk, which is an essential step to identify the functional units, to examine the interactions among different segments, and to find healthy, harmful, and lethal nucleotide sequences. All those results are beneficial in gene network construction. In addition, the fundamental details of gene mutations at molecular level help biologists to elaborate on the biological theories at the cellular and tissue levels, such as the theory of evolution. Moreover, by our method, biologists can distinguish the harmful and beneficial mutations and induce beneficial mutations during the evolution process in a proper way, which greatly helps to save rare species in danger. Furthermore, our solution to the optimal control problem proposed provides a new medical intervention to genetic diseases. Compared to the existing gene therapy, treatments by mutagens are safer by avoiding the side effect of virus infection. Lastly, our work also contributes to the construction of DNA computers. Calculation errors, the mispairings in the process of two single-stranded DNA segments, can be corrected at lowest cost by applying a correct mutagen sequence.

Further work can be done by extending codon-to-codon stochastic optimal control problem to gene-to-gene stochastic mutations. The distance reference between DNA segment with equal length can be defined as a weighted sum of the distance references between codons. Since certain combinations of amino acids are lethal, those high-risk states should be avoided. This goal can be achieved by either defining a preset trajectory or eliminating high-risk sequences in the state space. Another possibility is to examine system’s behavior under noisy measurements. Under this situation, the spontaneous mutations can be modeled as an additional random factor in our state updating equations, and another random noises should be added to the output equation.