A Model of Differentiation in Quantum Bioinformatics

1

A Model of Differentiation in Quantum Bioinformatics

2

Masanari Asanoa , Irina Basievab , Andrei Khrennikovb,c , Ichiro Yamato1

3 4 5 6 7 8

9

a

Liberal Arts Division, National Institute of Technology, Tokuyama College, Gakuendai, Shunan, Yamaguchi 745-8585 Japan b International Center for Mathematical Modeling in Physics and Cognitive Sciences Linnaeus University, V¨ axj¨ o-Kalmar, Sweden c National Research University of Information Technologies, Mechanics and Optics (ITMO), St. Petersburg 197101, Russia

Abstract Differentiation is a universal process found in various phenomena of nature. As seen in the example of cell differentiation, the creation diversity on individual’s character is caused by environmental interactions. In this paper, we try to explain its mechanism, which has been discussed mainly in Biology, by using the formalism of quantum physics. Our approach known as quantum bioinformatics shows that the temporal change of statistical state called decoherence fits to describe non-local phenomena like differentiation.

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Keywords: Differentiation; Quantum-like approach; Decoherence process

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1. Introduction

12 13 14 15 16 17 18 19 20 21 22 23

In this study, we focus on phenomenon of differentiation. It is simply explained as follow: A group of individual s with single character gradually splits up into some groups with different characters. Here, “individual” is an organism which can express various physical, chemical or biological activities, and we call its capacity “character.” The individual’s character is adapted to the environment; there exist many kinds of environmental factors which encourage individuals to change their characters. For individuals, interactions with environmental factors are incidental events, and then, the history of character changes in a time interval is expected to be different for each individual. As the result, individuals may get different characters, even if they have a common character at the beginning. We believe that such creation of diversity is a universal process found in various areas of

Preprint submitted to Elsevier

May 1, 2017

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nature. As a typical example, the mechanism of cell differentiation is presented, which is called epigenetics in Molecular Biology. In this phenomenon, the chromatin consisting of DNA and histones in a cell is regarded as one individual who experiences several kinds of chemical modifications such as DNA methylation and histone acetylation. Eventually, the history of these environmental interactions specifies the potentiality of gene expression, that is, cell’s character. Our aim is to describe a general mechanism of differentiation. To do this, we employ a mathematical formalism that has been used in quantum physics and especially in quantum information theory. Recently, it was widely used in psychology and decision making, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Also, it was applied to model behavioral of biological systems, see [12, 14, 15, 16, 17]. This approach known as quantum bioinformatics is applicable at all biological scales: from genome, proteins, cells to animals, ecological and social systems. As is well-known, the quantum physics was established to explain statistical properties of microscopic phenomena. Firstly, we introduce the state representation specific to the quantum physics, that is, the representation by density operator(matrix). A density operator(matrix) is defined on a Hilbert space (a complex vector space) and satisfies Hermiticity (self-adjointness), semi-definite and of trace one. From the definition, it has the following form. X ρ= Pi |xi ihxi |, i

44 45 46 47 48 49 50 51 52 53 54 55 56

By this ρ, a statistical distribution on physical states is encoded: {Pi } are P positive numbers satisfying i Pi = 1 and mean probabilities assigned for physical states. The physical states are represented by the terms {|xi ihxi |} which are called pure states. Mathematically, |xi ihxi | denotes ~xi~xTi , where ~xi is a normalized longitudinal vector and and ~xTi is its transposed one. The vector ~xi = |xi i is called state vector in the sense that its direction specifies a definite physical state (character). Especially, physical states corresponding to orthogonal vectors are assumed to be distinguishable through some physical quantity, that is, different physical values are to be measured for these states. Such orthogonal vectors are frequently given as eigenvectors of a Hermitian operator, whose eigenvalues correspond to real values measured on some physical quantity. Next, we focus on the following change of density operator, which is called

2

57

decoherence. |ΨihΨ| −→

X

Pk |Ck ihCk |.

k 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

The pure state |ΨihΨ| P at the left-hand side represents a single character denoted by Ψ, and k Pk |Ck ihCk | at the right-hand side represents a statisP tical mixture of characters {Ck }. The change from |ΨihΨ| to k Pk |Ck ihCk | means that the state transitions {Ψ 7→ Ck } are realized with the probabilities {Pk }. On the mechanism of stochastic transitions in decoherence, we present the following picture: Decoherence is generally a non-local phenomenon 1 , where chain-reaction transitions are occurring on a network connecting numerous systems. This picture is closely related to the theory of quantum measurement established by von Neumann, where an “apparatus” is considered as an environment for a microscopic system, and the following chain reaction is assumed: A state transition like Ψ 7→ Ck in the system is caused by another state transition in the apparatus. Under this causality, an existence of physical value that can be read out from the apparatus is crucial, because it means an “evidence” of the transition Ψ 7→ Ck . As a realistic problem, to provide an evidence observable for us, the apparatus should be a macroscopic system consisting of numerous elements. We stress here that creation of diversity in differentiation can be generally represented as a decoherence process, and this non-local phenomenon is to be captured in system(individual)-environment interactions. We also remark that the modern representation of ideas of von Neumann is given by the theory of open quantum systems describing interaction of a quantum system with its environment [18]. As was mentioned, we applied this theory and its generalization (theory of quantum adaptive systems) to biology by creating quantum bioinformatics [19]. In Sec. 2, we discuss the interactions between biological systems and their environments by using the state representation of density operator. It is crucial to point our that our “quantum-like” model will be reduced to the concept of quantum measurement, that is, it describes a differentiation phenomenon whose observable evidence becomes formed in the environment. This point is clearly explained by the simple numerical simulations in Sec. 3. 1

The terminology “a non-local phenomenon” means a phenomenon occurring in an open system that interacts with environment.

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88

89 90 91 92

2. A Mathematical Model of Differentiation 2.1. Character of Individual and Statistical Distribution The essence of differentiation is the temporal change of statical state on an individual’s character. In our model, one character is specified by a direction of vector in Hilbert space H: Character Ψ : |Ψi ∈ H.

93 94 95

This vector is normalized, that is, the inner product hΨ|Ψi is one. To describe the statistical distribution of various characters, we employ the form of density operator : The operator called pure state, |ΨihΨ|,

96

represents the existence of character Ψ, and the operator called mixed state, X X Pi |Ψi ihΨi | ( Pi = 1), i

97 98

i

represents the statistical mixture of characters {Ψi } with probabilities {Pi }. The characters {Ψi } are different from each other, that is, hΨi |Ψj i = 6 1 (i 6= j),

99 100 101 102 103 104 105 106 107 108 109 110 111 112

is satisfied. However, they need not be orthogonal, i.e. there can be a kind of overlap between different characters. As was mentioned in the introduction, the individual considered in our model is an organism that can express various activities. If each activity is realized through environmental interactions, it is a sort of behavioral strategy stable with respect to environment. For example, the system of DNA can provide various gene expressions by interacting with environmental elements (chemical modifications). Here, let us consider L kinds of strategies, which are regarded as events mutually exclusive, and assume that these strategies are expressed in characters denoted by {Ck }(k = 1, · · · , L). The exclusivity of events is ensured from the orthogonality of corresponding vectors {|Ck i}; hCk |Ck0 i = δk,k0 . Let H be given as L dimensional complex space CL . Then, {|Ck i} is an orthonormal basis in H, that is, generally, |Ψi ∈ H is expanded as L X |Ψi = ωk |Ck i . (1) k

4

113 114 115 116 117 118

P 2 From hΨ|Ψi = 1, the coefficients {ωk ∈ C} satisfy M k |ωk | = 1. We interpret this form as follow: The individual with character Ψ has the capacity to express L strategies on {Ck } with probabilities {Pk = |ωk |2 }. In other words, the character change Ψ 7→ Ck (the extraction of Ck ) is realizable with the probability Pk . We call this stochastic state transition the differentiation form Ψ to {Ck } and represent it as the shift of density operator |ΨihΨ| 7−→

L X

Pk |Ck ihCk |.

(2)

k=1 119 120

PL

Pk |Ck ihCk | at the right-hand side simply means the statistical mixture of characters {Ck }. Note, the pure state |ΨihΨ| at LHS is expanded as k=1

L X

Pk |Ck ihCk | +

122 123 124 125 126

127 128

P with using the basis {|Ck i}. The second term k6=k0 ωk ωk∗0 |Ck ihCk0 | means the difference between the two matrices. In quantum mechanics, it is called the interference term 2 , and the reduction of interference in Eq. (2) is called decoherence. It is frequently assumed that decoherence is to be occurred through environmental effects. This point is consistent with the mechanism of differentiation we consider. 2.2. Interaction with Environment Decoherence, see Eq. (2) is frequently represented by the following map; M(|ΨihΨ|) =

L X

Mk |ΨihΨ|Mk∗

130 131

=

L X

Pk |Ck ihCk |,

k=1

k=1 129

ωk ωk∗0 |Ck ihCk0 |,

k6=k0

k=1 121

X

where Mk is called measurement operator and defined by Mk = |Ck ihCk |, i.e. Mk is the projector on the vector |Ck i3 . This representation is mathematically simple, but rough: Certainly, the map M is useful for linking the initial 2

In quantum physics interference can be observed in the famous two slit experiment. The possibility to find a quantum-like analog of interference in cognitive science (including the experimental design) was discussed in the works of Khrennikov. In the framework of quantum bioinformatics interference phenomena were demonstrated (both theoretically and experimentally) for a variety of biological systems, e.g., destructive interference in lactose-glucose metabolism for E-coli bacteria [12] 3 In quantum information theory, the map M is a special case of quantum operation or quantum channel. [13]

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132 133 134 135 136 137 138 139 140 141 142

P state |ΨihΨ| and the final state M k=1 Pk |Ck ihCk |, but it does not provide the information about dynamics of the state interacting with the environment. To describe intermediate states in the decoherence process, we have to consider the essential cause of differentiation, that is, the accumulation of environmental interactions. Firstly, we assume the existence of environmental element that directly interacts with the individual. In the example of cell differentiation, the elements correspond to chemicals around the system of DNA. The expression of DNA is controlled through some interactions with them. The element’s character is also represented as vector in a new Hilbert space K. To simplify the discussion, K is defined as N dimensional complex space CN . Let us consider the normalized vector |φi ∈ K expanded as |φi =

N X

νi |ai i,

i=1 143 144 145

where {|ai i}(i = 1 · · · N ) is an orthonormal basis on K = CN . Also for the above form, we can give the interpretation related with differentiation: The element with character φ has the capacity to be differentiated into {ai }; |φihφ| 7−→

N X

Qi |ai ihai |,

(3)

i=1 146 147 148

where Qi = |νi |2 is the probability of character change φ 7→ ai . Mathematically, the LHS and RHS can be linked with the use of measurement operators ¯ i = |ai ihai |}; {M N N X X ¯ i |φihφ|M ¯ i∗ = M Qi |ai ihai |. (4) i=1

149 150 151 152

i=1

Let us consider the initial situation that the individual and environmental element have not interacted yet. Then, the composite system will be described by the following density operator, which is defined on the tensor product of vector spaces H ⊗ K. |ΨihΨ| ⊗ |φihφ| = |Ψ, φihΨ, φ|.

153 154

The vector |Ψ, φi = |Ψi ⊗ |φi ∈ H ⊗ K represents independence of the two ¯ i }, i.e. characters Ψ and φ. It is clear that even if the state transition by {M

6

155 156

the differentiation of Eq. (3) has occurred on the element, the individual’s character Ψ is invariant; |ΨihΨ| ⊗

N X

¯ i |φihφ|M ¯ ∗ = |ΨihΨ| ⊗ M i

i 157 158 159 160

161 162

N X

Qi |ai ihai |.

i

After the individual and environmental element have interacted, some correlation will be created between the characters of two systems. To describe the creation of correlation, we firstly introduce the following operation acting on the initial state. U |Ψ, φihΨ, φ|U ∗ , where U is a unitary operator that satisfies the condition U U ∗ = U ∗ U = I. In our modeling, the form of U=

L X

|Ck ihCk | ⊗ uk

(5)

k=1 163 164

is assumed. Here {uk } is a set of different unitary operators on K. The vector U |Ψ, φi is written as U |Ψi ⊗ |φi =

L X

hCk |Ψi|Ck i ⊗ uk |φi =

166 167 168 169 170 171

ωk |Ck i ⊗ |φk i.

(6)

k=1

k=1 165

L X

We used the relation hCk |Ψi = ωk and defined uk |φi = |φk i. Since |φk i 6= |φk0 i for k 6= k 0 generally, the vector of Eq. (6) cannot be decomposed into two independent vectors, like the initial vector |Ψi ⊗ |φi. Such a nonlocal informational structure, which is called entanglement in the terms of quantum mechanics, represents a unification of characters. Furthermore, we assume ¯ i } has occurred on the system of environthat the state transitions by {M mental element: N X

¯ i )U |Ψ, φihΨ, φ|U ∗ (I ⊗ M ¯ i∗ ). (I ⊗ M

i=1

7

172

¯ i = |ai ihai |, From M L X

¯ i )U |Ψi ⊗ |φi = (I ⊗ M

k=1 L X

=

¯ i |φk i ωk |Ck i ⊗ M ωk |Ck i ⊗ hai |φk i|ai i

k=1 L X

= (

ωk νi|k |Ck i) ⊗ |ai i

k=1

173 174

˜ i i ⊗ |ai i. = |Ψ (7) P 2 Here, νi|k = hai |φk i. (From hφk |φk i = 1, i |νi|k | = 1 is satisfied.) The P L ˜ ii = vector |Ψ k=1 ωk νi|k |Ck i is normalized as L

L

X X ˜ ii ωk νi|k ωk νi|k |Ψ qP qP |Ck i = = |Ck i, |Ψi i = q L L 2 2 ˜ ˜ k=1 hΨi |Ψi i k=1 k0 =1 |ωk0 | |νi|k0 | k0 =1 Pk0 Qi|k0 175

where Qi|k0 = |νi|k0 |2 . Thus, the form of Eq. (7) is rewritten as v u L uX p ¯ i )U |Ψi ⊗ |φi = t Pk0 Qi|k0 |Ψi i ⊗ |ai i = P rai |Ψi i ⊗ |ai i. (I ⊗ M k0 =1

176

(Note, P rai = N X

PL

k0 =1

Pk0 Qi|k0 satisfies

P

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P rai = 1.) As the result, we obtain

¯ i )U |Ψ, φihΨ, φ|U ∗ (I ⊗ M ¯ i∗ ) = (I ⊗ M

i=1 177

i

N X

P rai |Ψi ihΨi | ⊗ |ai ihai |.

(8)

i=1

¯ i acts only on the environmental system, but its In the above, the operator M effect is propagated to the individual since the characters of two systems are unified by U . As seen in the RHS, when the environmental element has the character ai , the individual has Ψi , that is, the correlation of two characters ¯ i } environmental interaction. is created. Hereafter, we call {U, M ¯ i were defined under the existence of enviNote, the operators U and M ¯ i } means the differentiation occurred in ronmental element. Especially, {M the environmental element. Of course, we can discuss its mechanism in more 8

185 186 187 188 189 190 191 192 193 194 195 196 197

198 199 200

¯ i will not be employed, but, then, detail, where the rough representation by M another differentiation occurring in the “environment of environment” will be considered, i.e. a new Hilbert space and projection operators are needed for the description. We consider numerous differentiations occurring in a chain reaction, which are possible in a biological system forming a broad network ¯ i } is a “rational way” to capture of organisms. Our approach using {U, M such non-local phenomenon. The crucial point in our model is that the statistical ensemble {Ψi , P rai }, ¯ i }, is an which is derived by introducing the environmental interaction {U, M intermediate state in the differentiation of Eq. (2): We expect that if the state transitions like Eq. (8) can occur repeatedly, the differentiation of Eq. (2) is realized eventually. In the next subsection, this picture is mathematically proven. 2.3. Process of Differentiation Let us define the following map with using the environmental interaction {U, Mi }. ! N X ¯ i )U (ρ ⊗ σ)U ∗ (I ⊗ M ¯ i∗ ) . Λ∗ (ρ) = trK (I ⊗ M (9) i=1

201 202 203 204

Here, ρ ∈ H is a density operator on individual’s character, σ = |φihφ| ∈ K, and trK (·) means the partial trace on K. This map represents the transition of individual’s character through the interaction with one environmental element. Actually, in the case of ρ = |ΨihΨ|, ∗

Λ (|ΨihΨ|) =

N X

P rai |Ψi ihΨi |

i=1 205 206

is obtained. Note, the map Λ∗ is rewritten as a quantum channel by using the terminology of quantum information theory; Λ∗ (ρ) =

N X

Ei ρEi∗ .

(10)

i=1 207

Here Ei is the linear operator defined by Ei =

L X

νi|k |Ck ihCk |.

k

9

(11)

208 209

For this quantum channel, which is called Kraus representation, one can easily find that the states {|Ck ihCk |} are invariant for the map Λ∗ ; Λ∗ (|Ck ihCk |) = |Ck ihCk |.

210

(12)

Further, one can check that for k 6= k 0 , ∗

Λ (|Ck ihCk0 |) =

N X

∗ νi|k νi|k 0 |Ck ihCk 0 | = hφk |φk 0 i|Ck ihCk 0 |,

(13)

i=1 211 212 213

P ∗ where hφk |φk0 i = N i=1 νi|k νi|k0 . These mathematical properties are closely related to the fact that repeatable transitions given by Λ∗ s realize the differentiation of Eq. (2): Let us consider the n interactions of Λ∗ , ρ(n) = Λ∗ (ρ(n − 1)) = · · · = Λ∗ (Λ∗ (· · · Λ∗ (ρ(0)) · · · )),

214

where ρ(0) = |ΨihΨ| =

L X

Pk |Ck ihCk | +

From the properties of Eqs. (12) and (13), ∗

ρ(1) = Λ (ρ(0)) =

L X

Pk Λ∗ (|Ck ihCk |) +

=

L X

Pk |Ck ihCk | +

ρ(2) = Λ (ρ(1)) =

X

ωk ωk∗0 hφk |φk0 i|Ck ihCk0 |,

L X

Pk Λ∗ (|Ck ihCk |) +

=

Pk |Ck ihCk | +

L X

ωk ωk∗0 hφk |φk0 iΛ∗ (|Ck ihCk0 |)

X

ωk ωk∗0 (hφk |φk0 i)2 |Ck ihCk0 |,

k6=k0

k=1

ρ(n) =

X k6=k0

k=1 L X

ωk ωk∗0 Λ∗ (|Ck ihCk0 |)

k6=k0

k=1 ∗

X k6=k0

k=1

Pk |Ck ihCk | +

X

ωk ωk0 (hφk |φk0 i)n |Ck ihCk0 |,

k,k0

k=1 216

ωk ωk∗0 |Ck ihCk0 |.

k6=k0

k=1 215

X

(14)

are derived. Since |hφk |φk0 i| < 1, lim ρ(n) =

n→∞

L X k=1

10

Pk |Ck ihCk |

(15)

217 218 219 220 221 222 223 224 225 226 227 228

is satisfied, that is, with increasing n, the statistical state ρ(n) approaches to the final state of the differentiation of Eq. (2). This shift is a sort of time evolution, because the parameter n indicates the number of environmental elements interacting with the individual, and it is assumed to increase with time. In the discussion so far, we did not answer the question of what statistical ensemble on individual’s characters exists in the intermediate state of differentiation process. To get the picture, we have to represent ρ(n) of Eq. (14) with the form of mixed state. From the mechanism of differentiation, the individual’s character correlates with the characters of environmental elements, i.e. {ai }. As an example, let us suppose the situation that the characters of two elements are transited to a1 or a2 (N = 2). Then, the four cases, (a1 , a1 ), (a1 , a2 ), (a2 , a1 ), (a2 , a2 ),

229 230 231 232

are considered. The notation (ai , aj ) means that the first element has the character ai and the second one has aj after the interactions. The individual’s characters correlating with the above four cases and the probabilities assigned for them can be derived from the following vectors. ˜ (a ,a ) i = E1 E1 |Ψi, |Ψ ˜ (a ,a ) i = E2 E2 |Ψi, |Ψ 1 1 2 2

233

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˜ (a ,a ) i = E2 E1 |Ψi, |Ψ ˜ (a ,a ) i = E1 E2 |Ψi, |Ψ 1 2 2 1 The vector |Ψi is the initial character, and E1 (E2 ) is the operator of Eq. (11), which constructs the map Λ∗ of Eq. (10). The individual’s character corre˜ (a ,a ) i. lating with (ai , aj ), say Ψ(ai ,aj ) , is specified by the direction of |Ψ i j ˜ (a ,a ) i; The probability assigned for Ψ(ai ,aj ) is estimated from the norm of |Ψ i j ˜ (a ,a ) |Ψ ˜ (a ,a ) i. Note, the relation |Ψ ˜ (a ,a ) i = |Ψ ˜ (a ,a ) i is satP r(ai ,aj ) = hΨ 1 2 2 1 i j i j isfied since the operators {Ei } are commutative, E2 E1 = E1 E2 . In our modeling, the individual’s character is correlated with only how many elements belong to each character4 . Therefore, introducing the notation {ni } = {n1 , n2 , · · · }, where ni = n(ai ) means the number of ai , we redefine the above four vectors as ˜ {2,0} i = |Ψ ˜ (a ,a ) i, |Ψ ˜ {0,2} i = |Ψ ˜ (a ,a ) i, |Ψ 1 1 2 2 4

This condition relates with the property that the individuals converge to a constant distribution of characters with the accumulation of the environmental interactions. The individual’s character will fluctuate eternally and not converge, if it is correlated with an ordering of the interactions.

11

244

245 246

˜ {1,1} i = |Ψ ˜ (a ,a ) i = |Ψ ˜ (a ,a ) i. |Ψ 1 2 2 1 We call {ni } history of interactions hereafter. Generally, the individual’s character with the history {ni } has the direction of ! N N L L Y Y X X Y ˜ {n } i = |Ψ E ni |Ψi = (νi|k )ni |Ck ihCk | |Ψi = ωk (νi|k )ni |Ck i, i

i

i=1 247

248

249 250

i=1

k

k=1

i=1

and its probability is given by   n ˜ {n } |Ψ ˜ {n } i. P r{ni } = hΨ i i n1 , n2 , · · · , nN   n (Note, is the multinomial coefficient calculated as QNn! n ! .) n1 , n2 , · · · , nN i=1 i As the result, we obtain the description of an intermediate state ρ(n) as a mixed state; X ρ(n) = P r{ni } |Ψ{ni } ihΨ{ni } |, (16) {ni }

251

where |Ψ{ni } i =

L X

ωk pP k

k=1

Q

ni i=1 (νi|k )

Pk

Q

ni i=1 (Qi|k )

|Ck i,

(17)

252

P r{ni } =

L X

Pk n!

N Y (Qi|k )ni i=1

k=1

ni !

.

(18)

254

(The vector |Ψ{ni } i is normalized.) In the next section, by simulating this statistical picture, we analyze the process of differentiation numerically.

255

3. Numerical Simulation

253

256 257 258 259

3.1. Case of H = C2 , K = C2 In this section, the process of differentiation is simulated. We consider the case of L, N = 2 (H = C2 and K = C2 ). Then, the statistical ensemble {Ψ{ni } , P r{ni } } is given by |Ψ{ni } i =

2 X k=1

ωk (ν1|k )n1 (ν2|k )n2 qP |Ck i, 2 n1 (Q )n2 P (Q ) 1|k 2|k k=1 k 12

(19)

260

2

P r{ni } 261 262 263 264 265

n! X Pk (Q1|k )n1 (Q2|k )n2 , = n1 !n2 ! k=1

see the definitions of Eqs. (17) and (18). ({ni } = {n1 , n2 } and n1 + n2 = n is always satisfied.) Our aim is to simulate how {Ψ{ni } , P r{ni } } is shifted for the number of interacted environmental elements n. Firstly, to evaluate the character Ψ{ni } of Eq. (19), we introduce the square of inner product of two vectors |C1 i and |Ψi: P1 (Q1|1 )n1 (Q2|1 )n2 |hC1 |Ψ{ni } i|2 = P2 , n1 (Q )n2 P (Q ) k 1|k 2|k k=1

266 267

(20)

(21)

which means the degree of similarity between characters C1 and Ψ{ni } . From the property, lim ρ(n) = P1 |C1 ihC1 | + P2 |C2 ihC2 |, n→∞

268 269 270 271

272 273

274

275

the value of |hC1 |Ψ{ni } i|2 is expected to approach to 0 or 1, if n becomes enough large. Here, the following problem arises: What condition of {ni } determines the allocation of C1 and C2 ? We can clarify it by rewriting Eq. (21) to 1  x   n (0 ≤ x ≤ 1). Q2|2 1−x Q1|2 P2 1 + P1 Q1|1 Q2|1 Here, the variable x corresponds to nn1 (n1 = 0, · · · , n). This function is a step function in the limit of n → ∞;  1 if x = nn1 > α 2 lim |hC1 |Ψ{ni } i| = (22) 0 if x = nn1 < α n→∞ The threshold α is the value of x satisfying     Q1|2 x Q2|2 1−x = 1, Q1|1 Q2|1 which is calculated as α=

log(Q2|1 /Q2|2 ) . log(Q1|2 Q2|1 /Q1|1 Q2|2 ) 13

276

The figure. 1 shows the behavior of |hC1 |Ψ{ni } i|2 for n1 /n at P1 = 0.7, P2 = 0.3,

277

Q1|1 = 0.5, Q2|1 = 0.5, Q1|2 = 0.3, Q2|2 = 0.7. 278 279 280 281 282 283 284 285 286 287 288 289 290 291

(23)

One can see that with increasing n, the characters {Ψ{ni } } approach to C1 log 7/5 ≈ 0.4. As was mentioned or C2 . The threshold α is estimated as log 7/3 in the introduction, when differentiation of the individual’s character has occurred, some “observable evidence” is formed in the environment. In our model, it will correspond to the quantity like n1 /n, which is expected to be observable at enough large n. Then, the result of n1 /n > α (n1 /n < α) can be regarded as the evidence of that the individual’s character converges to C1 (C2 ). The existence of threshold like α is crucially important in a phenomenon of differentiation. In the case of cell differentiation, for example, it will be found in the concentrations of chemicals, which modifies a specific part of DNA. Next, we focus on the probability P r{ni } of Eq. (20). Since the value of P r{ni } becomes very small at large n, the following probability is analyzed actually. X   P rn a < |hC1 |Ψ{ni } i|2 ≤ b = P r{ni } . (24) {ni }∈Hn [a,b]

292 293

294

295

Here, Hn [a, b] means the set of histories satisfying a < |hC1 |Ψ{ni } i|2 ≤ b (0 ≤ a < b ≤ 1) on the individual’s character Ψ{ni } ;  Hn [a, b] = {ni }|a < |hC1 |Ψ{ni } i|2 ≤ b . In the simulation, with using the probabilities,   P rn (l) = P rn 0.1(l − 1) < |hC1 |Ψ{ni } i|2 ≤ 0.1l , l = 1, · · · 10, the population, Nn (l) = 1000 × P rn (l),

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P is calculated with the same parameters settings of Eq. (23). ( 10 l=1 P rn (l) = 1 is satisfied.) The figure. 2 shows the behavior of {Nn (l), l = 1, · · · 10} for various n. At the beginning all of the 1000 individuals have same character √ (n = 0), √ Ψ. Since |Ψi = 0.7|C1 i + 0.3|C2 i, N0 (7) = 1000. One can find that with increasing n, the populations {Nn (l)} are shifted and approach to the result of N (1) = 300 and N (10) = 700. 14

Figure 1: The behaviors of |hC1 |Ψ{ni } i|2 of Eq. (21) for n1 /n, which are calculated at n = 5, 20, 50, 200: One can see that with increasing n, the individual’s character Ψ{ni } approaches to C1 for n1 /n > α ≈ 0.4 and C2 for n1 /n < α ≈ 0.4. (In this simulation, we set P1 = 0.7, P2 = 0.3, Q1|1 = 0.5, Q2|1 = 0.5, Q1|2 = 0.3 and Q2|2 = 0.7.)

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3.2. Case of H = C3 , K = C2 With using the same way discussed in Sec.3.1, we simulate the case of L = 3 and N = 2 (H = C3 and K = C2 ). The vector |Ψ{ni } i and the probability P r{ni } are written as |Ψ{ni } i =

3 X k=1

ωk (ν1|k )n1 (ν2|k )n2 qP |Ck i, 2 n n 1 2 k=1 Pk (Q1|k ) (Q2|k )

307

3

P r{ni } 308

n! X = Pk (Q1|k )n1 (Q2|k )n2 . n1 !n2 ! k=1

The figure. 3 shows the values of Pk (Q1|k )n1 (Q2|k )n2 |hCk |Ψ{ni } i| = P3 (k = 1, 2, 3), n1 n2 t=1 Pt (Q1|t ) (Q2|t ) 2

15

(25)

Figure 2: The shift of population of individuals: Nn (l) means the population whose character satisfy the condition 0.1(l − 1) < |hC1 |Ψ{ni } i|2 ≤ 0.1l. The total number of individuals is 1000.

309

for n1 /n, where we set P1 = 0.5, P2 = 0.3, P3 = 0.2,

310

Q1|1 = 0.5, Q2|1 = 0.5, 311

Q1|2 = 0.4, Q2|2 = 0.6, 312

Q1|3 = 0.3, Q2|3 = 0.7. 313 314

(26)

Since the individual’s character Ψ{ni } approaches to C1 , C2 or C3 with increasing n, in the limit of n → ∞, there exists two thresholds α, β satisfying lim |hC1 |Ψ{ni } i|2 = 1 If α ≤ n1 /n,

n→∞

lim |hC2 |Ψ{ni } i|2 = 1 If β < n1 /n < α,

n→∞

lim |hC3 |Ψ{ni } i|2 = 1 If n1 /n ≤ β.

n→∞

16

(27)

Figure 3: The behaviors of |hCk |Ψ{ni } i|2 (k = 1, 2, 3) of Eq. (25) for n1 /n: With increasing n, the character Ψ{ni } approaches to C1 for n1 /n ' 0.45, C2 for 0.45 ' n1 /n ' 0.35 and C3 for 0.35 ' n1 /n.

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319

In the simulated case, α ≈ 0.45 and β ≈ 0.35 are estimated. The figure.4 shows the shift of populations {Nn (l1 , l2 )(l1,2 = 1, 2, · · · 10)}, which is calculated in the same setting of Eq. (26). The population Nn (l1 , l2 ) is defined by Nn (l1 , l2 ) = 1000 × P rn (l1 , l2 ), where X

P rn (l1 , l2 ) =

P r{ni } .

{ni }∈Hn (l1 ,l2 ) 320

Hn (l1 , l2 ) in the above means the set of histories satisfying 0.1(l1 − 1) < |hC1 |Ψ{ni } i|2 ≤ 0.1l1 ∧ 0.1(l2 − 1) < |hC2 |Ψ{ni } i|2 ≤ 0.1l2 .

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One can see that the populations Nn (10, 1), Nn (1, 10) and Nn (1, 1) approach to 500, 300 and 200, respectively.

17

Figure 4: The shift of populations of individuals in differentiation: Nn (l1 , l2 ) means the population of characters satisfying 0.1(l1 − 1) < |hC1 |Xi|2 ≤ 0.1l1 and 0.1(l2 − 1) < |hC2 |Xi|2 ≤ 0.1l2 . One can see that the populations Nn (10, 1), Nn (1, 10) and Nn (1, 1) approach to 500, 300 and 200, respectively.

323

324 325 326

4. Summary and Conclusion In order to describe the dynamics of differentiation, we employed the state representation specific to the quantum theory, i.e. the density operator. It is defined in a finite Hilbert space and expanded as X ρ= ρij |Ci ihCj |, ij

327 328 329 330 331 332 333 334 335

with the orthogonal basis {|Ci i}. (The above form is mathematically equivalent to the matrix with components {ρij }.) As was explained in Sec. 2.1, the differentiation of individual’s character, where the initial character Ψ eventually achieve to the statistical ensemble of {Ck , Pk } is captured as the decoherence, i.e. the vanishment of non-diagonal part of the density matrix (the term of quantum interference), see Eq. (2). In Sec. 2.2 and 2.3, we discussed the process of differentiation, that is, we defined the intermediate states, which the individuals will experience for the achievement of {Ck , Pk }. Any intermediate state has the form of mixed 18

336

state like ρ=

X

Pi |Ψi ihΨi |,

i 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371

which corresponds to the statistical ensemble {Ψi , Pi }. It should be noted here that generally, the vectors {|Ψi i} are not orthogonal P with 0each0 other. Thus, the above ρ can be mathematically rewritten as j Qj |Ψj ihΨj | with using another set of vectors {|Ψ0j i} and probabilities {Qj }: one density matrix itself never identifies a unique statistical picture. Which ensemble is to be realized as an intermediate state? We believe, it depends on what mechanism is assumed in the background of state change. Actually, the intermediate ensemble {Ψ{ni } , P r{ni } } of Eq. (16) was defined based on the following idea. The differentiation occurs through the accumulation of environmental interactions, which are constructed by the unitary operator U of ¯ i } of Eq. (4). The operator U uniEq. (5) and the measurement operators {M ¯ i roughly fies the characters of individual and environmental element, and M represents the differentiation occurring on environmental system. Through ¯ i propagates to the network created by U , the effect of state transition by M the system of individual. The history of such interactions is denoted by {ni }, and it is reflected in the structures of {|Ψ{ni } i} and the values of probabilities {P r{ni } }, see Eq. (18) and Eq. (17). In the quantum information theory, the Lindblad equation [20, 21] is frequently used to describe the decoherence process phenomenologically. This equation is derived under the assumption that the system has “negligible influence” on the state of environment. On the other hand, we are concerned about the accumulation of these influences as the history of interactions. In Sec. 3, we numerically analyzed the behavior of ensemble {Ψ{ni } , P r{ni } }. As was shown in Fig. 2 or 4, {Ψ{ni } , P r{ni } } approaches to {Ck , Pk } with increasing the number of interacted environmental elements n. Furthermore, the figures 1 and 3 show that the history {ni } specifies which character the individual gets eventually. In other words, the differentiated character can be predicted if {ni } is observable: the system consisting of numerous environmental elements plays the role of measurement apparatus. These results suggest the necessity of more detailed statistical analysis for biological phenomena. The probabilities like {Pk } and {Qi|k } in Eq. (20) might become measurable, which will be useful information to predict the property of differentiation, that is, the evolution of ensembles {Ψ{ni } , P r{ni } } and its dependency on the history of environmental interactions {ni }. Here, someone might have the following question. Is it possible to sim19

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ulate the same results in some classical model? Many theorists have tried to explain biological phenomena by using various approaches based on the classical probability theory. In the paper of [22], for an example, a proliferative process of cancer cells was analyzed mathematically. New cancer cells, which are divided from old ones, transit (differentiate) into several phenotypes. The authors of [22] pointed out, the populations labeled by these phenotypes evolve like a Markov chain. We have to note that their aim is to capture the shift of population caused by the cell differentiation, but not to describe the mechanism of differentiation itself. We believe, not only the change of population, “the change of character” is also the essence of differentiation process. These two aspects should not be confused. In the framework of probability theory, an existence of character will be encoded as an element of sample space, however, it is just a “label” of population. On the other hand, in our approach, a character is represented as a direction of vector, and the simultaneous changes of the individual’s character and its population is described in the transition of density matrix. As a state representation, the density matrix is informationally-rich more than the form of probability distribution, and therefore, the simulation of our result is difficult for any classical approach. We stress this point as a contribution of the quantum bioinformatics. [1] Bagarello, F. (2012), Quantum Dynamics for Classical Systems: with Applications of the Number Operator, Wiley Ed., New York. 90, 015203. [2] Busemeyer, J. R. and Bruza, P. D. (2012), Quantum Models of Cognition and Decision, Cambridge Press, Cambridge. [3] Haven, E. (2008), “Private information and the ‘information function’: A survey of possible uses”, Theory and Decision, Vol. 64, pp. 193-228. [4] Khrennikov, A. (2010), Ubiquitous Quantum Structure: from Psychology to Finances, Springer, Berlin-Heidelberg-New York. [5] Khrennikov, A. (2015), “Quantum-like modeling of cognition”, Frontiers in Physics, Vol. 3, id. 77. [6] Asano M, Ohya M, Tanaka Y, Khrennikov A, Basieva I. (2011). Dynamics of entropy in quantum-like model of decision making: AIP Conference Proceedings, 63: 1327. 20

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[7] Asano M, Masanori O, Tanaka Y, Khrennikov A, Basieva I. (2011). Quantum-like model of brain’s functioning: Decision making from decoherence. Journal of Theoretical Biology, 281,1 : 56-64. [8] Asano M, Ohya M, Khrennikov A. (2011). Quantum-Like Model for Decision Making Process in Two Players Game - A Non-Kolmogorovian Model. Foundations of Physics; 41: 538-548. [9] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y. (2012). Quantum-like dynamics of decision-making. Physica A: Statistical Mechanics and its Applications; 391,5: 2083-2099. [10] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y. (2012). Quantum-like generalization of the Bayesian updating scheme for objective and subjective mental uncertainties. Journal of Mathematical Psychology; 56, 3: 166-175. [11] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y. (2016). A Quantum-like Model of Selection Behavior. Journal of Mathematical Psychology: DOI: 10.1016/j.jmp.2016.07.006 [12] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y, Yamato I. (2012). Quantum-like model of diauxie in Escherichia coli: Operational description of precultivation effect. Journal of Theoretical Biology; 314, 7: 130-137. [13] Accardi L, Ohya M. (1999). Compound channels, transition expectations, and liftings: Appl. Math. Optim., Vol.39, 33-59. [14] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y, Yamato I. (2013). A model of epigenetic evolution based on theory of open quantum systems. Syst Synth Biol 7: 161. [15] Asano M, Hashimoto T, Khrennikov A, Ohya M, Tanaka A. (2014). Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures. Physica Scripta, Volume 2014, Number T163 [16] Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y, Yamato I. (2015) Quantum Information Biology: From Information Interpretation

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of Quantum Mechanics to Applications in Molecular Biology and Cognitive Psychology. Found Phys, 45: 1362. [17] Asano M, Khrennikov A, Ohya M, Tanaka Y, Yamato I. (2016). Threebody system metaphor for the two-slit experiment and Escherichia coli lactose-glucose metabolism. Philosophical Transactions of the Royal Society A :DOI: 10.1098/rsta.2015.0243 [18] Ohya M, Volovich I. (2011). Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems. Springer. [19] Asano M, Khrennikov A, Ohya M, Tanaka Y, Yamato I. (2015). Quantum Adaptivity in Biology: From Genetics to Cognition. Springer. [20] Lindblad G. (1976). On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48 (2): 119. [21] Gorini V, Kossakowski A, Sudarshan E.C.G. (1976). Completely positive semigroups of N-level systems. J. Math. Phys. 17 (5): 821. [22] Piyush B. Gupta, Christine M. Fillmore, Guozhi Jiang, Sagi D. Shapira, Kai Tao, Charlotte Kuperwasser, Eric S. Lander. (2011). Stochastic State Transitions Give Rise to Phenotypic Equilibrium in Populations of Cancer Cells. Cell. 146(4): 633.

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