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c ICIC International ⃝2013 ISSN 1881-803X

ICIC Express Letters Volume 7, Number 7(tentative), July 2013

pp. 1–EL12-0921

pp. 2023-2028.

BIOMOLECULE-ASSISTED NATURAL COMPUTING APPROACHES FOR SIMPLE POLYNOMIAL ALGEBRA OVER FIELDS Tomonori Kawano1,2,3,4 1

Faculty and Graduate School of Environmental Engineering The University of Kitakyushu 1-1, Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan [email protected] 2

LINV@Kitakyushu Research Center Kitakyushu, Japan 3

LINV@Firenze University of Florence Sesto Fiorentino, Italy 4

Paris Interdisciplinary Energy Research Institute Paris, France

Received September 2012; accepted November 2012 Abstract. Biocomputing is a recently growing and highly interdisciplinary ﬁeld of research that investigates models and computational techniques inspired by biology and related sciences. Here, cyclic behavior (redox cycling) of puriﬁed horseradish peroxidase protein among native enzyme and its two electron-oxidized and single electron-oxidized intermediates known as Compounds I and II was algebraically expressed as a cyclic additive group Z3 = {C0 , C2 , C1 } = {C0 , 1C2 , 2C2 } = {0, 2, 1}, and a cyclic multiplicative group Z3∗ = {C1 , C2 } = {C1 , C21 } = {1, 2}, with C2 as the common generator. Above algebraically expressed features of the enzyme’s redox cycle was applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld derived from Z3 . Similarly, use of a pair of small DNA molecules was proposed for determining the coeﬃcients for additively and multiplicatively obtained polynomials over F , (Z2 ; +, ×); where Z2 = {C0 , C1 } = {0, 1}. Discussion includes the required designs of two distinct DNA molecules for performing binary logical conjunction (AND) and exclusive disjunction (XOR), upon polymerase-chain reactions. Keywords: Coeﬃcient ﬁeld, DNA, Enzyme, Natural computing, Polynomial ring

1. Introduction. Recently, a number of researchers are involved in development of natural computing systems which employ quantum [1], optical [2], chemical [3] and biological [4] phenomena as the direct media for manifesting the computation. Chieﬂy, studies handling the biological materials such as living cells and biomolecules as key components of computation attracted the biological scientists and informatics engineers [5,6]. Biocomputing is a recently growing and highly interdisciplinary ﬁeld of research that investigates the models and computational techniques inspired by biology and related sciences. Biocomputers are man-made biological networks whose goal is to probe and control the biological hosts, namely the cells and organisms, in which they operate [5]. In the near future, due to robust growth of the area, biocomputers will eventually enable disease diagnosis and treatment with single-cell precision, lead to “designer” cell functions for biotechnology, and bring about a new generation of biological measurement tools [5]. The state of the art “DNA-based biocomputer” now calculates a square root [6]. However, the number of reports on the applications of biomolecule-based computing models in the realistic problems is still limited to date. Therefore, in the present report, a novel 1

2

T. KAWANO

biomolecule-assisted algebraic operation model is proposed for additive and multiplicative operations of simple polynomials over ﬁelds. In the upper half of the report, puriﬁed horseradish peroxidase (HRP) was used as a model material. In the latter half of the report, use of newly engineered DNA molecules is proposed. 2. Biochemical Description of Peroxidase Redox Cycle. In plants, peroxidase family achieves a great deal of oxidation reactions essential for the cells, using hydrogen peroxide (H2 O2 ) as an acceptor of electron (e− ) and a variety of substrates as e− donors [7]. The redox cycles of peroxidases are largely analogous to those found in other hemoproteins. The overall inter-conversions among the native form, Compound I (Co-I), and Compound II (Co-II) of HRP are summarized in Figure 1(a). The conventional catalytic cycle for the oxidation of various substrates by HRP is coupled to the consumption of H2 O2 as follows: Native enzyme (3) + H2 O2 → Co-I (5) + H2 O (1) Co-I

(5)

+ S → Co-II

(4)

+P

(2)

Co-II + S + H → Native protein + H2 O + P (3) − where S and P are the substrate and product of its single e oxidation, respectively [7]. Numbers in the small brackets indicate the formal oxidation states of the prosthetic heme. Co-I and Co-II of HRP are considered to possess the hemes at ferryl states with and without additional porphyrin radicals, respectively [8,9]. Thus, Co-I and Co-II are analogous to the ferryl intermediates of human hemoglobin with and without additional globin radicals, respectively. Eﬀects of model substrates such as phenolics and aromatic amines [10,11] and chemical inhibitors such as nitric oxide [12], which can arrest the enzyme at (4)

+

(3)

Figure 1. Cyclic inter-conversions of horseradish peroxidase (HRP) among native and reactive intermediates. (a) Biochemical view on the redox cycling of HRP. (b) Simpliﬁed redox cycling model consisted of states (S) and transitions (T ). (c) Algebraic description of cyclic inter-conversions of HRP intermediates. (d) Modiﬁed algebraic description of cyclic interconversions of HRP intermediates. Each element in (c) was multiplied by 2 and renamed in (d). Presence of HRP intermediates can be spectroscopically monitored [10,11]. Absorption maxima for native enzyme, Co-I and Co-II are 500 and 639 nm; 577, 622 and 650 nm; and 527 and 556 nm, respectively [11]. For experimental conditions, see main text.

ICIC EXPRESS LETTERS, VOL.7, NO.7, 2013

3

speciﬁc oxidation state (either native form, Co-I and Co-II) has been intensively examined through spectroscopic analysis using HRP and other plant enzymes. 3. Algebraic Expression of Peroxidase Redox Cycle. 3.1. Preliminary mapping. As a simpliﬁed model is shown in Figure 1(b), native enzyme, Co-I and Co-II of HRP are zero-e− -oxidized (with inert FeIII heme), 2-e− -oxidized (with FeIV heme and neighboring cation radical), and 1-e− -oxidized (with FeIV heme) forms of enzyme, respectively. Let these intermediates with diﬀerent oxidation states (S) be S = {0ox , 2ox , 1ox } (4) Similarly, let the transitions of states (T ), namely, (α) 0ox → 2ox (2 e− oxidation, thus +2), (β) 2ox → 1ox (1 e− reduction, thus −1), and (γ) 1ox → 0ox (1 e− reduction, thus −1), be T = {α, β, γ} = { + 2, −1, −1} (5) Here, bijective mapping of (4) S = {0ox , 2ox , 1ox } into residue class Z3 = {C0 , C2 , C1 } and subjective mapping of (5) T = {α, β, γ} into Z3∗∗ = {C2 } were performed in order to algebraically express the cyclic nature of HRP. f1

S = {0ox , 2ox , 1ox } − → Z3 = {C0 , C2 , C1 }

(6)

Note that, by considering T as a residue class (mod. 3), T = {α, β, γ} = {+2, −1, −1} = {+2, +2, +2}. Here, the set of transition steps was mapped as C2 | Z3 , Z3∗∗ , thus, f2

T = {α, β, γ} − → Z3∗∗ = {C2 } = {+2}

(7)

As summarized in Figure 1(c), the cyclic behavior (redox cycling) of HRP protein was algebraically expressed by equivalently treating the elements in the state set {C0 , C2 , C1 } and the only element in the transition set {C2 }. 3.2. Cyclic groups. Note that newly obtained algebraic model (Figure 1(c)) shows both a cyclic additive group (Z3 ) and a cyclic multiplicative group (Z3∗ ), with C2 as the common generator as shown below. Z3 = {C0 , C2 , C1 } = {C0 , 1C2 , 2C2 } = {0, 2, 1}, C2 as the generator

(8)

Z3∗ = {C1 , C2 } = {C1 , C21 } = {1, 2}, C2 as the generator

(9)

3.3. Experimentally testified modifications. For ease of designing the experimental algorithms, elements in (8) Z3 , (9) Z3∗ and (7) Z3∗∗ were multiplied by C2 (f3 , f4 , f5 ) and thus further bijectively mapped into Z3′ , Z3′ ∗ and Z3′ ∗∗ , respectively (Figure 1(d)); and newly obtained elements in Z3′ , Z3′ ∗ and Z3′ ∗∗ were renamed as below. f3

Z3 = {C0 , C2 , C1 } − → Z3′ = {C0′ , C1′ , C2′ } f4

∗

f5

∗∗

→ Z3 ′ = {C2′ , C1′ } Z3∗ = {C1 , C2 } − → Z3 ′ Z3∗∗ = {C2 } −

= {C1′ }

(10) (11) (12)

= {C0′ , C1′ , C2′ } forms a novel additive cyclic group with C1′ as the generator Here, and Z3′ ∗ = {C1′ , C2′ } forms a novel multiplicative cyclic group with C2′ as the generator. C0′ , C1′ and C2′ in Z3′ now correspond to the native enzyme, Co-I and Co-II, respectively; and C1′ in Z3′ ∗∗ corresponds to the experimental step for state jumping. Z3′

4

T. KAWANO

4. HRP Reaction Applicable to Polynomial Algebra over F . In the following sections, the algebraically described features of HRP’s redox cycle were applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld, (Z3′ ; +, ×). Definition 4.1. Let F be a field and then a polynomial over F be defined as follows: F [X] = {f (x)|f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 , ai |F, n = 0, 1, 2, . . .}

(13)

Over F [X] ∋ f (x), g(x); f (x) and g(x) can be described as below. f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 n

n−1

(14)

g(x) = bn x + bn−1 x + . . . + b1 x + b0 Thus, additive (+) and multiplicative (×) operations can be expressed as below.

(15)

f (x) + g(x) = (an + bn )xn + (an−1 + bn−1 )xn−1 + . . . + (a1 + b1 )x + (a + b)

(16)

f (x) × g(x) = (an bn )x2n + (an bn−1 + an−1 bn )x2n−1

(17) + (an bn−2 + an−1 bn−1 + an−2 bn )x2n−2 + . . . + (a1 b0 + a0 b1 )x + a0 b Upon definition of addition (+) and multiplication (×) of polynomials, let (F [X]; +, ×) be a polynomial ring over F . Actual operations mimicked by HRP The cyclic nature HRP reaction (Figure 1) was applied to help determine the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over F (typical examples were selected from [13]). ′ ′ ′ Where Z3′ = {C0 , C1 , C2 } = {0, 1, 2}, let f (x) = x2 +2 and g(x) = 2x+1 be polynomials over Z3 . Here, polynomial addition f (x) + g(x) and multiplication f (x) × g(x) can be solved as follows: f (x) + g(x) = (x2 + 2) + (2x + 1) = x2 + 2x + 3 = x2 + 2x (mod. 3) Here, elimination of 3 (mod. 3) can be supported by the test with HRP cycle showing the presence of native enzyme (C0′ ) after addition of 1:1 molar ratio of H2 O2 over native enzyme (2 e− oxidation equivalent to operation C0′ + C1′ = C1′ ) and 2:1 molar ratio of salicylic acid over Co-I (successive 1 e− reduction steps equivalent to the operation C1′ + C1′ + C1′ = C0′ ) in the test reaction mixture. f (x) × g(x) = (x2 + 2)(2x + 1) = 2x3 + x2 + 4x + 2 = 2x3 + x2 + x + 2 (mod. 3) Here again, conversion of 4x to x (mod. 3) can be supported by the test with HRP cycle showing the presence of Co-I (C1′ ). 5. Use of Other Biomolecules. The above demonstration showed that HRP’s cyclic nature can be mathematically expressed as cyclic groups both additive and multiplicative, thus considered as F . Therefore, behavior of HRP can be the clue to arithmetic operation for determining the coeﬃcients of the polynomial rings over F additively and multiplicatively produced from a pair of polynomial rings f (x) and g(x) over F . Due to the nature of HRP, operations based on Z3 (Z3′ ) were chosen as the target of enzymeassisted application.

ICIC EXPRESS LETTERS, VOL.7, NO.7, 2013

5

However, for handling of Z2 -based problems with binary coeﬃcients which are much more frequently used in digitalized computation, alternative biomolecules other than heme proteins must be selected for widening the use of biomolecule-assisted approaches. 5.1. Proposed use of DNA. Here, the author proposes the use of a pair of engineered DNA molecules designed to perform the binary logical conjunction (AND) and exclusive disjunction (XOR) upon polymerase-chain reactions (PCR) for determining the coeﬃcients for additively and multiplicatively obtained polynomials over a coeﬃcient ﬁeld, (Z2 ; +, ×), where Z2 = {C0 , C1 } = {0, 1}.

Figure 2. Proposed structures of DNA molecules for arithmetic PCR 5.2. Required design for DNA molecules. For both additive and multiplicative operations of coeﬃcients with mod. 2, the arithmetic molecules must return 0 or 1 upon inputs (presence) of A and/or B which are also a set of binary numbers, 0 and 1. Figure 2 shows the designs of molecular tools, with which signals (reﬂecting the ampliﬁcation of DNA) are emitted after PCR. By AND gate molecule, multiplicative operations such as 1 × 1 = 1 with inputs (primer DNA A and B) corresponding to coeﬃcients ai and bi (0 and/or 1) in f (x) × g(x) can be performed. In turn, by XOR gate molecule, additive operations such as 1 + 1 = 0 (mod. 2) with inputs (primer DNA A and B) correspond to coeﬃcients ai and bi in f (x) + g(x) can be performed. By these steps, moleculeassisted determination of coeﬃcients in polynomials over a coeﬃcient ﬁeld, (Z2 ; +, ×), where Z2 = {C0 , C1 } = {0, 1}, can be manifested. Our group has recently designed and manufactured the above DNA molecules functioning as AND and XOR gates. Apart from the polynomial algebra, these molecules are also useful for simple digital arithmetic operations. Our ongoing project aims to speed-up the operations by newly developed hyper-fast PCR system. 6. Conclusion. Redox cycling of HRP among native form and catalytic intermediates (Co-I and Co-II) was algebraically expressed as a cyclic additive group Z3 = {C0 , C2 , C1 }, and a cyclic multiplicative group Z3∗ = {C1 , C2 }, with C2 as the common generator. This model was applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld derived from Z3 . Similarly, use of a pair of small DNA molecules was proposed for determining the coeﬃcients for additively and multiplicatively obtained polynomials over F , (Z2 ; +, ×); where Z2 = {C0 , C1 } = {0, 1}, thus digitalized. Our approach might be useful for further designing the models for biomolecule-assisted natural computing systems. Acknowledgment. This work was supported by a Grants-in-Aid for Scientiﬁc Research (Research Project Number: 23656495) and Regional Innovation Cluster Program 2012 by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

6

T. KAWANO

REFERENCES [1] D. R. Simon, On the power of quantum computation, SIAM J. Computing, vol.26, pp.1474-1483, 1997. [2] T. Kawano, Printable optical logic gates with CIELAB color coding system for Boolean operationmediated handling of colors, Proc. of the 6th Int. Conference on Genetic and Evolutionary Computing, Kitakyushu, Japan, pp.270-275, 2012. [3] T. Head, X. Chen, M. Yamamura and S. Gal, Aqueous computing: A survey with an invitation to participate, J. Comput. Sci. Technol., vol.17, no.6, pp.672-681, 2002. [4] A. Kameda, M. Yamamoto, H. Uejima, M. Hagiya, K. Sakamoto and A. Obuchi, Hairpin-based state machine and conformational addressing: Design and experiment, Natural Computing, vol.4, pp.103-126, 2005. [5] Y. Benenson, Biocomputers: From test tubes to live cells, Molecular Biosystems, vol.5, no.7, pp.675685, 2009. [6] Y. Benenson, Biocomputing: DNA computes a square root, Nature Nanotechnol., vol.6, no.8, pp.465467, 2011. [7] T. Kawano, Roles of the reactive oxygen species-generating peroxidase reactions in plant defense and growth induction, Plant Cell Rep., vol.21, no.9, pp.829-837, 2003. [8] R. Rutter, M. Valentine, M. P. Hendrich, L. P. Hager and P. G. Debrunner, Chemical nature of the porphyrin pi cation radical in horseradish peroxidase Compound I, Biochemistry, vol.22, pp.47694774, 1983. [9] C. E. Schulz, R. Rutter, J. T. Sage, P. G. Debrunner and L. P. Hager, Mossbauer and electron paramagnetic resonance studies of horseradish peroxidase and its catalytic intermediates, Biochemistry, vol.23, pp.4743-4754, 1984. [10] T. Kawano, S. Muto, M. Adachi, H. Hosoya and F. Lapeyrie, Spectroscopic evidence in support of horseradish peroxidase Compound II-catalyzed oxidation of salicylic acid but not of phenylethylamine, Biosci. Biotechnol. Biochem., vol.66, no.3, pp.651-654, 2002. [11] T. Kawano, S. Muto, M. Adachi, H. Hosoya and F. Lapeyrie, Spectroscopic evidence that salicylic acid converts a temporal inactive form of horseradish peroxidase (Compound III) to the irreversibly inactivated verdohemoprotein (P-670), Biosci. Biotechnol. Biochem., vol.66, no.3, pp.646-650, 2002. [12] A. Takayama, T. Kadono and T. Kawano, Heme redox cycling in soybean peroxidase: Hypothetical model and supportive data, Sens. Mater., vol.24, no.2, pp.87-97, 2012. [13] S. Ishimura, Easy Learning Series – Discrete Mathematics, Kyoritsu Shuppan Co. Ltd., Tokyo, 2007.

Lihat lebih banyak...
ICIC Express Letters Volume 7, Number 7(tentative), July 2013

pp. 1–EL12-0921

pp. 2023-2028.

BIOMOLECULE-ASSISTED NATURAL COMPUTING APPROACHES FOR SIMPLE POLYNOMIAL ALGEBRA OVER FIELDS Tomonori Kawano1,2,3,4 1

Faculty and Graduate School of Environmental Engineering The University of Kitakyushu 1-1, Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan [email protected] 2

LINV@Kitakyushu Research Center Kitakyushu, Japan 3

LINV@Firenze University of Florence Sesto Fiorentino, Italy 4

Paris Interdisciplinary Energy Research Institute Paris, France

Received September 2012; accepted November 2012 Abstract. Biocomputing is a recently growing and highly interdisciplinary ﬁeld of research that investigates models and computational techniques inspired by biology and related sciences. Here, cyclic behavior (redox cycling) of puriﬁed horseradish peroxidase protein among native enzyme and its two electron-oxidized and single electron-oxidized intermediates known as Compounds I and II was algebraically expressed as a cyclic additive group Z3 = {C0 , C2 , C1 } = {C0 , 1C2 , 2C2 } = {0, 2, 1}, and a cyclic multiplicative group Z3∗ = {C1 , C2 } = {C1 , C21 } = {1, 2}, with C2 as the common generator. Above algebraically expressed features of the enzyme’s redox cycle was applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld derived from Z3 . Similarly, use of a pair of small DNA molecules was proposed for determining the coeﬃcients for additively and multiplicatively obtained polynomials over F , (Z2 ; +, ×); where Z2 = {C0 , C1 } = {0, 1}. Discussion includes the required designs of two distinct DNA molecules for performing binary logical conjunction (AND) and exclusive disjunction (XOR), upon polymerase-chain reactions. Keywords: Coeﬃcient ﬁeld, DNA, Enzyme, Natural computing, Polynomial ring

1. Introduction. Recently, a number of researchers are involved in development of natural computing systems which employ quantum [1], optical [2], chemical [3] and biological [4] phenomena as the direct media for manifesting the computation. Chieﬂy, studies handling the biological materials such as living cells and biomolecules as key components of computation attracted the biological scientists and informatics engineers [5,6]. Biocomputing is a recently growing and highly interdisciplinary ﬁeld of research that investigates the models and computational techniques inspired by biology and related sciences. Biocomputers are man-made biological networks whose goal is to probe and control the biological hosts, namely the cells and organisms, in which they operate [5]. In the near future, due to robust growth of the area, biocomputers will eventually enable disease diagnosis and treatment with single-cell precision, lead to “designer” cell functions for biotechnology, and bring about a new generation of biological measurement tools [5]. The state of the art “DNA-based biocomputer” now calculates a square root [6]. However, the number of reports on the applications of biomolecule-based computing models in the realistic problems is still limited to date. Therefore, in the present report, a novel 1

2

T. KAWANO

biomolecule-assisted algebraic operation model is proposed for additive and multiplicative operations of simple polynomials over ﬁelds. In the upper half of the report, puriﬁed horseradish peroxidase (HRP) was used as a model material. In the latter half of the report, use of newly engineered DNA molecules is proposed. 2. Biochemical Description of Peroxidase Redox Cycle. In plants, peroxidase family achieves a great deal of oxidation reactions essential for the cells, using hydrogen peroxide (H2 O2 ) as an acceptor of electron (e− ) and a variety of substrates as e− donors [7]. The redox cycles of peroxidases are largely analogous to those found in other hemoproteins. The overall inter-conversions among the native form, Compound I (Co-I), and Compound II (Co-II) of HRP are summarized in Figure 1(a). The conventional catalytic cycle for the oxidation of various substrates by HRP is coupled to the consumption of H2 O2 as follows: Native enzyme (3) + H2 O2 → Co-I (5) + H2 O (1) Co-I

(5)

+ S → Co-II

(4)

+P

(2)

Co-II + S + H → Native protein + H2 O + P (3) − where S and P are the substrate and product of its single e oxidation, respectively [7]. Numbers in the small brackets indicate the formal oxidation states of the prosthetic heme. Co-I and Co-II of HRP are considered to possess the hemes at ferryl states with and without additional porphyrin radicals, respectively [8,9]. Thus, Co-I and Co-II are analogous to the ferryl intermediates of human hemoglobin with and without additional globin radicals, respectively. Eﬀects of model substrates such as phenolics and aromatic amines [10,11] and chemical inhibitors such as nitric oxide [12], which can arrest the enzyme at (4)

+

(3)

Figure 1. Cyclic inter-conversions of horseradish peroxidase (HRP) among native and reactive intermediates. (a) Biochemical view on the redox cycling of HRP. (b) Simpliﬁed redox cycling model consisted of states (S) and transitions (T ). (c) Algebraic description of cyclic inter-conversions of HRP intermediates. (d) Modiﬁed algebraic description of cyclic interconversions of HRP intermediates. Each element in (c) was multiplied by 2 and renamed in (d). Presence of HRP intermediates can be spectroscopically monitored [10,11]. Absorption maxima for native enzyme, Co-I and Co-II are 500 and 639 nm; 577, 622 and 650 nm; and 527 and 556 nm, respectively [11]. For experimental conditions, see main text.

ICIC EXPRESS LETTERS, VOL.7, NO.7, 2013

3

speciﬁc oxidation state (either native form, Co-I and Co-II) has been intensively examined through spectroscopic analysis using HRP and other plant enzymes. 3. Algebraic Expression of Peroxidase Redox Cycle. 3.1. Preliminary mapping. As a simpliﬁed model is shown in Figure 1(b), native enzyme, Co-I and Co-II of HRP are zero-e− -oxidized (with inert FeIII heme), 2-e− -oxidized (with FeIV heme and neighboring cation radical), and 1-e− -oxidized (with FeIV heme) forms of enzyme, respectively. Let these intermediates with diﬀerent oxidation states (S) be S = {0ox , 2ox , 1ox } (4) Similarly, let the transitions of states (T ), namely, (α) 0ox → 2ox (2 e− oxidation, thus +2), (β) 2ox → 1ox (1 e− reduction, thus −1), and (γ) 1ox → 0ox (1 e− reduction, thus −1), be T = {α, β, γ} = { + 2, −1, −1} (5) Here, bijective mapping of (4) S = {0ox , 2ox , 1ox } into residue class Z3 = {C0 , C2 , C1 } and subjective mapping of (5) T = {α, β, γ} into Z3∗∗ = {C2 } were performed in order to algebraically express the cyclic nature of HRP. f1

S = {0ox , 2ox , 1ox } − → Z3 = {C0 , C2 , C1 }

(6)

Note that, by considering T as a residue class (mod. 3), T = {α, β, γ} = {+2, −1, −1} = {+2, +2, +2}. Here, the set of transition steps was mapped as C2 | Z3 , Z3∗∗ , thus, f2

T = {α, β, γ} − → Z3∗∗ = {C2 } = {+2}

(7)

As summarized in Figure 1(c), the cyclic behavior (redox cycling) of HRP protein was algebraically expressed by equivalently treating the elements in the state set {C0 , C2 , C1 } and the only element in the transition set {C2 }. 3.2. Cyclic groups. Note that newly obtained algebraic model (Figure 1(c)) shows both a cyclic additive group (Z3 ) and a cyclic multiplicative group (Z3∗ ), with C2 as the common generator as shown below. Z3 = {C0 , C2 , C1 } = {C0 , 1C2 , 2C2 } = {0, 2, 1}, C2 as the generator

(8)

Z3∗ = {C1 , C2 } = {C1 , C21 } = {1, 2}, C2 as the generator

(9)

3.3. Experimentally testified modifications. For ease of designing the experimental algorithms, elements in (8) Z3 , (9) Z3∗ and (7) Z3∗∗ were multiplied by C2 (f3 , f4 , f5 ) and thus further bijectively mapped into Z3′ , Z3′ ∗ and Z3′ ∗∗ , respectively (Figure 1(d)); and newly obtained elements in Z3′ , Z3′ ∗ and Z3′ ∗∗ were renamed as below. f3

Z3 = {C0 , C2 , C1 } − → Z3′ = {C0′ , C1′ , C2′ } f4

∗

f5

∗∗

→ Z3 ′ = {C2′ , C1′ } Z3∗ = {C1 , C2 } − → Z3 ′ Z3∗∗ = {C2 } −

= {C1′ }

(10) (11) (12)

= {C0′ , C1′ , C2′ } forms a novel additive cyclic group with C1′ as the generator Here, and Z3′ ∗ = {C1′ , C2′ } forms a novel multiplicative cyclic group with C2′ as the generator. C0′ , C1′ and C2′ in Z3′ now correspond to the native enzyme, Co-I and Co-II, respectively; and C1′ in Z3′ ∗∗ corresponds to the experimental step for state jumping. Z3′

4

T. KAWANO

4. HRP Reaction Applicable to Polynomial Algebra over F . In the following sections, the algebraically described features of HRP’s redox cycle were applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld, (Z3′ ; +, ×). Definition 4.1. Let F be a field and then a polynomial over F be defined as follows: F [X] = {f (x)|f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 , ai |F, n = 0, 1, 2, . . .}

(13)

Over F [X] ∋ f (x), g(x); f (x) and g(x) can be described as below. f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 n

n−1

(14)

g(x) = bn x + bn−1 x + . . . + b1 x + b0 Thus, additive (+) and multiplicative (×) operations can be expressed as below.

(15)

f (x) + g(x) = (an + bn )xn + (an−1 + bn−1 )xn−1 + . . . + (a1 + b1 )x + (a + b)

(16)

f (x) × g(x) = (an bn )x2n + (an bn−1 + an−1 bn )x2n−1

(17) + (an bn−2 + an−1 bn−1 + an−2 bn )x2n−2 + . . . + (a1 b0 + a0 b1 )x + a0 b Upon definition of addition (+) and multiplication (×) of polynomials, let (F [X]; +, ×) be a polynomial ring over F . Actual operations mimicked by HRP The cyclic nature HRP reaction (Figure 1) was applied to help determine the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over F (typical examples were selected from [13]). ′ ′ ′ Where Z3′ = {C0 , C1 , C2 } = {0, 1, 2}, let f (x) = x2 +2 and g(x) = 2x+1 be polynomials over Z3 . Here, polynomial addition f (x) + g(x) and multiplication f (x) × g(x) can be solved as follows: f (x) + g(x) = (x2 + 2) + (2x + 1) = x2 + 2x + 3 = x2 + 2x (mod. 3) Here, elimination of 3 (mod. 3) can be supported by the test with HRP cycle showing the presence of native enzyme (C0′ ) after addition of 1:1 molar ratio of H2 O2 over native enzyme (2 e− oxidation equivalent to operation C0′ + C1′ = C1′ ) and 2:1 molar ratio of salicylic acid over Co-I (successive 1 e− reduction steps equivalent to the operation C1′ + C1′ + C1′ = C0′ ) in the test reaction mixture. f (x) × g(x) = (x2 + 2)(2x + 1) = 2x3 + x2 + 4x + 2 = 2x3 + x2 + x + 2 (mod. 3) Here again, conversion of 4x to x (mod. 3) can be supported by the test with HRP cycle showing the presence of Co-I (C1′ ). 5. Use of Other Biomolecules. The above demonstration showed that HRP’s cyclic nature can be mathematically expressed as cyclic groups both additive and multiplicative, thus considered as F . Therefore, behavior of HRP can be the clue to arithmetic operation for determining the coeﬃcients of the polynomial rings over F additively and multiplicatively produced from a pair of polynomial rings f (x) and g(x) over F . Due to the nature of HRP, operations based on Z3 (Z3′ ) were chosen as the target of enzymeassisted application.

ICIC EXPRESS LETTERS, VOL.7, NO.7, 2013

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However, for handling of Z2 -based problems with binary coeﬃcients which are much more frequently used in digitalized computation, alternative biomolecules other than heme proteins must be selected for widening the use of biomolecule-assisted approaches. 5.1. Proposed use of DNA. Here, the author proposes the use of a pair of engineered DNA molecules designed to perform the binary logical conjunction (AND) and exclusive disjunction (XOR) upon polymerase-chain reactions (PCR) for determining the coeﬃcients for additively and multiplicatively obtained polynomials over a coeﬃcient ﬁeld, (Z2 ; +, ×), where Z2 = {C0 , C1 } = {0, 1}.

Figure 2. Proposed structures of DNA molecules for arithmetic PCR 5.2. Required design for DNA molecules. For both additive and multiplicative operations of coeﬃcients with mod. 2, the arithmetic molecules must return 0 or 1 upon inputs (presence) of A and/or B which are also a set of binary numbers, 0 and 1. Figure 2 shows the designs of molecular tools, with which signals (reﬂecting the ampliﬁcation of DNA) are emitted after PCR. By AND gate molecule, multiplicative operations such as 1 × 1 = 1 with inputs (primer DNA A and B) corresponding to coeﬃcients ai and bi (0 and/or 1) in f (x) × g(x) can be performed. In turn, by XOR gate molecule, additive operations such as 1 + 1 = 0 (mod. 2) with inputs (primer DNA A and B) correspond to coeﬃcients ai and bi in f (x) + g(x) can be performed. By these steps, moleculeassisted determination of coeﬃcients in polynomials over a coeﬃcient ﬁeld, (Z2 ; +, ×), where Z2 = {C0 , C1 } = {0, 1}, can be manifested. Our group has recently designed and manufactured the above DNA molecules functioning as AND and XOR gates. Apart from the polynomial algebra, these molecules are also useful for simple digital arithmetic operations. Our ongoing project aims to speed-up the operations by newly developed hyper-fast PCR system. 6. Conclusion. Redox cycling of HRP among native form and catalytic intermediates (Co-I and Co-II) was algebraically expressed as a cyclic additive group Z3 = {C0 , C2 , C1 }, and a cyclic multiplicative group Z3∗ = {C1 , C2 }, with C2 as the common generator. This model was applied to help determining the coeﬃcients in polynomials formed after additive and/or multiplicative operations between polynomial rings f (x) and g(x) over a coeﬃcient ﬁeld derived from Z3 . Similarly, use of a pair of small DNA molecules was proposed for determining the coeﬃcients for additively and multiplicatively obtained polynomials over F , (Z2 ; +, ×); where Z2 = {C0 , C1 } = {0, 1}, thus digitalized. Our approach might be useful for further designing the models for biomolecule-assisted natural computing systems. Acknowledgment. This work was supported by a Grants-in-Aid for Scientiﬁc Research (Research Project Number: 23656495) and Regional Innovation Cluster Program 2012 by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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T. KAWANO

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