A model for analyzing phenomena in multicellular organisms with multivariable polynomials: Polynomial life

Transcription

1 International Journal of Biomathematics Vol. 11, No. 1 (2018) (10 pages) c World Scientific Publishing Company DOI: /S A model for analyzing phenomena in multicellular organisms with multivariable polynomials: Polynomial life Hiroshi Yoshida Department of Mathematics, Kyushu University Motooka 744, Nishi-ku, Fukuoka , Japan yoshida.hiroshi@kyudai.jp Received 16 January 2016 Accepted 11 September 2017 Published 26 December 2017 Most of life maintains itself through turnover, namely cell proliferation, movement and elimination. Hydra s cells, for example, disappear continuously from the ends of tentacles, but these cells are replenished by cell proliferation within the body. Inspired by such a biological fact, and together with various operations of polynomials, I here propose polynomial-life model toward analysis of some phenomena in multicellular organisms. Polynomial life consists of multicells that are expressed as multivariable polynomials. A cell is expressed as a term of polynomial, in which point (m, n) is described as a term x m y n and the condition is described as its coefficient. Starting with a single term and following reductions by set of polynomials, I simulate the development from a cell to a multicell. In order to confirm uniqueness of the eventual multicell-pattern, Gröbner base can be used, which has been conventionally used to ensure uniqueness of normal form in the mathematical context. In this framework, I present various patterns through the polynomial-life model and discuss patterns maintained through turnover. Cell elimination seems to play an important role in turnover, which may shed some light on cancer or regenerative medicine. Keywords: Regeneration; turnover; multivariable polynomials; polynomial life. Mathematics Subject Classification 2010: 92B05 1. Introduction Many fully-developed metazoan tissues remain in a state of flux throughout life, noted Pellettieri and Alvarado [8]. As they pointed out, the multicellular organism maintains itself through cell proliferation, movement and elimination, which is sometimes called turnover. Since turnover is a fundamental phenomenon in life, much remains to be studied on turnover. Cell movement and elimination as well as proliferation seem to play an important role in maintaining tissues like hydra [2]. Taking regeneration as an instance, balance among proliferation, movement and elimination looks essential; actually lack of this balance sometimes gives rise to cancer instead of regeneration [4]. Further, aging and many diseases seem to involve

2 H. Yoshida an imbalance between cell death and cell division [8]. Hence, systematic analysis of turnover might contribute to the understanding of regenerative medicine as well as some serious diseases. As an example of turnover, hydra s cells are observed moving toward the body ends, where they vanish, while cell division (proliferation) or growth throughout the body enables a hydra to live for a long time in constant size and form [2]. Another example of turnover comes from planarians, whose cells are in a state of continuous flux, actually old differentiated cells are exchanged with stem cell descendants [8]. Aside from turnover in real life, I have performed a few theoretical analyses on turnover [14, 15], where I modeled a cell chain with heterodimers, parametrized how these heterodimers are redistributed during cell division, and derived a condition to facilitate turnover. Through these studies, I have found that the condition for turnover indicates a gradient across cells, which plays an important role in regeneration as the conventional, positional information model also suggests [13]. Extending these studies, I here study a method to analyze turnover with the aid of the polynomial-life model, where a multicell is expressed as a polynomial. Using reduction of a polynomial with a set of polynomials, I describe proliferation, movement and elimination of cells, thereby analyzing turnover effectively. For comparison with the polynomial-life model, I here make a brief view of cellular automata (CA) [12] and rewriting systems such as Lindenmayer system (Lsystem) [6, 7]. CA were introduced as the space over which a self-replicating machine works [11]. Typical CA consist of square cells, each of which is in one of the finite number of states. The state of each cell evolves according to the state of itself and its neighbors. All cell states evolve simultaneously in (synchronous) CA. A rewriting system is a finite set of rules u v, where u and v are words [10, Chap. 4]. A typical rewriting system changes only one letter in the word per step, and this transformation depends on neighboring letters. By contrast, an L-system is a parallel rewriting system that was introduced by Lindenmayer for modeling the development of plants such as algae [6, 7]. Parallel means that all of the letters changeable by the rules are transformed per step. Further, the polynomial-life model is compared with CA and rewriting systems including L-systems. The main difference between my model and the other systems mentioned above is that the next cell condition is decided by the outer (environmental) rule (the set of polynomial) and the ordering of cells, while in the other models, the next cell state is decided mainly by neighboring cells states. Under some conditions, the cell proliferates in rewriting systems including L-systems and the polynomial-life model, but it does not proliferate in typical CA. Another difference between L-systems and the polynomial-life model is that the polynomial-life model can directly deal with the rearrangement of cell positions in two- or more-dimensional cell groups with the aid of term order, while L-systems have some difficulties even in two-dimensional (2D) cell sheet; actually, 2D cell sheet must be expressed as branching-structure or needs translation into 2D figures with turtle interpretation [9]

3 A model for analyzing phenomena Through the polynomial-life model, cell elimination looks like playing an important role in turnover, which might shed some light on regenerative medicine by avoiding cancer. 2. Model In this section, I propose a polynomial-life model toward analysis of turnover phenomena in multicells. First, I explain how to express a multicell as a polynomial, and then I shall show how several operations for polynomials designate cell-proliferation, -arrangement and -elimination Multicell expressed as a polynomial In this work, I assume that a multicell consists of some cells with attributes: a position and a condition. Here, 2D sheets of cells are mainly considered so that each cell has an attribution of point (m, n),m,n N. Point(m, n) is described as a term x m y n, and further the condition is described as (c n1 1,cn2 2,...)(n i N), so that each cell is expressed as a term x m y n c n1 1 cn2 2 over Q[x, y, c 1,c 2,...]. I here assume that there exists only one cell at each position, hence, a multicell is expressed as a polynomial, a sum of terms defined as c m,n x m y n c n1 1 cn2 2, (2.1) m,n where c m,n Q designates an additional condition for a cell at position (m, n). Note that c m,n is also called coefficient of a term in the polynomial context. As an example, a polynomial (1+x 2 +x 4 +x 6 )(1+y 2 +y 4 )(c 1 +c 2 y+x(c 2 +c 1 y)) expresses a multicell as illustrated in Fig. 1. This polynomial consists of terms: {c 1 x 7 y 5,c 2 x 6 y 5,c 1 x 5 y 5,c 2 x 4 y 5,...}, meaning that it consists of a cell of c 1 1 c0 2 at Fig. 1. A multicell expressed as a polynomial, (1+x 2 +x 4 +x 6 )(1+y 2 +y 4 )(c 1 +c 2 y+x(c 2 +c 1 y)). In this checkerboard pattern, a point with c 1 is painted white, while a point with c 2 is painted gray. The point at the leftmost and bottom is (0, 0) and the one at the rightmost and top is (7, 5). For example, (1, 1) is painted white according to the term c 1 xy, while (3, 2) is gray due to c 2 x 3 y

4 H. Yoshida point (7, 5), one of c 0 1 c1 2 at (6, 5), one of c1 1 c0 2 every additional condition of the cells is 1. at (4, 5) and so on. In this multicell, 2.2. Operations for polynomials I here show some operations for polynomials, based on [3, Chap. 2] and extending [16]; first, I explain how to order terms of polynomials, which is called term order, and then I define head term and head coefficient with the term order. Finally, I will show polynomial reduction with respect to the head term. Term order. A term order is a total order that satisfies two conditions: t s ut us and t 1fortermss, t, u. I here use two kinds of term order: lexicographical order (in short, lex order ) and degree reverse lexicographical order (in short, drl order ) as defined in the following: Assume that the order among the variables: x 1,x 2,...,x m (m 2) is x 1 x 2 x m and let n =(n 1,n 2,...,n m ) be the degree vector of a term x n1 1 xn2 2 x nm m, which is also denoted by x n in the sequel. lex order. When α and β are degree vectors, x α x β if and only if for some i (1 i m), α 1 = β 1, α 2 = β 2,...,α i 1 = β i 1,α i >β i holds. For example, in the case of two variables x, y with x y, arelation1 y y 2 x xy xy 2 holds. drl order. When α and β are degree vectors, x α x β if and only if m i=1 α i > m i=1 β i or ( m i=1 α i = m i=1 ) and for some i (1 i m) α m = β m,α m 1 = β m 1,..., α i+1 = β i+1,α i <β i holds. For example, in the case of three variables x, y, z with x y z, arelation1 z y x z 2 yz xz y 2 xy x 2 z 3 holds. With a term order, terms can be ordered in a unique way. In the case of three variables x, y, z with x y z, terms in a polynomial f =4x 4 +3z 5 + x 3 z +7yz 4 + x 2 y 2 are ordered as follows: x 4 x 3 z x 2 y 2 yz 4 z 5, yz 4 z 5 x 4 x 2 y 2 x 3 z, using lex order, using drl order. In each sequence, the first term is called head term, denoted by HT here, hence HT(f) =x 4 with lex order and HT(f) =yz 4 with drl order. Accordingly, head coefficient, denoted by HC, is the coefficient of the head term, and head monomial, denoted by HM, is HC HM, so that HC(f) =4andHM(f) =4x 4 with lex order, HC(f) =7andHM(f) =7yz 4 with drl order. Reduction. Next, I show reduction of a polynomial f by a polynomial g with some term order. If some term, t, off is divisible by HT(g), f is transformed into f mg( h), where m is a term such that t = mhm(g). This transformation is called reduction and is represented as f g h. For example, in the case of two variables: x, y with x y, lex order, and polynomials: f =3x 2 y 3 + y 2 +3,g=2xy 2, f is

5 A model for analyzing phenomena Table 1. Relation between cell and polynomial operations. Cell Polynomial operation Example A cell a term a cell with condition c at (2, 3): cx 2 y 3 A multicell a polynomial Update of cell condition reduction Ordering of cells term order lex or drl order Erasing of cells reduction with x n or y m Erasing cells with x 2ory 3: reduction with {x 2,y 3 } Note: To eliminate cells with x 2andy 3, {x 2 y 3 } is used. transformed into f 3x 2 y 3 /(2xy)g =3xy 2 + y 2 +3( h 1 ) because 3x 2 y 3,aterm of f, is divisible by H(g) =xy. Such divisibleness is denoted by xy 3x 2 y 3 in the sequel. Further, h 1 g y 2 +3y +3( h 2 ) because of xy 3xy 2.h 2 cannot, however, be transformed anymore because h 2 has no terms divisible by HT(g) =xy. Such a polynomial is called the normal form with respect to g. Likewise, a polynomial, f, is transformed by a set of polynomials, G if f g f for some g G, whichis denoted by f G f.further,f G f is defined as the reflexive transitive closure of G. Some relations between cell and polynomial operations are shown in Table 1. Please also note that arrangement of 2D cell sheet is effectively described by term order, thereby rearrangement after cell division being surrounded by cells in every direction is also described with little difficulty. Gröbner basis. In general, the normal form of a polynomial with respect to G sometimes differs, depending on applying order of polynomials in G for reduction. In this paper, such a difference causes some troubles in interpreting a polynomial as a multicell, which is mentioned in the previous subsection. To overcome this, in some cases, I use a Gröbner basis of G, which assures the uniqueness of the normal form [1, Theorem 5.35]. For example, when f = xy 2 x, g 1 = xy +1,g 2 = y 2 1, using lex order with x y, bothf g1 x y and f g2 0 hold, and hence both of x y and 0 are normal forms of f. AGröbner basis of {g 1,g 2 } with respect to lex order with x y can be calculated as G = {x + y, y 2 1}. Using any applying order of any elements of G, f is reducted into 0, that is, f G 0. The uniqueness of the normal form with Gröbner basis means the unique form of a multicell, thereby the eventual form of a multicell after some transitions can be confirmed. To calculate Gröbner bases, I used Mathematica or Singular [5] in this work. 3. Results and Discussion I first introduce a simplified pattern, which was inspired by the retina, using the polynomial-life model, and then show more complex patterns with additional operations being exploited. I finally discuss the turnover phenomenon from the viewpoint of the polynomial-life model

6 H. Yoshida 3.1. Simplified retina pattern To generate a simple pattern, inspired by the retina, I started with a single cell denoted by c 0, which means a cell of condition c 0 at point (0, 0). The set of polynomials for reduction was S = {c 0 c 1 c 2,c 1 (1 + x 4 )(1 + y 4 )(1 + x 8 )(1 + y 8 ),c 2 (x 2 + y 2 )(x + y + xy + x 2 y + xy 2 ),c 3 (1 + x 4 )(1 + y 4 ),c 4 (1 + x 8 )(1 + y 8 )}, by which c 0 was eventually transformed into (1 + x 4 )(1 + x 8 )(1 + y 4 )(1 + y 8 ) (x 2 + y 2 )(x + y + xy + x 2 y + xy 2 ) after 18 steps. This polynomial can be depicted as Fig. 2(a) according to the Model section. I further added an eliminating rule E = {x 10,y 10 } to the above set S and started with c 0, from which I eventually obtained a pattern as illustrated in Fig. 2(b). As mentionedintable1,sete eliminates cells whose x- ory-position is equal to 10 or more. Using this framework, various patterns can be obtained; for example, reduction of c 0 with set {x 2 y 2 1} S yielded Fig. 2(c). I confirmed uniqueness of the normal forms, corresponding to the eventual patterns, by reduction with Gröbner basis. The Gröbner basis of set S with respect to c 0 c 1 c 2 x y with lex order is {c 2 x 3 x 2 y x 3 y,..., x 5 y 16 x 9 y 16 x 13 y 16 }. One reduction of c 0 with this basis immediately yielded the polynomial, meaning the eventual pattern (Fig. 2(a)). Thus, using a Gröbner basis often makes steps of reduction to reach the eventual pattern ( normal form in the polynomial context) much shorter Some more complex patterns with additional operations So far, we have seen some patterns generated with reductions of polynomials only. These are relatively simple patterns such as periodic ones due to finiteness (a) (b) (c) Fig. 2. A pattern of the simplified retina: the grayscale is painted depending on the value of coefficients of terms; 1 corresponds to black, and 0 to white, after substituting with c 0 = 1,c 1 =1,andc 2 = 2. (a) The figure corresponds to a polynomial (1 + x 4 )(1 + x 8 )(1 + y 4 )(1 + y 8 )(x 2 + y 2 )(x + y + xy + x 2 y + xy 2 ), starting from c 0 with set S only. (b) Starting from c 0 with set S E, apolynomialx 9 y 8 + x 9 y xy 2 + y 7 + y 3 was obtained, corresponding to the right. (c) The pattern from reduction of c 0 with S {x 2 y 2 1}

7 A model for analyzing phenomena (a) Fig. 3. A more complex pattern: in these figures, every cell is painted gray since every coefficient of xy termsis1. (a) Q 7 i=1 (1 + x2i + y 2i ) and (b) a pattern generated with combination of the three operations. of polynomial reduction. In order to generate more complex patterns, additional operations look necessary. I here adopted polynomial multiplication, x y replacement and reflection with respect to some line, which mean multiplication by some polynomial, exchange between variables x and y in polynomials and reflection that yields line symmetry with the original image. Polynomial multiplication can produce fractal-like patterns as an example illustrated in Fig. 3(a). Exchange between x and y can be done by simple substitution of letters (x and y) in formulas; further, line-symmetry reflection with respect to x = m(m N) was here implemented with polynomial reduction and subsequent letter substitution. For a polynomial, f, the following two procedures yield a polynomial corresponding to the reflective image that is line symmetric to x = m: (i) reduction of z 2m f with {mx 1}, a (ii) substitution z with x in the normal form of (i). Combination of the above three operations can yield some more complex patterns. For example, let p be x(1 + x 3 + x 6 + x 2 y + x 4 y + x 3 y 2 ), and let r and a(m) beoperationsofx y replacement and line-symmetry reflection with respect to x = m, respectively. Further, let h(m) beoperation1+r + a(m)r + ra(m)r, then calculate p 2 = (1 + x 32 h(4) + x 2 32 )p, p 3 =(1+x 33 h(13) + x 2 33 )p 2, and p 4 =(1+x 34 h(40) + x 2 34 )p 3. (1 + r)p 4 yields a pattern as illustrated in Fig. 3(b). (b) 3.3. Toward turnover Using the above-mentioned operations for polynomials, I here surveyed some patterns that show turnover. For this purpose, polynomial reduction with {x m,y n } (m, n N) and polynomial multiplication were used. a Any term order will do

8 H. Yoshida (a) Fig. 4. Regeneration Experiment I: t 1,t 2 and t 3 are painted white, gray and black, respectively. When the cell types are mingled, the cell is painted grayscale as a mingled color. (a, left) The unit pattern. (b, right) The cell-type pattern after some steps. This pattern is maintained through both elimination and proliferation. (a) (b) (b) (c) Fig. 5. Regeneration Experiment II: t 1,t 2 and t 3 are painted white, gray and black, respectively. When the cell types are mingled, the cell is painted grayscale as a mingled color. (a, left) The unit pattern. (b, right) The cell-type pattern after some steps. (c, below) Without elimination, cells increase in number rapidly

9 A model for analyzing phenomena I started with a unit pattern described with a polynomial t 1 xy(1 + x)(1 + y)+ t 2 (x + x 2 + y 3 (x + x 2 )+y + y 2 + x 3 (y + y 2 )) + t 3 (1 + x 3 + y 3 + x 3 y 3 ), where t 1,t 2 and t 3 represent cell types. This unit pattern is illustrated in Fig. 4(a). Then, the following two operations were applied repeatedly: (i) multiplication by polynomial (1 + x m1 + x m2 + )(1 + y n1 + y n2 + )(m 1 < m 2 <,n 1 < n 2 < ), (ii) polynomial reduction with {x m,y n }. Figure 4(b) shows the obtained pattern in the case of multiplication by (1 + x 3 + x 7 + x 10 + x 14 )(1 + y 4 + y 10 + y 13 )and reduction with {x 9,y 7 }. This pattern consists of 669 cells, which are eliminated on the boundary, represented with the polynomial reduction, but are supplied with cells through proliferation, with the polynomial multiplication. Another example is shown in Fig. 5. The unit pattern (Fig. 5(a)) is derived from (1 + t 1 x + t 2 x 2 )(1 + y + t 3 y 2 ). Multiplication by (1 + x 3 + x 6 + x 9 + x 12 )(1 + y 3 + y 5 + y 7 ) and reduction with {x 7,y 8 } followed repeatedly, which yielded a pattern with 418 cells illustrated in Fig. 5(b). Such maintenance of cells together with elimination and proliferation is called turnover in biology. It looks like that such turnover appears whatever degree of x or y in the multiplication and the reduction is used. Further, in patterns to show turnover, elimination looks essential to prevent unlimited proliferation of cells. Actually, without elimination corresponding to polynomial reduction, cells increase in number rapidly. For example, Fig. 5(c) shows a pattern with 4158 cells after only two steps without elimination in the case of Fig. 5(b). Such rapid increase of cell number seems to correspond to cancers behavior in real tissues [4]. Hence, from the view of the polynomial-life model, incessant removal of cells might be effective to placate cancer Conclusion In this work, multicells were modeled as multivariable polynomials. Cell elimination is likely to play an important role in turnover where it avoids unlimited proliferation. This result might contribute to regenerative medicine, where cancer, corresponding to unlimited proliferation, sometimes bringsaboutturnoverinstead of regeneration. Acknowledgment This work was supported by Mext Kakenhi Grant No References [1] T. Becker and V. Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra (Springer-Verlag, Heidelberg, 1993). [2] R. D. Campbell, Cell movements in hydra, Amer. Zool. 14 (1974) [3] D. A. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. (Springer- Verlag, New York, Secaucus, NJ, USA, 2007). [4] D. R. Green and G. I. Evan, A matter of life and death, Cancer Cell 1 (2002)

10 H. Yoshida [5] G.M.GreuelandG.Pfister,A Singular Introduction to Commutative Algebra, 2nd edn. (Springer, 2007). [6] A. Lindenmayer, Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs, J. Theor. Biol. 18(3) (1968) [7] A. Lindenmayer, Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs, J. Theor. Biol. 18(3) (1968) [8] J. Pellettieri and A. S. Alvarado, Cell turnover and adult tissue homeostasis: From humans to planarians, Ann. Rev. Genet. 41 (2007) [9] P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants (Springer- Verlag, New York, 1990). [10] G. Rozenberg and A. Salomaa, Handbook of Formal Languages (Springer-Verlag, 1997). [11] J. von Neumann and A. W. Burks, Theory of Self-Reproducing Automata (University of Illinois Press, Urbana, 1966). [12] S. Wolfram, Universality and complexity in cellular automata, Physica D 10 (1984) [13] L. Wolpert, Positional information and pattern formation in development, Dev. Genet. 15 (1994) [14] H. Yoshida, A condition for regeneration of a cell chain inspired by the Dachsous-Fat system, J. Math. Ind. 3 (2011) [15] H. Yoshida, A pattern to regenerate through turnover, Biosystems 110 (2012) [16] H. Yoshida, A model towards analysis of regenerating patterns using multivariable polynomials Polynomial life, in Proc. 21st Int. Symp. Artificial Life and Robotics (B-Con Plaza, Beppu, Japan, 2016), pp