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A pattern to regenerate through turnover
BioSystems 110 (2012) 43–50
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A pattern to regenerate through turnover Hiroshi Yoshida ∗ Faculty of Mathematics, Kyushu University, Ito, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
a r t i c l e
i n f o
Article history: Received 31 March 2011 Received in revised form 28 April 2012 Accepted 3 August 2012 Keywords: Turnover Dachsous–Fat system Regeneration Continuous flux
a b s t r a c t Tissues of animals and plants are maintained through balanced cell growth, movement, and elimination. Although cells are exchanged perpetually, the whole structure of the tissue is maintained. This form of maintenance is called cell turnover. Here I propose a bio-inspired model of patterns that regenerate through turnover. This model is derived from the Dachsous–Fat system, which has recently attracted much attention because it is considered to facilitate regeneration in insect legs. In this model, I parameterized the manner of the redistribution of Dachsous and Fat during cell division, and then derived equations in the parameters that enable the patterns to regenerate and maintain themselves through turnover. I extended the equations derived in the one-dimensional model into a two-dimensional model. Finally, I discuss a possible relationship between regeneration and turnover. © 2012 Elsevier Ireland Ltd. All rights reserved.
1. Introduction Richard D. Campbell, who carried out many experiments in Hydra, noted the following (Campbell, 1974, p. 523): “The cells and tissues of hydra are in continuous flux. The polyp undergoes perpetual growth and tissue loss, coupled by balanced cell renewal patterns involving all cell types.” Here I propose a bio-inspired model of patterns that regenerate in flux; in other words, patterns whose elements are perpetually exchanged even as the whole structure is maintained. As Campbell indicated, a Hydra’s tissues move toward the body extremities, where they vanish, while cell division or growth occurs throughout the body column (Campbell, 1967; Gilbert, 2010; Berking, 2003). Thus, a Hydra can live for a long time with a constant size and form by balancing the loss and growth of tissues. Through cell growth, movement, and elimination, tissues are well maintained; this is referred to as cell turnover. Another example of cell turnover occurs in planarians, whose cells are in a state of continuous flux. In fact, old differentiated cells are replaced by the progeny of dividing adult stem cells (neoblasts) (Pellettieri and Alvarado, 2007). At the organ level, the intestine is maintained through turnover, wherein the three states of cells, stem cells, semidifferentiated cells, and fully differentiated cells, are orchestrated to update and maintain function on the surface (d’Onofrio and Tomlinson, 2007). Hence, cell turnover may be a key means of tissue homeostasis for many tissues. Cell turnover cannot be observed explicitly, but it seems essential for tissues to maintain or regenerate themselves. Hence, cell turnover was incorporated into the model and a condition for patterns to regenerate through turnover was derived. The model was
∗ Tel.: +81 92 802 4402; fax: +81 92 802 4405. E-mail address: [email protected] 0303-2647/$ – see front matter © 2012 Elsevier Ireland Ltd. All rights reserved. http://dx.doi.org/10.1016/j.biosystems.2012.08.001
inspired by the Dachsous–Fat (Ds–Ft) signaling system, which is considered to facilitate regeneration in cricket legs (Bando et al., 2009). The Ds–Ft system is also thought to be a process underlying the steepness hypothesis, which states that leg size and regeneration are regulated through a gradient across cells (Lawrence et al., 2008; Bando et al., 2011). In addition to experimental studies, the author has studied regeneration phenomena through numerical simulation of connected chaotic elements (Yoshida et al., 2005), where cell types are regarded as various types of attractors within chemicals’ state space, providing a unique view of cockroach leg regeneration (Yoshida and Kaneko, 2009). The author has also modeled a multicellular organism using a Lindenmayer system (L-system) to derive a condition for the coexistence of various cell types (Yoshida et al., 2011), wherein an algebraic method was used to analyze the model with parameters. Here, I focused on molecules between cells, rather than within cells, and have parameterized how the molecules behave during cell division. Previous work Yoshida (2011) dealt with molecules between cells and analyzed a condition for cells to regenerate when any two in the cell chain are extracted and developed. Extending the previous work, I incorporated cell loss into the model described in the present work, and analyzed patterns that maintain and regenerate themselves through balanced loss and growth: namely, through turnover. I first constructed a one-dimensional model that was described by a cell chain based on the Ds–Ft heterodimer system. I set each cell having a value on each side. I next parameterized the redistribution ratio of values during cell division, and derived equations from the parameters that enable the pattern to be maintained through turnover. I extended the equations obtained for the one-dimensional model to a two-dimensional model. Finally, I discuss a possible relationship between regeneration and turnover.
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2. Model I propose a model that is bio-inspired by the Ds–Ft system, which has been considered to facilitate regeneration in cricket legs. First, I introduce the steepness hypothesis, which the Ds–Ft system is deemed to substantiate. The steepness hypothesis is a model that describes a condition for a cell chain to halt cell proliferation. As illustrated in Fig. 1(a), the gradient of a given chemical is assumed to become less steep as the length of a given cell chain becomes longer. When the gradient reaches a given threshold value, the cell chain is assumed to halt its proliferation. Fig. 1(b) shows a gradient of Ds and Ft molecules across cells, which has been postulated to explain regeneration phenomena in cricket legs. It is argued that the cricket starts to regenerate its excised legs by proliferating cells and gives up regeneration (cell proliferation) when the gradient of Ds–Ft heterodimers reaches a certain threshold (Bando et al., 2009, 2011). In addition, Fig. 1(c) further illustrates reverse intercalary regeneration with the help of a gradient of Ds–Ft heterodimers. When a distally amputated leg is grafted with a proximally amputated one, reverse intercalary regeneration occurs so that value continuity is recovered. Just after the graft, a steep (reverse) Ds–Ft gradient is formed at the junction and this keeps cells proliferating until the gradient is less than the threshold, resulting in recovery of the amputated portion. The Ds–Ft heterodimeric system is thus considered to be involved in regeneration and to be a materialization of the steepness hypothesis. I accordingly propose a model, where each cell has a value on each side and the cell grows or vanishes according to these values. Because the phenomenon of cell growth and elimination is called cell turnover, the word turnover will be used in the sequel. First, I modeled a one-dimensional cell chain that can regenerate through turnover. Fig. 2 illustrates this model in which a single cell divides to become a cell chain of a certain length and then both the extremities vanish perpetually, while the inherent chain pattern is maintained. The inset shows that each cell is assumed to have a value on each of its sides (left: lv ; right: rv ) and that the value on the newly created wall after cell division is assigned as: p1 lv + p2 rv ,
(1)
where p1 and p2 are parameters denoting the redistribution ratio. Such a redistribution of the value on the cell wall is an extension of the steepness hypothesis (Lawrence et al., 2008), where a newly created cell (not cell wall) takes up an intermediate positional value from its neighbors, corresponding to p1 = p2 = 1/21 in (1). Although the manner of this redistribution is not well understood, here the Ds–Ft redistribution (1) is adopted for a mathematical simplification. Next, the development and turnover of the cell chain is defined as follows. (i)
(ii)
1
Start with a single cell that has values of a and b on its left and right sides, respectively. In the sequel, let n denote the cell number at which the cell chain begins the turnover process. For each cell (ith cell), enumerate the cell number between the ith cell and both ends of the cell chain. Let ln,i and rn,i be the cell numbers between the ith cell and the leftmost and rightmost cells, respectively. If ln,i + rn,i > n holds and the cell is situated at either of the ends, this cell vanishes. Otherwise, the cell divides into two. At this time, let lv,i and rv,i be the values on the left and right sides of the ith cell, respectively. The value on the newly
In this case, regeneration in the cricket leg can be explained as shown in Fig. 1(c).
(iii)
created cell wall is set to p1 lv,i + p2 rv,i according to Formula (1). If ln,i ≤ rn,i , the new cell is situated on the right of the original cell; otherwise, it is situated on the left, which ensures the turnover because older cells are moved closer to the ends of the cell chain. Perform the previous procedure for every cell simultaneously, and repeat this procedure. The state where some cells begin vanishing is named turnover phase.
I here make a brief view of cellular automata (CA) and rewriting systems including the L-system. CA were introduced as the space over which a self-replicating machine works (von Neumann, 1966). Typical CA consist of square cells, each of which is in one of a finite number of states, such as 0 and 1. The grid of cells can be n (≥1) dimensional, but a two-dimensional grid is usually used. The state of each cell evolves according to the state of itself and its neighbors (in two-dimensional CA, the eight cells surrounding a central cell, the so-called Moore neighborhood, and the four cells orthogonally surrounding a central cell, the so-called Neumann neighborhood, are commonly used). All cell states evolve simultaneously in (synchronous) CA (Schiff, 2008). A rewriting system is a finite set of rules u → v, where u and v are words (Rozenberg and Salomaa, 1997, Chapter 4). A typical rewriting system transforms only one letter in the word per step, and this transformation depends on neighboring letters.2 By contrast, a L-system is a parallel rewriting system that was introduced by Aristid Lindenmayer for modeling the development of plants such as algae (Lindenmayer, 1968a,b). Parallel means that all of the letters changeable by the rules are transformed per step. L-systems have lately been utilized as a code to produce computer graphics of various plants (Prusinkiewicz and Lindenmayer, 1990). I further briefly compare this model with CA and rewriting systems including L-systems. The main difference between this model and the other systems described above is that the next state of the cell is determined by the values between cells, not by the cells’ state. Under some conditions, the cell proliferates in rewriting systems including L-systems and this model, but it does not proliferate in typical CA. Another difference between L-systems and this model is that this model deals with the arithmetic (addition, subtraction, multiplication, and division of numbers) of values between cells. Some L-systems also perform arithmetic for modeling concentrations in the tissue, such as nutrients and inhibitors (Lindenmayer et al., 1974). Further a map L-system simulate two-dimensional cellular layers (Prusinkiewicz and Lindenmayer, 1990, Chapter 7), but they have difficulty in handling the changes in neighboring relations of cells, which was dealt with in this model. As an example of this model, I start with one cell having a value of 2 on the left wall and of 1 on the right. After one cell division, the newly created cell wall has a value of 2p1 + p2 according to Formula (1). After two divisions, the four-cell chain has values (2, p22 + 2p1 (1 + p2 ), 2p1 + p2 , p2 + p1 (2p1 + p2 ), 1) from the left to the right. During cell division, Procedure (ii) assures that the old cells are situated closer to either of the ends of the cell chain, while newly created cells are situated in an inner position. If n (the cell number at which the cell chain begins turnover) is set at, say, 6, both the ends start to vanish after three cell divisions, bringing the cell chain into the turnover phase. I summarize the procedure in Fig. 3. Starting with a single cell with values a and b on the ends, each cell divides into two until the cell number reaches n. Then, the cell chain moves into turnover, during which cells on both ends vanish while the inner cells proliferate. In this turnover phase, regeneration of a pattern is evaluated
2 Some rewriting systems have no dependency of neighboring letters, and such independency is sometimes called context-free.
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Fig. 1. The steepness hypothesis and the Ds–Ft system. (a) The steepness hypothesis. The linear gradient of a certain chemical is assumed and is shown by the grayscale shading. The slope of the gradient becomes less steep as the cell chain becomes longer and the cell proliferates until the slope declines below some threshold value. (b) A schematic illustration of the Ds–Ft system. The Ds–Ft heterodimeric bridges are formed by Ds and Ft molecules between cells during cell division. The gradient of Ds–Ft heterodimers across the cells is considered to provide cells with a polarity or regeneration cue (Strutt, 2009). In this figure, the number of Ds–Ft heterodimers decreases from left to right and this gradient substantiates steepness in the steepness hypothesis. (c) An explanation of reverse intercalary regeneration in the cricket leg. When a distally amputated leg (with the eighth value in the figure) is grafted with a proximally amputated one (with the fourth value), reverse intercalary regeneration occurs such that the positional–value continuity is recovered. Just after the grafting, a steep (reserve) Ds–Ft gradient is formed at the junction. This steep gradient keeps cells proliferating until the gradient is below the threshold, resulting in recovery of the 7-6-5th cells.
divide
lv
p1
p2
rv
.....
p1 lv + p2rv
........
turnover
vanish
vanish
Fig. 2. Schematic illustration of the model. A single cell divides to become a cell chain of a certain length, and then both its extremities vanish along with cell division, progressing into the turnover phase. The turnover phase may consist of a few different patterns. The figure inset depicts how a cell divides and creates a new wall.
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H. Yoshida / BioSystems 110 (2012) 43–50
(i) Initial
a
b
(ii) Divide and Create a new wall
a
m
b
m=ap1 + bp 2
m1=bp22+ap1 (1+p2)
m1 m2 m3 b m2=ap1 + bp2
a
m3=ap12+bp2 (1+p1 ) ..... dividing
...
new
original
...
If l n,i
> r n,i
rn,i
ln,i .....
.......
turnover
................
.....
n
vanish
vanish .....
Fig. 3. The development and turnover phases of a cell chain. (i) The values of the ends are set as a and b. (ii) Each cell divides into two and the value on the newly created cell wall is set according to Formula (1) until the number of cells reaches n. In cell division, the original (old) cell is situated on the side with the shorter distance from either of the cell chain ends, while the newly created cell is situated closer to the center part of the cell chain. This positioning ensures cell turnover. When the cell number reaches n, cells at both ends vanish, going into the turnover phase; this might contain a few subphases.
by confirming whether the values are maintained along the cell chain during the turnover state. Note that the turnover phase can be composed of two or more steps with distinct patterns; in such cases, the regeneration is defined as a periodic reproduction of the pattern in the turnover phase.
(ap1 + ap1 p2 + ap1 p22 + ap1 p32 + bp42 , ap1 + ap1 p2 + ap1 p22 + bp32 , ap1 + ap1 p2 + bp22 , ap21 + ap1 p2 + ap21 p2 + bp22 + bp1 p22 , ap1 + bp2 , ap21 + bp1 p2 + ap21 p2 + bp22 + bp1 p22 , ap21 + bp2 +bp1 p2 , ap31 + bp2 + bp1 p2 + bp21 p2 , ap41 + bp2 + bp1 p2 + bp21 p2 +bp31 p2 ),
3. Methods Here, I describe the condition for regeneration through turnover as a set of algebraic equations. The derivation and analysis of the equations are shown in Section 4. Before that, I briefly explain an obstacle in the analysis of algebraic equations and describe a method to overcome it. Imagine the following set of equations: {x3 − 2x2 y + xy − 2y2 − 4x + 8y = 0, 2xy − 4y2 − 3x + 6y = 0} . The solution to the above equations can be decomposed into two solutions of smaller and simpler form: {x − 2y = 0} and {x2 + y − 4 =0, 2y − 3 =0}.
One cannot decide x and y based on the former, but can decide x(= ±
5/2) and
y(= 3/2) based on the latter. It is therefore useful and usually necessary to perform this kind of decomposition in analyzing complicated algebraic equations because we cannot know the decidability of solutions in advance. Such decomposition of algebraic equations is referred to as prime ideal decomposition, which has been used recently to analyze some systems (Romanovski et al., 2007; Yoshida et al., 2010). Here the prime ideal decomposition is calculated by using the routine minAssChar in Singular software (Greuel et al., 2005).
4. Results I obtained a condition for a pattern to maintain and regenerate itself through turnover. As mentioned in Section 2, I started with a single cell that has a value of a on its left side and b on its right. For instance, I set the cell number for starting the turnover (denoted by n) as eight, and developed the cell chain up to five steps, obtaining the following vector:
(2) which designates values from the leftmost to rightmost cells. Another five steps yielded the following vector: (ap21 + 2ap21 p2 + bp1 p22 + 2ap21 p22 + ap1 p32 + bp1 p32 + ap21 p32 + bp42 + bp1 p42 , ap21 + 2ap21 p2 + ap1 p22 + bp1 p22 + ap21 p22 + bp32 + bp1 p32 , ap21 + ap1 p2 + ap21 p2 + bp22 + bp1 p22 , ap31 + ap1 p2 + ap21 p2 + ap31 p2 + bp22 + bp1 p22 + bp21 p22 , ap1 + bp2 , ap21 + bp1 p2 + ap21 p2 + bp1 p22 + ap21 p22 + bp32 + bp1 p32 , ap21 + bp1 p2 + ap21 p2 + bp22 + bp1 p22 , ap31 + ap21 p2 + bp21 p2 + ap31 p2 + bp22 + 2bp1 p22 + bp21 p22 , ap41 + ap21 p2 + ap31 p2 + bp31 p2 + ap41 p2 + bp22 + 2bp1 p22 + 2bp21 p22 + bp31 p22 ).
(3)
The condition for a pattern to maintain itself through turnover can be calculated by Vector (2) − Vector (3) = 0. This condition itself was so complicated that, as mentioned in Section 3, prime ideal decomposition was applied to obtain eight components of
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Fig. 4. A two-dimensional model. Each cell has four values, one each for its left, top, right, and bottom sides. Start with a single cell having values a, b, c, and d. Each cell divides alternately in the horizontal and vertical directions. The inset shows the redistribution ratio of values. In the same manner as the one-dimensional model, after cell division a newly created cell is situated toward the thicker cell layer. When the width (height) of the cell group reaches m (n), it progresses to the turnover phase.
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(b)
(a) horizontal
The turnover phase step 19
step 22
vertical
step 25
step 28
step 29
=step 19
Fig. 5. A two-dimensional example with a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8. (a) At step 19, a monotonous gradient appears in both the horizontal and vertical directions. The white of the grayscale used here corresponds to the smallest value (−4) and the black to the largest value (4). The upper figure (horizontal) shows the gradient of the values on the right side of each cell and the bottom figure (vertical) shows the gradient of the values on the top side. (b) In the turnover phase, the pattern has a period of 10 steps. To make it clear, the gradient pattern is mapped onto a smile pattern. The smile pattern at step 19 becomes the same after 10 iterations.
simpler form: (I) a = b = 0, (II) b = p1 = 0, (III) p1 − 1 = p2 = 0, (IV) a = p2 = 0, (V) p1 = p2 = 0, (VI) p1 p2 − 1 = ap21 + bp2 = bp22 + ap1 = 0, (VII) p1 + p2 − 1 = a − b = 0, (VIII) p1 = p2 − 1 =0. Of these eight components, only (VI) provided a heterogeneous pattern; the other components generate trivial cell chain patterns, such as 0 . . . 0 and 1 . . . 1. Under component (VI), the vector (2) (of course equal to (3)) became: (−bp22 (1 + p2 + p32 ), −bp22 (1 + p22 ), −bp32 , −bp32 , − b(−1 + p2 )p2 , b, b, b(1/p2 + p2 ), b(1 + 1/p22 + p2 )).
(4)
As another example, I set n = 12 and developed a single cell with (a, b). The cell number of the cell chain could be expressed by the series {1, 2, 4, 8, 14, 12, 12, 12, 12, 14, 12, 12, 12, 12, 12, 12, 14, 12, 12, 12, 12, 12, 12, 14, . . . }, which means that after the 10th step the cell chain moved into the turnover phase with a period of seven.3 Hence, a condition for a pattern to maintain itself through turnover was derived by comparing the pattern at the 10th step with the one at the 17th. The obtained condition that makes the patterns at the 10th and 17th steps equal was revealed to be the same as the eight components, using prime ideal decomposition. Likewise, I set n = 18 and developed a single cell with (a, b). The number in the cell chain
3 The series contains “14”; this is more than n(= 12) because the two cells exist at the ends of the chain, but they vanish at the next step. Such cells appear in the following series also.
became the series {1, 2, 4, 8, 16, 20, 18, 18, 18, 18, 18, 20, (18)8 , 20, (18)9 , 20, (18)9 , 20, . . . }, where the superscript denotes repetition. This series means that after the 21st step, the cell chain progressed into the turnover phase with a period of 10. The condition that makes the patterns at the 21st and 31st steps equal was again the same as the eight components, using prime ideal decomposition. I further confirmed for n = 8, 10, 12, . . ., 30 that the cell number of the chain became the series {1, 2, 4, 8, . . ., n + 2, nm , n + 2, . . ., n + 2, nn/2 , n + 2, . . . }, where the superscript denotes repetition and m was some number between n/2 −2 and n/2. The series eventually showed a period of n/2 +1, and the condition for the cell pattern to recover after n/2 +1 steps during the turnover phase was found to be the same as the eight components, only the sixth (VI) of which gave a nontrivial pattern. As discussed below, the sixth condition (VI) enables the cell chain to maintain its pattern through turnover for an even number, n, in general. 5. Discussion I have obtained a condition for patterns to maintain themselves through turnover, which is the sixth component (VI) in Section 4. This condition can be reduced to: p1 = 1/p2 ,
a = −bp32 .
(5)
Under this condition, the pattern has a gradient of values when p2 > 0. In the case of n = 8, the obtained pattern Vector (4) can also be described as: b(v1 , v2 , v3 , v3 , v4 , 1, 1, v5 , v6 ),
(6)
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where v1 ≤ v2 ≤ v3 ≤ v4 ≤ 1 ≤ v5 ≤ v6 when p2 > 0 because 1 + p2 + p32 ≥ 1 + p22 ≥ p2 , p32 > (−1 + p2 )p2 , 1 < 1/p2 + p2 , and 1/p2 + p2 < 1 + 1/p22 + p2 .4 In addition, the manner in which a pattern can maintain itself through turnover is as follows. Under condition (5), the cell chain starts with a cell having (− p3 , 1) at both its ends. In the sequel, denote p2 by p and set b as 1 for simplicity. After one cell division, the newly created cell wall has a value of (− p3 )/p + 1 × p = p − p2 , which gives the cell chain values of (− p3 , p − p2 , 1) from the leftmost to rightmost. After another cell division, the cell chain acquires values of (− p3 , − p3 , p − p2 , 1, 1), where the second value −p3 comes from −p3 /p + (p − p2 ) × p and the fourth 1 comes from (p − p2 )/p + 1 × p. The values at the next step become (− p3 , − p2 (1 + p), − p3 , − p3 , p − p2 , 1, 1, p + 1/p, 1), whose central part (− p3 , p − p2 , 1) is always maintained. This is why the pattern of values is maintained during the turnover phase. In addition, according to the cell number n, which denotes the number of cells at start of the turnover phase, the length(s) of the cell chain in the turnover are determined. One of the values in the turnover phase in the case of n is (− p2 (1 + p + p2 + · · · + pm ), − p2 (1 + p + p2 + · · · + pm−1 ), . . ., − p2 (1 + p2 + p3 ), − p2 (1 + p2 ), − p3 , p − p2 , 1, (1 + p2 )/p, (1 + p2 + p3 )/p2 , . . ., (1 + p2 + · · · + pm−1 )/pm−2 , (1 + p2 + · · · + pm )/pm−1 ), where m = n/2. Like Vector (6), this vector shows a gradient when p > 0. In order to show one possible extension of the one-dimensional model, I extend the condition (5) and the gradient pattern (6) into a two-dimensional model as illustrated in Fig. 4. Each cell is assumed to divide alternately in the horizontal and vertical directions; this simple dividing manner is adopted for a mathematical simplification even though the actual cell may divide toward any of more directions. As an extended condition of (5), I assume the following relationship between the parameters: p1 = 1/p2 ,
a = −bp32 ,
p3 = 1/p4 ,
c = −dp34 ,
(7)
where a, b, c, and d are the values on the left, right, top, and bottom sides, respectively, and p1 , p2 (p3 , p4 ) is a pair of the distribution ratio of values in the horizontal (vertical) direction. The cell group moves into the turnover phase when the width reaches m or the height reaches n. For example, using a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8, Fig. 5(a) shows the gradient of values created in the horizontal and vertical directions in the same manner as the one-dimensional chain (see (6)). At step 19, which is in the turnover phase, both the horizontal and vertical gradients become (− 4, − 3, − 2, − 1, 0, 1, 2, 3, 4). I map these gradients onto a given pattern. For instance, Fig. 5(b) illustrates a smile pattern by mapping cells with (two-dimensional) gradient points, (± 2, 3), (± 3, 0), (± 2, − 1), (± 1, − 2), (0, 1), (0, − 2) onto black and others onto white. From steps 19 to 22, the smile pattern wears away outward, and then it recovers up to step 29. This shows that although the components are perpetually changing into newer cells, its (value) pattern is maintained through turnover. Here I exemplify the result using a specific pair of parameters (a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8). This result can be extended to a more general case as long as p2 > 0 and Eq. (7) holds. The gradient appears at least once in the turnover phase, as shown in Fig. 5(b). To conclude, I have obtained the condition for a cell chain to maintain its pattern through turnover. Under this condition, a oneor two-dimensional gradient appeared. In conventional models of regeneration, such as the positional information and the steepness theories (Wolpert, 1994; Lawrence et al., 2008), a gradient across
2
4 All these inequalities are reducible to an inequality, p22 − p2 + 1 = (p2 − 1/2) + 3/4 > 0.
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cells was assumed, providing a cue for regeneration. This is consistent with the gradient that results from a condition in which patterns maintain themselves through turnover in my model. 6. Conclusion In this work, I incorporated turnover into the model and obtained a condition in which one-dimensional patterns maintain themselves. I extended the obtained condition into a twodimensional model. This model indicates that it is possible to design some patterns that can maintain and regenerate themselves through turnover. The gradient emerging from the turnover condition is consistent with the conventional model for regeneration. Acknowledgments I thank the editor and the anonymous reviewers for helpful comments that improved this manuscript. I am grateful to Prof. Sumihare Noji and Dr. Tetsuya Bando for valuable discussions regarding cricket legs and I had warm encouragement of Ms. Atsuko Sono. The author was supported by MEXT KAKENHI Grant Number 23124509. This study was also supported in part by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Japan Science and Technology Agency (JST). References Bando, T., Mito, T., Maeda, Y., Nakamura, T., Ito, F., Watanabe, T., Ohuchi, H., Noji, S., 2009. Regulation of leg size and shape by the Dachsous/Fat signalling pathway during regeneration. Development 136 (13), 2235–2245. Bando, T., Mito, T., Nakamura, T., Ohuchi, H., Noji, S., 2011. Regulation of leg size and shape: Involvement of the Dachsous-Fat signaling pathway. Dev. Dyn. 240 (5), 1028–1041. Berking, S., 2003. A model for budding in hydra: pattern formation in concentric rings. J. Theor. Biol. 222, 37–52. Campbell, R.D., 1967. Tissue dynamics of steady state growth in Hydra littoralis II. patterns of tissue movement. J. Morphol. 121, 19–28. Campbell, R.D., 1974. Cell movements in Hydra. Am. Zool. 14, 523–535. d’Onofrio, A., Tomlinson, I.P., 2007. A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt. J. Theor. Biol. 244 (3), 367–374. Gilbert, S.F., 2010. Developmental biology, 9th ed. Sinauer Associates, Inc., Sunderland, USA. Greuel, G.M., Pfister, G., Schonemann, H., 2005. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, http://www.singular.uni-kl.de. Lawrence, P.A., Struhl, G., Casal, J., 2008. Do the protocadherins Fat and Dachsous link up to determine both planar cell polarity and the dimensions of organs? Nat. Cell Biol. 10 (12), 1379–1382. Lindenmayer, A., 1968a. Mathematical models for cellular interactions in development. I. filaments with one-sided inputs. J. Theor. Biol. 18 (3), 280–299. Lindenmayer, A., 1968b. Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. J. Theor. Biol. 18 (3), 300–315. Lindenmayer, A., 1974. Adding continuous components to L-systems. In: Rozenberg, G., Salomaa, A. (Eds.), L-Systems, Vol. 15 of Lecture Notes in Computer Science. Springer, pp. 53–68. Pellettieri, J., Alvarado, A.S., 2007. Cell turnover and adult tissue homeostasis: from humans to planarians. Annu. Rev. Genet. 41, 83–105. Prusinkiewicz, P., Lindenmayer, A., 1990. The Algorithmic Beauty of Plants. SpringerVerlag, New York. Romanovski, V.G., Chen, X., Hu, Z., 2007. Linearizability of linear systems perturbed by fifth degree homogeneous polynomials. J. Phys. A: Math. Theor. 40, 5905–5919. Rozenberg, G., 1997. In: Salomaa, A. (Ed.), Handbook of Formal Languages. Vol. 1 of Word, Language, Grammar. Springer-Verlag, Berlin Heidelberg. Schiff, J., 2008. Cellular Automata: A Discrete View of the World. WileyInterscience Series in Discrete Mathematics and Optimization. New Jersey, Wiley-Interscience. Strutt, D., 2009. Gradients and the specification of planar polarity in the insect cuticle. Cold Spring Harb Perspect Biol 1 (a000489). von Neumann, J., 1966. In: Burks, W. (Ed.), Theory of Self-reproducing Automata. University of Illinois Press, Urbana. Wolpert, L., 1994. Positional information and pattern formation in development. Dev. Genet. 15, 485–490.
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H. Yoshida / BioSystems 110 (2012) 43–50
Yoshida, H., 2011. A condition for regeneration of a cell chain inspired by the Dachsous-Fat system. J. Math-for-Ind. 3, 93–98. Yoshida, H., Furusawa, C., Kaneko, K., 2005. Selection of initial conditions for recursive production of multicellular organisms. J. Theor. Biol. 233, 501–514. Yoshida, H., Kaneko, K., 2009. Unified description of regeneration by coupled dynamical systems theory: intercalary/segmented regeneration in insect legs. Dev. Dyn. 238, 1974–1983.
Yoshida, H., Kimura, K., Yoshida, N., Tanaka, J., Miwa, Y., 2010. Algebraic approaches to underdetermined experiments in biology. IPSJ Trans. Bioinform. 3, 62–69. Yoshida, H., Miwa, Y., Kaneko, M., 2011. Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation. Appl. Algerbra Eng. Commun. 22 (2), 147–164.
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BioSystems journal homepage: www.elsevier.com/locate/biosystems
A pattern to regenerate through turnover Hiroshi Yoshida ∗ Faculty of Mathematics, Kyushu University, Ito, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
a r t i c l e
i n f o
Article history: Received 31 March 2011 Received in revised form 28 April 2012 Accepted 3 August 2012 Keywords: Turnover Dachsous–Fat system Regeneration Continuous flux
a b s t r a c t Tissues of animals and plants are maintained through balanced cell growth, movement, and elimination. Although cells are exchanged perpetually, the whole structure of the tissue is maintained. This form of maintenance is called cell turnover. Here I propose a bio-inspired model of patterns that regenerate through turnover. This model is derived from the Dachsous–Fat system, which has recently attracted much attention because it is considered to facilitate regeneration in insect legs. In this model, I parameterized the manner of the redistribution of Dachsous and Fat during cell division, and then derived equations in the parameters that enable the patterns to regenerate and maintain themselves through turnover. I extended the equations derived in the one-dimensional model into a two-dimensional model. Finally, I discuss a possible relationship between regeneration and turnover. © 2012 Elsevier Ireland Ltd. All rights reserved.
1. Introduction Richard D. Campbell, who carried out many experiments in Hydra, noted the following (Campbell, 1974, p. 523): “The cells and tissues of hydra are in continuous flux. The polyp undergoes perpetual growth and tissue loss, coupled by balanced cell renewal patterns involving all cell types.” Here I propose a bio-inspired model of patterns that regenerate in flux; in other words, patterns whose elements are perpetually exchanged even as the whole structure is maintained. As Campbell indicated, a Hydra’s tissues move toward the body extremities, where they vanish, while cell division or growth occurs throughout the body column (Campbell, 1967; Gilbert, 2010; Berking, 2003). Thus, a Hydra can live for a long time with a constant size and form by balancing the loss and growth of tissues. Through cell growth, movement, and elimination, tissues are well maintained; this is referred to as cell turnover. Another example of cell turnover occurs in planarians, whose cells are in a state of continuous flux. In fact, old differentiated cells are replaced by the progeny of dividing adult stem cells (neoblasts) (Pellettieri and Alvarado, 2007). At the organ level, the intestine is maintained through turnover, wherein the three states of cells, stem cells, semidifferentiated cells, and fully differentiated cells, are orchestrated to update and maintain function on the surface (d’Onofrio and Tomlinson, 2007). Hence, cell turnover may be a key means of tissue homeostasis for many tissues. Cell turnover cannot be observed explicitly, but it seems essential for tissues to maintain or regenerate themselves. Hence, cell turnover was incorporated into the model and a condition for patterns to regenerate through turnover was derived. The model was
∗ Tel.: +81 92 802 4402; fax: +81 92 802 4405. E-mail address: [email protected] 0303-2647/$ – see front matter © 2012 Elsevier Ireland Ltd. All rights reserved. http://dx.doi.org/10.1016/j.biosystems.2012.08.001
inspired by the Dachsous–Fat (Ds–Ft) signaling system, which is considered to facilitate regeneration in cricket legs (Bando et al., 2009). The Ds–Ft system is also thought to be a process underlying the steepness hypothesis, which states that leg size and regeneration are regulated through a gradient across cells (Lawrence et al., 2008; Bando et al., 2011). In addition to experimental studies, the author has studied regeneration phenomena through numerical simulation of connected chaotic elements (Yoshida et al., 2005), where cell types are regarded as various types of attractors within chemicals’ state space, providing a unique view of cockroach leg regeneration (Yoshida and Kaneko, 2009). The author has also modeled a multicellular organism using a Lindenmayer system (L-system) to derive a condition for the coexistence of various cell types (Yoshida et al., 2011), wherein an algebraic method was used to analyze the model with parameters. Here, I focused on molecules between cells, rather than within cells, and have parameterized how the molecules behave during cell division. Previous work Yoshida (2011) dealt with molecules between cells and analyzed a condition for cells to regenerate when any two in the cell chain are extracted and developed. Extending the previous work, I incorporated cell loss into the model described in the present work, and analyzed patterns that maintain and regenerate themselves through balanced loss and growth: namely, through turnover. I first constructed a one-dimensional model that was described by a cell chain based on the Ds–Ft heterodimer system. I set each cell having a value on each side. I next parameterized the redistribution ratio of values during cell division, and derived equations from the parameters that enable the pattern to be maintained through turnover. I extended the equations obtained for the one-dimensional model to a two-dimensional model. Finally, I discuss a possible relationship between regeneration and turnover.
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2. Model I propose a model that is bio-inspired by the Ds–Ft system, which has been considered to facilitate regeneration in cricket legs. First, I introduce the steepness hypothesis, which the Ds–Ft system is deemed to substantiate. The steepness hypothesis is a model that describes a condition for a cell chain to halt cell proliferation. As illustrated in Fig. 1(a), the gradient of a given chemical is assumed to become less steep as the length of a given cell chain becomes longer. When the gradient reaches a given threshold value, the cell chain is assumed to halt its proliferation. Fig. 1(b) shows a gradient of Ds and Ft molecules across cells, which has been postulated to explain regeneration phenomena in cricket legs. It is argued that the cricket starts to regenerate its excised legs by proliferating cells and gives up regeneration (cell proliferation) when the gradient of Ds–Ft heterodimers reaches a certain threshold (Bando et al., 2009, 2011). In addition, Fig. 1(c) further illustrates reverse intercalary regeneration with the help of a gradient of Ds–Ft heterodimers. When a distally amputated leg is grafted with a proximally amputated one, reverse intercalary regeneration occurs so that value continuity is recovered. Just after the graft, a steep (reverse) Ds–Ft gradient is formed at the junction and this keeps cells proliferating until the gradient is less than the threshold, resulting in recovery of the amputated portion. The Ds–Ft heterodimeric system is thus considered to be involved in regeneration and to be a materialization of the steepness hypothesis. I accordingly propose a model, where each cell has a value on each side and the cell grows or vanishes according to these values. Because the phenomenon of cell growth and elimination is called cell turnover, the word turnover will be used in the sequel. First, I modeled a one-dimensional cell chain that can regenerate through turnover. Fig. 2 illustrates this model in which a single cell divides to become a cell chain of a certain length and then both the extremities vanish perpetually, while the inherent chain pattern is maintained. The inset shows that each cell is assumed to have a value on each of its sides (left: lv ; right: rv ) and that the value on the newly created wall after cell division is assigned as: p1 lv + p2 rv ,
(1)
where p1 and p2 are parameters denoting the redistribution ratio. Such a redistribution of the value on the cell wall is an extension of the steepness hypothesis (Lawrence et al., 2008), where a newly created cell (not cell wall) takes up an intermediate positional value from its neighbors, corresponding to p1 = p2 = 1/21 in (1). Although the manner of this redistribution is not well understood, here the Ds–Ft redistribution (1) is adopted for a mathematical simplification. Next, the development and turnover of the cell chain is defined as follows. (i)
(ii)
1
Start with a single cell that has values of a and b on its left and right sides, respectively. In the sequel, let n denote the cell number at which the cell chain begins the turnover process. For each cell (ith cell), enumerate the cell number between the ith cell and both ends of the cell chain. Let ln,i and rn,i be the cell numbers between the ith cell and the leftmost and rightmost cells, respectively. If ln,i + rn,i > n holds and the cell is situated at either of the ends, this cell vanishes. Otherwise, the cell divides into two. At this time, let lv,i and rv,i be the values on the left and right sides of the ith cell, respectively. The value on the newly
In this case, regeneration in the cricket leg can be explained as shown in Fig. 1(c).
(iii)
created cell wall is set to p1 lv,i + p2 rv,i according to Formula (1). If ln,i ≤ rn,i , the new cell is situated on the right of the original cell; otherwise, it is situated on the left, which ensures the turnover because older cells are moved closer to the ends of the cell chain. Perform the previous procedure for every cell simultaneously, and repeat this procedure. The state where some cells begin vanishing is named turnover phase.
I here make a brief view of cellular automata (CA) and rewriting systems including the L-system. CA were introduced as the space over which a self-replicating machine works (von Neumann, 1966). Typical CA consist of square cells, each of which is in one of a finite number of states, such as 0 and 1. The grid of cells can be n (≥1) dimensional, but a two-dimensional grid is usually used. The state of each cell evolves according to the state of itself and its neighbors (in two-dimensional CA, the eight cells surrounding a central cell, the so-called Moore neighborhood, and the four cells orthogonally surrounding a central cell, the so-called Neumann neighborhood, are commonly used). All cell states evolve simultaneously in (synchronous) CA (Schiff, 2008). A rewriting system is a finite set of rules u → v, where u and v are words (Rozenberg and Salomaa, 1997, Chapter 4). A typical rewriting system transforms only one letter in the word per step, and this transformation depends on neighboring letters.2 By contrast, a L-system is a parallel rewriting system that was introduced by Aristid Lindenmayer for modeling the development of plants such as algae (Lindenmayer, 1968a,b). Parallel means that all of the letters changeable by the rules are transformed per step. L-systems have lately been utilized as a code to produce computer graphics of various plants (Prusinkiewicz and Lindenmayer, 1990). I further briefly compare this model with CA and rewriting systems including L-systems. The main difference between this model and the other systems described above is that the next state of the cell is determined by the values between cells, not by the cells’ state. Under some conditions, the cell proliferates in rewriting systems including L-systems and this model, but it does not proliferate in typical CA. Another difference between L-systems and this model is that this model deals with the arithmetic (addition, subtraction, multiplication, and division of numbers) of values between cells. Some L-systems also perform arithmetic for modeling concentrations in the tissue, such as nutrients and inhibitors (Lindenmayer et al., 1974). Further a map L-system simulate two-dimensional cellular layers (Prusinkiewicz and Lindenmayer, 1990, Chapter 7), but they have difficulty in handling the changes in neighboring relations of cells, which was dealt with in this model. As an example of this model, I start with one cell having a value of 2 on the left wall and of 1 on the right. After one cell division, the newly created cell wall has a value of 2p1 + p2 according to Formula (1). After two divisions, the four-cell chain has values (2, p22 + 2p1 (1 + p2 ), 2p1 + p2 , p2 + p1 (2p1 + p2 ), 1) from the left to the right. During cell division, Procedure (ii) assures that the old cells are situated closer to either of the ends of the cell chain, while newly created cells are situated in an inner position. If n (the cell number at which the cell chain begins turnover) is set at, say, 6, both the ends start to vanish after three cell divisions, bringing the cell chain into the turnover phase. I summarize the procedure in Fig. 3. Starting with a single cell with values a and b on the ends, each cell divides into two until the cell number reaches n. Then, the cell chain moves into turnover, during which cells on both ends vanish while the inner cells proliferate. In this turnover phase, regeneration of a pattern is evaluated
2 Some rewriting systems have no dependency of neighboring letters, and such independency is sometimes called context-free.
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Fig. 1. The steepness hypothesis and the Ds–Ft system. (a) The steepness hypothesis. The linear gradient of a certain chemical is assumed and is shown by the grayscale shading. The slope of the gradient becomes less steep as the cell chain becomes longer and the cell proliferates until the slope declines below some threshold value. (b) A schematic illustration of the Ds–Ft system. The Ds–Ft heterodimeric bridges are formed by Ds and Ft molecules between cells during cell division. The gradient of Ds–Ft heterodimers across the cells is considered to provide cells with a polarity or regeneration cue (Strutt, 2009). In this figure, the number of Ds–Ft heterodimers decreases from left to right and this gradient substantiates steepness in the steepness hypothesis. (c) An explanation of reverse intercalary regeneration in the cricket leg. When a distally amputated leg (with the eighth value in the figure) is grafted with a proximally amputated one (with the fourth value), reverse intercalary regeneration occurs such that the positional–value continuity is recovered. Just after the grafting, a steep (reserve) Ds–Ft gradient is formed at the junction. This steep gradient keeps cells proliferating until the gradient is below the threshold, resulting in recovery of the 7-6-5th cells.
divide
lv
p1
p2
rv
.....
p1 lv + p2rv
........
turnover
vanish
vanish
Fig. 2. Schematic illustration of the model. A single cell divides to become a cell chain of a certain length, and then both its extremities vanish along with cell division, progressing into the turnover phase. The turnover phase may consist of a few different patterns. The figure inset depicts how a cell divides and creates a new wall.
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(i) Initial
a
b
(ii) Divide and Create a new wall
a
m
b
m=ap1 + bp 2
m1=bp22+ap1 (1+p2)
m1 m2 m3 b m2=ap1 + bp2
a
m3=ap12+bp2 (1+p1 ) ..... dividing
...
new
original
...
If l n,i
> r n,i
rn,i
ln,i .....
.......
turnover
................
.....
n
vanish
vanish .....
Fig. 3. The development and turnover phases of a cell chain. (i) The values of the ends are set as a and b. (ii) Each cell divides into two and the value on the newly created cell wall is set according to Formula (1) until the number of cells reaches n. In cell division, the original (old) cell is situated on the side with the shorter distance from either of the cell chain ends, while the newly created cell is situated closer to the center part of the cell chain. This positioning ensures cell turnover. When the cell number reaches n, cells at both ends vanish, going into the turnover phase; this might contain a few subphases.
by confirming whether the values are maintained along the cell chain during the turnover state. Note that the turnover phase can be composed of two or more steps with distinct patterns; in such cases, the regeneration is defined as a periodic reproduction of the pattern in the turnover phase.
(ap1 + ap1 p2 + ap1 p22 + ap1 p32 + bp42 , ap1 + ap1 p2 + ap1 p22 + bp32 , ap1 + ap1 p2 + bp22 , ap21 + ap1 p2 + ap21 p2 + bp22 + bp1 p22 , ap1 + bp2 , ap21 + bp1 p2 + ap21 p2 + bp22 + bp1 p22 , ap21 + bp2 +bp1 p2 , ap31 + bp2 + bp1 p2 + bp21 p2 , ap41 + bp2 + bp1 p2 + bp21 p2 +bp31 p2 ),
3. Methods Here, I describe the condition for regeneration through turnover as a set of algebraic equations. The derivation and analysis of the equations are shown in Section 4. Before that, I briefly explain an obstacle in the analysis of algebraic equations and describe a method to overcome it. Imagine the following set of equations: {x3 − 2x2 y + xy − 2y2 − 4x + 8y = 0, 2xy − 4y2 − 3x + 6y = 0} . The solution to the above equations can be decomposed into two solutions of smaller and simpler form: {x − 2y = 0} and {x2 + y − 4 =0, 2y − 3 =0}.
One cannot decide x and y based on the former, but can decide x(= ±
5/2) and
y(= 3/2) based on the latter. It is therefore useful and usually necessary to perform this kind of decomposition in analyzing complicated algebraic equations because we cannot know the decidability of solutions in advance. Such decomposition of algebraic equations is referred to as prime ideal decomposition, which has been used recently to analyze some systems (Romanovski et al., 2007; Yoshida et al., 2010). Here the prime ideal decomposition is calculated by using the routine minAssChar in Singular software (Greuel et al., 2005).
4. Results I obtained a condition for a pattern to maintain and regenerate itself through turnover. As mentioned in Section 2, I started with a single cell that has a value of a on its left side and b on its right. For instance, I set the cell number for starting the turnover (denoted by n) as eight, and developed the cell chain up to five steps, obtaining the following vector:
(2) which designates values from the leftmost to rightmost cells. Another five steps yielded the following vector: (ap21 + 2ap21 p2 + bp1 p22 + 2ap21 p22 + ap1 p32 + bp1 p32 + ap21 p32 + bp42 + bp1 p42 , ap21 + 2ap21 p2 + ap1 p22 + bp1 p22 + ap21 p22 + bp32 + bp1 p32 , ap21 + ap1 p2 + ap21 p2 + bp22 + bp1 p22 , ap31 + ap1 p2 + ap21 p2 + ap31 p2 + bp22 + bp1 p22 + bp21 p22 , ap1 + bp2 , ap21 + bp1 p2 + ap21 p2 + bp1 p22 + ap21 p22 + bp32 + bp1 p32 , ap21 + bp1 p2 + ap21 p2 + bp22 + bp1 p22 , ap31 + ap21 p2 + bp21 p2 + ap31 p2 + bp22 + 2bp1 p22 + bp21 p22 , ap41 + ap21 p2 + ap31 p2 + bp31 p2 + ap41 p2 + bp22 + 2bp1 p22 + 2bp21 p22 + bp31 p22 ).
(3)
The condition for a pattern to maintain itself through turnover can be calculated by Vector (2) − Vector (3) = 0. This condition itself was so complicated that, as mentioned in Section 3, prime ideal decomposition was applied to obtain eight components of
H. Yoshida / BioSystems 110 (2012) 43–50
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Fig. 4. A two-dimensional model. Each cell has four values, one each for its left, top, right, and bottom sides. Start with a single cell having values a, b, c, and d. Each cell divides alternately in the horizontal and vertical directions. The inset shows the redistribution ratio of values. In the same manner as the one-dimensional model, after cell division a newly created cell is situated toward the thicker cell layer. When the width (height) of the cell group reaches m (n), it progresses to the turnover phase.
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(b)
(a) horizontal
The turnover phase step 19
step 22
vertical
step 25
step 28
step 29
=step 19
Fig. 5. A two-dimensional example with a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8. (a) At step 19, a monotonous gradient appears in both the horizontal and vertical directions. The white of the grayscale used here corresponds to the smallest value (−4) and the black to the largest value (4). The upper figure (horizontal) shows the gradient of the values on the right side of each cell and the bottom figure (vertical) shows the gradient of the values on the top side. (b) In the turnover phase, the pattern has a period of 10 steps. To make it clear, the gradient pattern is mapped onto a smile pattern. The smile pattern at step 19 becomes the same after 10 iterations.
simpler form: (I) a = b = 0, (II) b = p1 = 0, (III) p1 − 1 = p2 = 0, (IV) a = p2 = 0, (V) p1 = p2 = 0, (VI) p1 p2 − 1 = ap21 + bp2 = bp22 + ap1 = 0, (VII) p1 + p2 − 1 = a − b = 0, (VIII) p1 = p2 − 1 =0. Of these eight components, only (VI) provided a heterogeneous pattern; the other components generate trivial cell chain patterns, such as 0 . . . 0 and 1 . . . 1. Under component (VI), the vector (2) (of course equal to (3)) became: (−bp22 (1 + p2 + p32 ), −bp22 (1 + p22 ), −bp32 , −bp32 , − b(−1 + p2 )p2 , b, b, b(1/p2 + p2 ), b(1 + 1/p22 + p2 )).
(4)
As another example, I set n = 12 and developed a single cell with (a, b). The cell number of the cell chain could be expressed by the series {1, 2, 4, 8, 14, 12, 12, 12, 12, 14, 12, 12, 12, 12, 12, 12, 14, 12, 12, 12, 12, 12, 12, 14, . . . }, which means that after the 10th step the cell chain moved into the turnover phase with a period of seven.3 Hence, a condition for a pattern to maintain itself through turnover was derived by comparing the pattern at the 10th step with the one at the 17th. The obtained condition that makes the patterns at the 10th and 17th steps equal was revealed to be the same as the eight components, using prime ideal decomposition. Likewise, I set n = 18 and developed a single cell with (a, b). The number in the cell chain
3 The series contains “14”; this is more than n(= 12) because the two cells exist at the ends of the chain, but they vanish at the next step. Such cells appear in the following series also.
became the series {1, 2, 4, 8, 16, 20, 18, 18, 18, 18, 18, 20, (18)8 , 20, (18)9 , 20, (18)9 , 20, . . . }, where the superscript denotes repetition. This series means that after the 21st step, the cell chain progressed into the turnover phase with a period of 10. The condition that makes the patterns at the 21st and 31st steps equal was again the same as the eight components, using prime ideal decomposition. I further confirmed for n = 8, 10, 12, . . ., 30 that the cell number of the chain became the series {1, 2, 4, 8, . . ., n + 2, nm , n + 2, . . ., n + 2, nn/2 , n + 2, . . . }, where the superscript denotes repetition and m was some number between n/2 −2 and n/2. The series eventually showed a period of n/2 +1, and the condition for the cell pattern to recover after n/2 +1 steps during the turnover phase was found to be the same as the eight components, only the sixth (VI) of which gave a nontrivial pattern. As discussed below, the sixth condition (VI) enables the cell chain to maintain its pattern through turnover for an even number, n, in general. 5. Discussion I have obtained a condition for patterns to maintain themselves through turnover, which is the sixth component (VI) in Section 4. This condition can be reduced to: p1 = 1/p2 ,
a = −bp32 .
(5)
Under this condition, the pattern has a gradient of values when p2 > 0. In the case of n = 8, the obtained pattern Vector (4) can also be described as: b(v1 , v2 , v3 , v3 , v4 , 1, 1, v5 , v6 ),
(6)
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where v1 ≤ v2 ≤ v3 ≤ v4 ≤ 1 ≤ v5 ≤ v6 when p2 > 0 because 1 + p2 + p32 ≥ 1 + p22 ≥ p2 , p32 > (−1 + p2 )p2 , 1 < 1/p2 + p2 , and 1/p2 + p2 < 1 + 1/p22 + p2 .4 In addition, the manner in which a pattern can maintain itself through turnover is as follows. Under condition (5), the cell chain starts with a cell having (− p3 , 1) at both its ends. In the sequel, denote p2 by p and set b as 1 for simplicity. After one cell division, the newly created cell wall has a value of (− p3 )/p + 1 × p = p − p2 , which gives the cell chain values of (− p3 , p − p2 , 1) from the leftmost to rightmost. After another cell division, the cell chain acquires values of (− p3 , − p3 , p − p2 , 1, 1), where the second value −p3 comes from −p3 /p + (p − p2 ) × p and the fourth 1 comes from (p − p2 )/p + 1 × p. The values at the next step become (− p3 , − p2 (1 + p), − p3 , − p3 , p − p2 , 1, 1, p + 1/p, 1), whose central part (− p3 , p − p2 , 1) is always maintained. This is why the pattern of values is maintained during the turnover phase. In addition, according to the cell number n, which denotes the number of cells at start of the turnover phase, the length(s) of the cell chain in the turnover are determined. One of the values in the turnover phase in the case of n is (− p2 (1 + p + p2 + · · · + pm ), − p2 (1 + p + p2 + · · · + pm−1 ), . . ., − p2 (1 + p2 + p3 ), − p2 (1 + p2 ), − p3 , p − p2 , 1, (1 + p2 )/p, (1 + p2 + p3 )/p2 , . . ., (1 + p2 + · · · + pm−1 )/pm−2 , (1 + p2 + · · · + pm )/pm−1 ), where m = n/2. Like Vector (6), this vector shows a gradient when p > 0. In order to show one possible extension of the one-dimensional model, I extend the condition (5) and the gradient pattern (6) into a two-dimensional model as illustrated in Fig. 4. Each cell is assumed to divide alternately in the horizontal and vertical directions; this simple dividing manner is adopted for a mathematical simplification even though the actual cell may divide toward any of more directions. As an extended condition of (5), I assume the following relationship between the parameters: p1 = 1/p2 ,
a = −bp32 ,
p3 = 1/p4 ,
c = −dp34 ,
(7)
where a, b, c, and d are the values on the left, right, top, and bottom sides, respectively, and p1 , p2 (p3 , p4 ) is a pair of the distribution ratio of values in the horizontal (vertical) direction. The cell group moves into the turnover phase when the width reaches m or the height reaches n. For example, using a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8, Fig. 5(a) shows the gradient of values created in the horizontal and vertical directions in the same manner as the one-dimensional chain (see (6)). At step 19, which is in the turnover phase, both the horizontal and vertical gradients become (− 4, − 3, − 2, − 1, 0, 1, 2, 3, 4). I map these gradients onto a given pattern. For instance, Fig. 5(b) illustrates a smile pattern by mapping cells with (two-dimensional) gradient points, (± 2, 3), (± 3, 0), (± 2, − 1), (± 1, − 2), (0, 1), (0, − 2) onto black and others onto white. From steps 19 to 22, the smile pattern wears away outward, and then it recovers up to step 29. This shows that although the components are perpetually changing into newer cells, its (value) pattern is maintained through turnover. Here I exemplify the result using a specific pair of parameters (a = −1, b = 1, c = 1, d = −1, p1 = p2 = p3 = p4 = 1, m = n = 8). This result can be extended to a more general case as long as p2 > 0 and Eq. (7) holds. The gradient appears at least once in the turnover phase, as shown in Fig. 5(b). To conclude, I have obtained the condition for a cell chain to maintain its pattern through turnover. Under this condition, a oneor two-dimensional gradient appeared. In conventional models of regeneration, such as the positional information and the steepness theories (Wolpert, 1994; Lawrence et al., 2008), a gradient across
2
4 All these inequalities are reducible to an inequality, p22 − p2 + 1 = (p2 − 1/2) + 3/4 > 0.
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cells was assumed, providing a cue for regeneration. This is consistent with the gradient that results from a condition in which patterns maintain themselves through turnover in my model. 6. Conclusion In this work, I incorporated turnover into the model and obtained a condition in which one-dimensional patterns maintain themselves. I extended the obtained condition into a twodimensional model. This model indicates that it is possible to design some patterns that can maintain and regenerate themselves through turnover. The gradient emerging from the turnover condition is consistent with the conventional model for regeneration. Acknowledgments I thank the editor and the anonymous reviewers for helpful comments that improved this manuscript. I am grateful to Prof. Sumihare Noji and Dr. Tetsuya Bando for valuable discussions regarding cricket legs and I had warm encouragement of Ms. Atsuko Sono. The author was supported by MEXT KAKENHI Grant Number 23124509. This study was also supported in part by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Japan Science and Technology Agency (JST). References Bando, T., Mito, T., Maeda, Y., Nakamura, T., Ito, F., Watanabe, T., Ohuchi, H., Noji, S., 2009. Regulation of leg size and shape by the Dachsous/Fat signalling pathway during regeneration. Development 136 (13), 2235–2245. Bando, T., Mito, T., Nakamura, T., Ohuchi, H., Noji, S., 2011. Regulation of leg size and shape: Involvement of the Dachsous-Fat signaling pathway. Dev. Dyn. 240 (5), 1028–1041. Berking, S., 2003. A model for budding in hydra: pattern formation in concentric rings. J. Theor. Biol. 222, 37–52. Campbell, R.D., 1967. Tissue dynamics of steady state growth in Hydra littoralis II. patterns of tissue movement. J. Morphol. 121, 19–28. Campbell, R.D., 1974. Cell movements in Hydra. Am. Zool. 14, 523–535. d’Onofrio, A., Tomlinson, I.P., 2007. A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt. J. Theor. Biol. 244 (3), 367–374. Gilbert, S.F., 2010. Developmental biology, 9th ed. Sinauer Associates, Inc., Sunderland, USA. Greuel, G.M., Pfister, G., Schonemann, H., 2005. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, http://www.singular.uni-kl.de. Lawrence, P.A., Struhl, G., Casal, J., 2008. Do the protocadherins Fat and Dachsous link up to determine both planar cell polarity and the dimensions of organs? Nat. Cell Biol. 10 (12), 1379–1382. Lindenmayer, A., 1968a. Mathematical models for cellular interactions in development. I. filaments with one-sided inputs. J. Theor. Biol. 18 (3), 280–299. Lindenmayer, A., 1968b. Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. J. Theor. Biol. 18 (3), 300–315. Lindenmayer, A., 1974. Adding continuous components to L-systems. In: Rozenberg, G., Salomaa, A. (Eds.), L-Systems, Vol. 15 of Lecture Notes in Computer Science. Springer, pp. 53–68. Pellettieri, J., Alvarado, A.S., 2007. Cell turnover and adult tissue homeostasis: from humans to planarians. Annu. Rev. Genet. 41, 83–105. Prusinkiewicz, P., Lindenmayer, A., 1990. The Algorithmic Beauty of Plants. SpringerVerlag, New York. Romanovski, V.G., Chen, X., Hu, Z., 2007. Linearizability of linear systems perturbed by fifth degree homogeneous polynomials. J. Phys. A: Math. Theor. 40, 5905–5919. Rozenberg, G., 1997. In: Salomaa, A. (Ed.), Handbook of Formal Languages. Vol. 1 of Word, Language, Grammar. Springer-Verlag, Berlin Heidelberg. Schiff, J., 2008. Cellular Automata: A Discrete View of the World. WileyInterscience Series in Discrete Mathematics and Optimization. New Jersey, Wiley-Interscience. Strutt, D., 2009. Gradients and the specification of planar polarity in the insect cuticle. Cold Spring Harb Perspect Biol 1 (a000489). von Neumann, J., 1966. In: Burks, W. (Ed.), Theory of Self-reproducing Automata. University of Illinois Press, Urbana. Wolpert, L., 1994. Positional information and pattern formation in development. Dev. Genet. 15, 485–490.
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Yoshida, H., 2011. A condition for regeneration of a cell chain inspired by the Dachsous-Fat system. J. Math-for-Ind. 3, 93–98. Yoshida, H., Furusawa, C., Kaneko, K., 2005. Selection of initial conditions for recursive production of multicellular organisms. J. Theor. Biol. 233, 501–514. Yoshida, H., Kaneko, K., 2009. Unified description of regeneration by coupled dynamical systems theory: intercalary/segmented regeneration in insect legs. Dev. Dyn. 238, 1974–1983.
Yoshida, H., Kimura, K., Yoshida, N., Tanaka, J., Miwa, Y., 2010. Algebraic approaches to underdetermined experiments in biology. IPSJ Trans. Bioinform. 3, 62–69. Yoshida, H., Miwa, Y., Kaneko, M., 2011. Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation. Appl. Algerbra Eng. Commun. 22 (2), 147–164.