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Regenerative growth curves
Regenerative Growth Curves EBERHARD OTTO VOIT,* HERMANN JOSEF ANTON, AND JURGEN BLECKER Zoologisches Institut der Universitiit zu Kijln, I. Lehrstuhl: Experimentelle Morphologic, Weyertal I1 9, 5000 Ktiln 41, Bundesrepublik Deutschland
ABSTRACT Regenerative growth curves are analyzed with a new mathematical function. The function includes known regenerative growth laws as special cases but also accounts for the simultaneous normal growth of the organism. The function is embedded in a general system of growth descriptions that was derived earlier from underlying mechanisms, The major advantage of the function is that the regenerative growth can be mathematically separated from the normal growth. The separation is shown to be biologically meaningful. The regenerative growth function is used to analyze experimental data on limb regeneration in salamanders (Triturw, Salamandra). The function describes adult regeneration equally well and larval regeneration significantly better than the regenerative growth functions used before. The mathematical analysis shows quantitatively how regeneration rate and capacity differ in larvae and adults. The method of analysis can be used to quantify experimental conditions affecting regeneration and to use regeneration as a vertebrate bioindicator in ecological
studies.
INTRODUCTION The regeneration of parts of the body is a very complicated process whose many details are being intensively studied but still not yet fully understood (for reviews see [25], [23], [9], [26]). Although the regeneration processes are governed by biochemical, physiological, developmental, and many more mechanisms, the increase in volume or length of a regenerating part, on the whole, follows a rather smooth curve (e.g. [15], [l], [14], [9]). Generally, the curve starts slow with a lag phase in which processes of wound healing, dedifferentiation, and accumulation of mesodermal blastema take place; it then increases rapidly during the phases of blastema proliferation and differentiation, and finally reaches a shallow plateau when morphological regeneration is completed and only the current normal size still has to be regained (Figure 1).
*Present address: Dept. The University of Michigan, MATHEMATICAL
of Microbiology and Immunology, Ann Arbor, MI 48109, U.S.A.
BIOSCIENCES
131253-269
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
6643 Medical
Science
(1985)
II,
253 0025.5564/85/$03.30
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E. 0. VOIT, H. .I. ANTON, AND J. BLECKER
days of regeneration FIG. 1. Growth curve of a regenerating postmetamorphic juvenile newt (Triturus r~/guris). The symbols represent measurements [l]; the line was calculated with Equation (0
Because such a curve somehow summarizes the effects of all underlying mechanisms, its analysis may help us to understand better some of the details of regeneration. One important part of such an analysis is to determine the mathematical function that corresponds to the regenerative growth curve. The knowledge of this function can yield insight into the regeneration process, its lag phase, half-time, and final value, and into possible interrelations between regenerative and simultaneous normal growth. In another part of the analysis one may ask how the regenerative growth curve of an animal depends on its species and its developmental stage, and how it varies among comparable animals. More specific questions deal with abnormal modes of regeneration that may have been caused naturally or by experimental conditions. It seems difficult to answer such questions from observations only. Here, a mathematical method will be presented to analyze regenerative growth curves and to help interpret experimental data. There are already mathematical functions to describe regenerative growth [24, 3, 201, but they do not consider that the regenerating animal may be growing at the same time. Particularly in young animals, however, this seems to be an important factor. The earlier growth functions will therefore be generalized by superimposing normal growth on the regenerative growth. The new regenerative growth function is applied to experimental data on limb regeneration in newts (Tritums vulgaris, T. alpestris) and the salamander (Salamandra salamandra). The data can be found in Blecker [4], but will also be published elsewhere for easier access (Anton, Blecker and Voit, in preparation). Because the analysis yields numerical results, it can be used to quantify experimental or natural conditions affecting regeneration. For instance, it
REGENERATIVE
255
GROWTH
could allow the ecologist to use the regeneration of appendages in amphibia as a bioindicator for environmental studies on 0, shortage, chemical pollution, elevated water temperature due to reactor cooling, or other deviations from normal conditions. THE REGENERATIVE
GROWTH
FUNCTION
Spencer and Coulombe [24] compared normal and regenerative liver tissue. They found that the allometric growth law [lo] L=aBJ’
growth of
(1)
is appropriate to describe how in a growing organism the liver weight L depends on the body weight B. Here a and p are constants. In contrast, the regenerative liver growth was better represented by the exponential function L= cL,.(l-
e-h’)
(2)
or by the hyperbola
which describe how the maximum liver weight L, is regained. c, X, k are constants; k is the time required to achieve half of the maximum weight. Both curves begin with a steep increase and then gradually become shallower. According to other authors (e.g. [9, pp. 259 ff.]), regenerative growth begins with a lag phase during which the wound is being closed and the stump becomes organized (e.g. [2], [ll]). The regenerative growth curve is hence sigmoid rather than logarithmic, exponential, or simple hyperbolic. Baranowitz et al. [3] described the tail regeneration in a lizard and a newt species by the Gompertz growth law [7]
(4) R, is the length of the regenerating tail at the “day of initiation,” which is defined as the “day post-amputation on which the proliferation of cells first follows a Gompertz pattern” [3]. R, is the final tail length, t, the time after initiation, b a constant between 0 and 1. Baranowitz et al. [3] claim that the Gompertz fit is “almost perfect.” Unfortunately, they included neither very early nor very late length measurements of the regenerating tail, so the Gompertz function only fits the rather unspecific middle part of the sigmoid regenerative growth curve, and not the transitions to the plateau phases at the beginning and at the end of the regeneration process.
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E. 0. VOIT, H. J. ANTON,
Savageau [20] introduced
the generalized
R(t)
=
AND J. BLECKER
hyperbola
R,.(t+a)’ b+(t+u)”
which is capable of representing hyperbolic as well as sigmoid curves. R, is the final value to which R(t) converges if t increases. u, b, c are nonnegative parameters. None of these equations considers that the regenerating organism, as a whole, may be growing simultaneously. Therefore, we will replace the constant R, by a function of time that accounts for the superimposed normal growth. Our generalized regenerative growth function is
R(t)
(t+a)=
=R,(t).
b+(t+u)’
a, 6, and c are nonnegative parameters. We will call R,(t) the “normal (growth) component” and (t + a)‘/[ b + (t + u)~ ] the “regenerative (growth) component.” The normal component, R,(t), is the current regeneration target; that is, at each time point, the regenerating part of the organism tries to regain that size, R,(t), that it would have had during normal development at that time point. Depending on the parameter values b and c, the regenerative component can exhibit different S-shaped curves (Figure 2); the parameter b is responsible for the steepness; the parameter c, for the curvature between the lag phase and the growth phase, and between the growth phase and the final plateau.
10
FIG. 2. zomponent:
20
The values of the parameters b and c determine the shape of the regenerative ab = 1, c = 4; @/I =100,000, c = 6; 06 = 50, c = 2. (See text for details.)
REGENERATIVE GROWTH
257
The parameter a in Equations (5) and (6) shifts the time scale of the regenerative component. Because at time t = 0 regeneration just begins, R (0) should be equal to zero, and hence, u is equal to zero by definition. That implies in the biological interpretation that R(t) does not describe, for instance, the whole length of a regenerating limb or the whole weight of a regenerating organ, but only the tissue that actually is regenerated at time t. A stump or the rest of an organ at time t = 0 is not included in R(t) and has to be added if Equation (6) is applied to experimental data for an entire limb or organ. In the analysis of experimental data on limb regeneration in newts, the normal component R,(t) was chosen to be a linear or a logarithmic function. The linear function includes R,(t) = constant, which is identical with Equation (5). The logarithmic function was selected from among the nonlinear functions because it reflects the decelerating growth phase that is often observed in animal growth after the initial exponential growth phase. A logistic equation or a general growth equation [20] would have been an alternative choice. With a linear normal component R,(t), the regenerative growth function can be written as R(t)
= R&l+&+
(7)
,with nonnegative parameters R,, g,,, b, and c. In comparison with Equation (5), the current regeneration target at time t = 0, R,, grows linearly with the normal-growth parameter g,. In logarithmic form, the regenerative growth function reads
R(r)=[s+pln(tcq)].&.
In contrast to the time-shift parameter a in Equation (6), the time-shift parameter q refers to the normal growth and therefore cannot be set equal to zero by definition. The normal component in Equations (5), (7), or (8) uses one (R,), two ( R,, g,), or three (s, p, q) parameters, respectively. These parameters may or may not be observable. For instance, in the limb regeneration of newts after amputation, R, is the length of the amputated part. In Equation (8), the length I, of the amputated appendage can be used to express the parameter q in terms of I,, S, and p by setting 1, = s + p ln(O+ q), q = exp[(I, - s)/p]. The parameters g,, p, and q that reflect the normal growth of the organism can be estimated separately from control experiments or, in limb regeneration, from the unaffected contralateral limb. In contrast, the parameters b
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E. 0. VOIT, H. J. ANTON,
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and c cannot be observed in that sense. In a data analysis, all values of parameters that cannot be observed or obtained separately must be determined by mathematical methods. In the present analysis, this was done with a computer search procedure which estimates that set of parameter values yielding the least sum of squared differences (SSQ) between data and the mathematical function whose parameter values are being optimized. The search procedure used here was the Levenberg-Marquardt algorithm [12, 141 in the implementation of the software library IMSL [5]. Most of the data were fitted with a constant, a linear, and a logarithmic normal component. In most cases, one of the three functions clearly yielded the best fit as seen from the least SSQ, but in some cases, Equation (8) produced slightly better results than Equations (5) and (7). The question whether Equation (8) was only better because it contained one or two more parameters was answered with the MSE, test [16], which objectively decides if an improved fit is worth the higher number of parameters. If in the analyses of the following sections one fit is said to be superior to another, that refers to the result of the MSE, test.
ANALYSIS
OF LIMB REGENERATION
IN SALAMANDERS
In the following, we will use Equations (5), (7), and (8) to analyze increase in length of regenerating forelimbs of salamanders (Triturus pestris, T. vulgaris, Salamandra salamandra). Salamandra larvae were collected in the
the al-
vicinity of Wuppertal, W. Germany; the newts were obtained from laboratory spawnings. The animals were maintained in the dark throughout the experimental period at a constant temperature of 22f0.5”C and only brought into daylight for operations and observations. In all specimens, the right forelimb was amputated through the proximal third of the stylopodium. We amputated larvae (4 T. alp., 9 T. vulg., 3 S. salam.) between two and four weeks before metamorphosis, 3 S. salam. one month after metamorphosis, and 15 T. alp. and 5 T. v&g. one year and 10 T. alp. two years after metamorphosis. We also studied 5 T. alp. and 3 T. vu/g. larvae that were one or two years old but postponed metamorphosis. Such abnormalities are mostly due to a defect in the pituitary gland caused by incorrect invagination and induction during early development. The regenerating and the contralateral limbs were observed and drawn with a stereomicroscope having a Leitz drawing prism attached. Observations were made every other day during the first three weeks and subsequently every 3 to 7 days, until the regenerative growth had reached the final plateau phase. Here we will only mathematically analyze the increase in length of the regenerating limb. More experimental details and further results are or will be published elsewhere ([4]; Anton, Voit, and Blecker, in preparation).
REGENERATIVE
1.
259
GROWTH
THE NORMAL
GROWTH
COMPONENT
Depending on an animal’s age, the shape of the regenerative growth curve varies from a steeply inclining curve in larvae to a shallower curve in adults with a prominent lag phase (Figure 3). The longer lag phase in postmetamorphic animals is caused by the extension of the regression phase due to advanced ossification and could also be affected by the darkness (cf. [13]). The mathematical analysis of about 50 regenerative growth curves showed that Equation (8) was superior to Equations (5) and (7) when data came from larvae, whereas Equations (5) and (7) were optimal in describing postmetamorphic regeneration. This finding agrees with our experimental observations showing that normal growth of larval limbs is about logarithmic [corresponding to the logarithmic term in Equation (S)], whereas postmetamorphic limb growth is nearly linear with a very slight slope or none at all [corresponding to the linear and constant numerators in Equations (5) and (7), respectively]. When the growth of the unaffected contralateral limb was recorded, the parameter values for the optimal fit to the data could be determined in two ways: either all parameter values of the normal and the regenerative component were optimized simultaneously with the computer search algorithm mentioned in the last section, or the growth of the contralateral limb was approximated by a logarithmic function in larvae or by a linear function in postmetamorphic animals and these functions were used as normal compo-
5
c 0
U(t) R(t)
0
0
th
days of regeneration
Regenerative growth curves of a larval @ and a one-year-old @ newt (Triturtu ufpestris). The symbols represent the measured data; R(t) was calculated with Equation (8) in @ and Equation (7) in @; U(f) is the normal growth of the unaffected contralateral limb; th = half-time; S = length of the stump at amputation. (See text for further details.)
E. 0. VOIT, H. J. ANTON,
260
AND J. BLECKER
nents. Both methods yielded fits of about the same quality and even similar parameter values for the regenerative component. This implies that the parameter values of the normal and the regenerative components can be determined independently of each other. It can be concluded that neither component affects the other much, and therefore the mathematical separation of the two components is also biologically meaningful. This conclusion is further supported by the analysis of the regeneration of postmetamorphic forelimbs: for some of these cases the constant, linear, and logarithmic normal component yielded fits with about the same residual error SSQ. Again, about the same parameter values were found for the regenerative component. For instance, the best fits to the data for the forelimb regeneration of a two-year-old T. alpestris were obtained with the following parameter values: Equation Equation
(5) : (7):
Equation
(8):
SSQ =1.4386, b =1796, c =1.980, R, = 5,444; SSQ=1.4316, b=1557, c=1.943, R,= 5.461, g, = 0.000074; SSQ= 1.4343, b = 1452, c = 1.919, s = 0.0408, p=1.02242, q=1.15x1023.
(That the parameter values for b and c are in fact very similar will be shown in the next section, in particular, in Figure 4; three encircled symbols, VVV). In all three cases the residual error is about the same; the MSE, test favored Equation (5) over Equation (7) over Equation (8). It should be mentioned that the slope g, in Equation (7) is very small, and hence the linear function is about constant. The parameter values s, p, and q of the normal component in Equation (8) shift the logarithmic function in such a way that during the regeneration period the normal component is also approximately constant. This indicates that Equation (5) in this case is the appropriate description, as the MSE, test also shows. 2.
THE REGENERATIVE
COMPONENT
It is well known that the regeneration rate and capacity are age dependent and decrease if an animal grows older (e.g. [S, 17, 271). However, it is difficult to describe or measure this fact quantitatively, because nonadult animals regenerate and grow at the same time. As demonstrated above, the normal and the regenerative component of a regenerative growth curve can be separated mathematically. It is hence possible to study the regeneration rate and capacity of an animal while the superimposed normal growth is excluded. The regenerative component is characterized by the two parameters b and c which determine its shape [cf. Equations (5), (7), (8) and Figure 21. In order to compare the regenerative components of different animals, the values of b and c were considered as coordinates of points in a semilogarithmic plot, each point representing one regenerate (Figure 4).
REGENERATIVE
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GROWTH
b . 108 5x105
10” 5x10’
IO’ 5x10=
lo3 500
100 50
10
1 FIG. 4.
Representation
2
1.5
of the regenerative
3
2.5 component
3.5 by the parameters
4
c h and
c’.
Each symbol represents one (h. c) pair and hence one regenerate. The three encircled symbols T correspond to best fits with Equations 5.7 and 8 to the same regenerate (cf. last section). T.
---I
ulp.
Larvae
1 year old 2 years old Abnormal larvae
: . .
T.
vulg.
s. sulun1.
0
0
0
0
A
262
E. 0. VOIT, H. J. ANTON, @
lo
so
0
@
10
10
@
100
120 days
FIG.
Trirurus
5.
Some
alpesrris.
AND J. BLECKER
of
HO regeneration
regenerative components of larvae and postmetamorphic The letters refer to the letters in Figure 4.
animals
of
Clearly, the points of larvae and of postmetamorphic animals are clustered in different parts of the plot, as indicated by the dotted “clouds.” Interpreted biologically, this means that the mode of regeneration is age dependent even if normal growth is excluded. It is striking that the clouds are elongated in directions like those marked by t, = 10,20,. These lines contain the (h, c) pairs of all regenerative components with the half-time t,;’ that is, each regenerate whose b, c-pair lies on the line t, has at time t, reached half of its “regeneration due”. That means, t, is the time when R(t), which is equal to the length of the regenerating limb diminished by the length S of the remaining stump, reaches half the length U(t) of the unaffected contralateral limb diminished by S: that is, R (t,,) = [U( t,,- S]/2 (cf. Figure 3).’ Along the t,-lines, the b, c pairs refer to differently shaped curves with more or less distinct lag phases (the points marked by letters A through F correspond to the curves in Figure 5). Interpreted biologically, the b, c pairs represent the animals’ individuality within the same developmental stage. The half-time t, allows us to compare the regenerative components of different animals with a single parameter. Although th does not consider the shape of the regenerative component, which somehow reflects the individual
‘The half-time is t,, = h”’ as calculated by setting 0.5 = tl;/(h + th) (c > 0). ‘With the given definition of the half-time, th is an observable quantity. For instance, limb regeneration after amputation is studied, R(f), U(r), and S are at hand.
if
REGENERATIVE
263
GROWTH
S salamandra
i ‘I:vulgaris
. .
* . .
.
l
.AA . :
T alpestris
l* :T , 10
1
I 30
* . .
. .
.
.
l
I
I 50
1 60
I 70
FIG. 6. Half-times fh of animals of different species and ages. 0 larvae, v 2 years old, A abnormal larvae. In all species, larvae and postmetamorphic strictly separated.
I80 th
+ 1 year old, animals are
among larvae and postmetamorphic animals of different ages, the half-time t, was found to be appropriate in distinguishing between larvae and postmetamorphic animals. In agreement with the known fact that the regeneration rate decreases in older animals, t, is much shorter in larvae than in postmetamorphic animals: In larvae of T. alpestris, I,, was found to be between 6.5 and 17; in one-year-old postmetamorphic T. afpestris, between 31 and 42; and in two-year-olds, between 34 and 72.3 Similar results were obtained for the other species analyzed (see Figure 6). Our data are not yet sufficient to determine the mathematical form for the fading of the regeneration rate. A plot of the t,-values versus the animals’ age (Figure 7) may suggest a nonlinear, perhaps logarithmic function. It also is possible that t, increases linearly with a steep slope before metamorphosis and later is again linear but with shallow incline. It could even be that t, is about constant in the larval stage and also constant, with a higher level, after metamorphosis. In any event, the mathematical analysis suggests what kind of experiments have to be done in order to elucidate the age dependence of the regeneration rate. Besides the normally developed newts, five T. alpestris and three T. vulgaris were studied that were one or two years old but still in the larval stage. The t,-values of these abnormal larvae were found among the larval as well as the postmetamorphic half-times (see Figure 6), which shows their variation
3t,, is a measure for the regeneration rufe. If r,, is high, it takes a long time to regenerate half of the lost part; it may also mean that the animal will not regain a full-sized part, an effect known from observations (e.g. [27]). th can then be considered as a measure for the regenerative ccrpacit~~.
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E. 0. VOIT, H. J. ANTON,
AND J. BLECKER
th
.
70 60
50
8 .
40
h
30
0
20 s 6
10 LL
-2
0-Y
2
4
6
8
10
12
14
16
18
20
Age (months after metamorphosis) FIG. 7.
Dependence
of the half-times
th on the animals’
ages. 0 = T. alpesrris,
0 = T.
uulguris.
ambivalent status also on a mathematical level. The b, c pairs of these animals appeared within the larval and the postmetamorphic clouds and between the clouds of the (b, c) plot (Figure 4). DISCUSSION 1.
METHOD
OF ANALYSIS
Many regenerative growth curves show a sigmoid shape that can be described mathematically by very different functions. The choice of a particular function should therefore be based on reasons derived from the process that is being analyzed. Otherwise, the mathematical formulation may fit the experimental data nicely but might not yield any further insight in the biological process. For instance, an arctangent function has a sigmoid graph and might fit some data from regeneration. However, it seems difficult to connect the parameters of that mathematical function with mechanisms involved in the regeneration process. This example shows the crux of many mathematical models and analyses, which is that the agreement between experimental result and mathematical model does not prove the validity of the model. Experimental data may prove a mathematical model to be wrong, but because two models with different structures can yield the same output, it can always happen that the model works differently than the process being studied but still shows the same reaction to a given input. The risk of using a wrong model becomes lower if (i) the structure of the model can be derived
REGENERATIVE
GROWTH
265
from experimental knowledge, and (ii) many details, results, conclusions and predictions from the model agree with the process being analyzed. The mathematical model used in this paper was chosen for two reasons: (i) The model includes as special cases the regenerative growth functions by Spencer and Coulombe [24] and by Savageau [20] that were successfully applied earlier to experimental data; and (ii) because the model [Equations (7) and (S)] can be written as power-law system. A power-law system is a set of nonlinear differential equations that are derived from very general, nonrestricitive assumptions on the underlying mechanisms of biological or chemical processes (e.g. [21]). More specifically, the power-law system can be shown to be a “natural” description for any growth processes [18-211. The model (6) contains two components that determine the regenerative growth: the normal and the regenerative component. The normal component was kept very general in Equation (6), but was restricted in Equations (7) and (8) to linear and logarithmic functions. Both functions may only be approximations to the “real” growth process, but seem to be reasonable in limb regeneration of newts as seen from the observed normal growth of unaffected limbs [4]. However, other functions R,(t) in Equation (6) may be more appropriate for the analysis of other regeneration processes. For the regenerative component we chose the generalized hyperbolic function proposed by Savageau [20], which also includes the regenerative function used by Spencer and Coulombe [24]. As an alternative, we could have tried other growth laws, for instance the Gompertz law, which Baranowitz et al. [3] applied to data from regenerating tails in a newt and a lizard species, or the general power-law equation [20], which includes all laws mentioned as special cases. However, since we felt that other growth laws could hardly improve the fit over the simple two-parameter hyperbolic function [cf. Equation (7) and Figures 1 and 31, we only used the latter as regenerative component. Due to its two parameters, the regenerative component can mimic a variety of different sigmoid curves (Figure 2). That the S-shape of a regenerative growth curve is mainly due to its regenerative component rather than to the normal component can be concluded from adult animals that do not grow any more but whose regenerative growth curves are still sigmoid. A crucial property of the model (6) is that normal and regenerative components are separable. There is no proof of the validity of this assumption, and one could imagine that the two components interacted in such a way that they could not be separated mathematically. However, there are indications that the two components are independent and superimposed: (i) In old animals, constant, linear, and logarithmic normal components sometimes yielded similarly good fits to the data. In all three cases, the parameter values for the regenerative component were about the same, which indicates that the regenerative component does not balance out different normal components.
266
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AND J. BLECKER
(ii) In larvae, the values for the normal growth parameters could be obtained from records on the growth of the unaffected contralateral limb of the same animal. The parameter values for the regenerative component were very similar, whether those parameter values were used for the normal component or the parameter values of both components were optimized simultaneously. (iii) Besides the conclusions from this mathematical analysis, it has long been known that in tadpoles regeneration occurs faster when more has to be regenerated and, more precisely, that independent of the amputation level, the same percentage of the lost part is always being regenerated at a given time [6, 91. The same phenomenon was later found in lizard and newt tails and in fish fins (for references see [ll, 31). This strongly suggests that in different animals and different amputation levels the same regenerative component is merely multiplied by a constant factor that reflects the species, the size of the lost part, and the animal’s individual metabolism. In Equation (7), this factor is R,.
These findings imply that the separation of the regeneration process into two independent, superimposed components for normal and regenerative growth is not only mathematically possible but also biologically meaningful. 2.
BIOLOGICAL
RESULTS
It is well known that in newts the rate and capacity of limb regeneration highly depend on an animal’s age [8; 9, p. 146; 17; 23; 27; 26, p. 71. The mathematical analysis presented here yields the same result but in much more detail. Besides the description of the regeneration process with Equations (7) and (8), some insight was obtained by studying the crucial parameters h and c of the regenerative component and the half-time t,. As a simple, one-parameter measure, t, represents the regeneration rate and its age-dependent fading. The value of t, was found to increase with age from about 12 days in larvae to about 35 or 45 days in adult newts; similar half-times were found in different newt species and in the salamander. This agrees with observations that larvae regenerate twice as fast as adults, if not more (e.g. [23, 261). The functional dependence of t, on age could not be derived, due to a shortage of sufficient data. In any event, the mathematical analysis, and in particular the introduction of t,, as a measure for regeneration rate and capacity, show which experiments would have to be done and how they could be evaluated in order to determine how regeneration changes with age. From the existing data, t, can be thought to increase rapidly during the first months of life and slower afterwards, according to a logarithmic or S-shaped function. This would agree with the often observed decelerating growth in animals after an initial exponential growth phase. The function could also consist of two
REGENERATIVE
GROWTH
267
linear phases with different slopes and/or different additive terms before and after metamorphosis. It would not seem unreasonable to explain the discontinuity by metamorphosis. The half-time is a characteristic of regenerative growth curves but does not represent the curve completely, e.g. very different curve shapes can reach half of the final value at the same time. In contrast, the parameter-value pair (b, c) contains all information about the regenerative component and so implies its shape. The mathematical analysis of our data showed that the (b, c) pairs varied considerably among all animals studied and even within groups of “comparable” animals, for instance, among larvae of the same species and age. However, the groups of larvae and of postmetamorphic animals turned out to be strictly separate in the (b, c) plot (Figure 4). The clouds of different species overlap somewhat but might be separable by statistical means. The separation in clouds shows that the representation of the regenerative growth component by its (b, c) pair reflects the developmental age as well as the individual variability among the animals of the same group. The latter could hardly be seen from the half-times and so demonstrates the cost of simplifying the regenerative component from two parameters to one. The analysis of some one- and two-year-old larvae, which normally would have already gone through metamorphosis much earlier, showed that their (b, c) pairs lie in the larval or in the postmetamorphic cloud or between the clouds (Figure 4). The exact conditions of these abnormal larvae are not known, but it could be that the different (b, c) pairs reflect different degrees of defects of their pituitary glands. In any event, the example demonstrates how from the (b, c) pair analysis animals of uncertain origin or with unusual properties can be classified. This way, one could also quantitatively study the influences of temperature (cf. [22]), light-dark regime (cf. [13]), or other experimental or environmental conditions that could be relevant for limb regeneration. As an example of experimentally affected regeneration, we will analyze elsewhere the regeneration of chimeric newts (Anton and Voit, in preparation). In our analysis, we have only studied the regeneration of amputated limbs. It would now be interesting to apply the same methods to other regeneration processes, particularly to the regeneration of the tail, which is also being intensively studied (for references see [9, 3, 261). One could then compare our quantitative results on the limb with those obtained by other authors on the tail (e.g. [3]), and also the (b, c) parameter values and the t,-values for different limbs and the tail within the same animal. This would show whether similar qarameter values were found in all regenerating parts of the same organism or whether their regenerative growth satisfied the allometric growth law [cf. Equation (l)]. Such a similarity or relationship would lead to the assumption of a central control instance goveming the regeneration process.
268
E. 0. VOIT, H. J. ANTON,
Otherwise,
AND J. BLECKER
if the parameter values showed no similarities, one perhaps would
rather surmise that the regenerating
stump would react more autonomously
and that its cells would possess the necessary information the regeneration process.
for the control of
REFERENCES 1
H. J. Anton, Die Regeneration von chimarischen Extremitlten bei den Urodelen unter besonderer Beriicksichtigung der Frage nach der Herkunft und den Differenzierungsleistungen des Regenerationsblastems, Inaugural-Dissertation, Koln, 1953.
2
H. J. Anton, Autoradiographische Untersuchungen liber den EiweiRstoffwechsel der Extremitltenregeneration der Urodelen. Wilh. Roux Arch. 161:49-88 (1968). S. A. Baranowitz, P. F. A. Maderson, and T. G. Connelly, Lizard and newt regeneration: A quantitative study, J. Exp. Zool. 210:17-37 (1979).
3 4
5
6 I 8 9 10 11
12 13 14 15 16 17 18 19
J. Blecker, Vergleichende Untersuchungen liber die tltenregeneration bei larvalen, jungen metamorphosierten Triturus, Staatsexamensarbeit, Koln, 1984. K. M. Brown and J. E. Dennis, Derivative free analogues
bei tail
Dauer der Vorderextremiund liberjahrigen larvalen of the Levenberg-Marquardt
and Gauss algorithms for nonlinear least squares approximations, Numer. Math. 18:289-297 (1972). M. M. Ellis, The relation of the amount of tail regenerated to the amount removed in tadpoles of Rana clamitans, J. Exp. Zool. 7(3):421-455 (1909). B. Gompertz, On the nature of the function expressive of the law of human mortality, Philos. Trans. Roy. Sot. London 36:513-585 (1825). P. A. Goodwin, A comparison of regeneration rates and metamorphosis in Triturus and Amblystoma, Growth 10:75-87 (1946). R. J. Goss, Principles of Regeneratton, Academic, New York, 1969. J. S. Huxley, Problems in Relative Growth, Dial, New York, 1932. L. E. Iten and S. V. Bryant, Forelimb regeneration from different levels of amputation in the newt, Notophthalmus viridescens: Length, rate, and stages. Wilh. Roux’ Arch. 173:263-282 (1973). K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2:164-168 (1944). C. E. Maier and M. Singer, The effect of light on forelimb regeneration in the newt, J. Exp. Zool. 202:241-244 (1977). D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. SIAM 11(2):431-441 (1963). C. Mettetal, Lo R.kggh&ation des Membres chez In Salumandre et le Trrton, Librairie “Union,” Strasbourg, 1939. J. Neter and W. Wasserman, Applied Linear Sfntistical Models. Richard D. Irwin, Homewood, Ill., 1974. W. H. Pritchett and J. N. Dent, The role of size in the rate of limb regeneration in the adult newt, Growth 36:275-289 (1972). M. A. Savageau, Growth of complex systems can be related to the properties of their underlying determinants, Proc. Nat. Acad. Sci. U.S. A. 76(11):5413-5417 (1979). M. A. Savageau, Allometric morphogenesis of complex systems: Derivation of the basic equation from first principles, Proc. Nat. Acad. Sci. U.S. A. 76(12):6023-6025 (1979).
REGENERATIVE 20
M. A. Savageau, Growth Math. Biosci. 48:267-278
21
M. A. Savageau
22 23
269
GROWTH equations: (1980).
A general
and E. 0. Voit, Power-law
equation
approach
and a survey of special cases,
to modeling
Theory, J. Ferment. Technol. 60(3):221-228 (1982). M. K. Schauble and M. R. Nentwig, Temperature and prolactin newt forelimb regeneration, J. Exp. Zool. 187:335-344 (1974). M. Singer, Limb Regeneration in the Vertebrates, Addison-Wesley No. 6, 1973, pp. 6.1-6.27.
biological as control Module
systems,
I.
factors
in
in Biology,
24
R. P. Spencer and M. J. Coulombe, Growth 30x277-284 (1966).
25
C. S. Thornton, (1968).
26
H. Wallace, Vertabrafe Limb Regeneration, Wiley, Chichester, N.Y., 1981. V. N. Zamaraev, Does regenerative capacity decrease with age? Soviet J. Dev. Bio[. 4:495-503 (1974).
27
Advances
Quantitation
in limb regeneration,
of hepatic Adv.
growth
and regeneration
in Morphogenesis
71205-249
ABSTRACT Regenerative growth curves are analyzed with a new mathematical function. The function includes known regenerative growth laws as special cases but also accounts for the simultaneous normal growth of the organism. The function is embedded in a general system of growth descriptions that was derived earlier from underlying mechanisms, The major advantage of the function is that the regenerative growth can be mathematically separated from the normal growth. The separation is shown to be biologically meaningful. The regenerative growth function is used to analyze experimental data on limb regeneration in salamanders (Triturw, Salamandra). The function describes adult regeneration equally well and larval regeneration significantly better than the regenerative growth functions used before. The mathematical analysis shows quantitatively how regeneration rate and capacity differ in larvae and adults. The method of analysis can be used to quantify experimental conditions affecting regeneration and to use regeneration as a vertebrate bioindicator in ecological
studies.
INTRODUCTION The regeneration of parts of the body is a very complicated process whose many details are being intensively studied but still not yet fully understood (for reviews see [25], [23], [9], [26]). Although the regeneration processes are governed by biochemical, physiological, developmental, and many more mechanisms, the increase in volume or length of a regenerating part, on the whole, follows a rather smooth curve (e.g. [15], [l], [14], [9]). Generally, the curve starts slow with a lag phase in which processes of wound healing, dedifferentiation, and accumulation of mesodermal blastema take place; it then increases rapidly during the phases of blastema proliferation and differentiation, and finally reaches a shallow plateau when morphological regeneration is completed and only the current normal size still has to be regained (Figure 1).
*Present address: Dept. The University of Michigan, MATHEMATICAL
of Microbiology and Immunology, Ann Arbor, MI 48109, U.S.A.
BIOSCIENCES
131253-269
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
6643 Medical
Science
(1985)
II,
253 0025.5564/85/$03.30
254
E. 0. VOIT, H. .I. ANTON, AND J. BLECKER
days of regeneration FIG. 1. Growth curve of a regenerating postmetamorphic juvenile newt (Triturus r~/guris). The symbols represent measurements [l]; the line was calculated with Equation (0
Because such a curve somehow summarizes the effects of all underlying mechanisms, its analysis may help us to understand better some of the details of regeneration. One important part of such an analysis is to determine the mathematical function that corresponds to the regenerative growth curve. The knowledge of this function can yield insight into the regeneration process, its lag phase, half-time, and final value, and into possible interrelations between regenerative and simultaneous normal growth. In another part of the analysis one may ask how the regenerative growth curve of an animal depends on its species and its developmental stage, and how it varies among comparable animals. More specific questions deal with abnormal modes of regeneration that may have been caused naturally or by experimental conditions. It seems difficult to answer such questions from observations only. Here, a mathematical method will be presented to analyze regenerative growth curves and to help interpret experimental data. There are already mathematical functions to describe regenerative growth [24, 3, 201, but they do not consider that the regenerating animal may be growing at the same time. Particularly in young animals, however, this seems to be an important factor. The earlier growth functions will therefore be generalized by superimposing normal growth on the regenerative growth. The new regenerative growth function is applied to experimental data on limb regeneration in newts (Tritums vulgaris, T. alpestris) and the salamander (Salamandra salamandra). The data can be found in Blecker [4], but will also be published elsewhere for easier access (Anton, Blecker and Voit, in preparation). Because the analysis yields numerical results, it can be used to quantify experimental or natural conditions affecting regeneration. For instance, it
REGENERATIVE
255
GROWTH
could allow the ecologist to use the regeneration of appendages in amphibia as a bioindicator for environmental studies on 0, shortage, chemical pollution, elevated water temperature due to reactor cooling, or other deviations from normal conditions. THE REGENERATIVE
GROWTH
FUNCTION
Spencer and Coulombe [24] compared normal and regenerative liver tissue. They found that the allometric growth law [lo] L=aBJ’
growth of
(1)
is appropriate to describe how in a growing organism the liver weight L depends on the body weight B. Here a and p are constants. In contrast, the regenerative liver growth was better represented by the exponential function L= cL,.(l-
e-h’)
(2)
or by the hyperbola
which describe how the maximum liver weight L, is regained. c, X, k are constants; k is the time required to achieve half of the maximum weight. Both curves begin with a steep increase and then gradually become shallower. According to other authors (e.g. [9, pp. 259 ff.]), regenerative growth begins with a lag phase during which the wound is being closed and the stump becomes organized (e.g. [2], [ll]). The regenerative growth curve is hence sigmoid rather than logarithmic, exponential, or simple hyperbolic. Baranowitz et al. [3] described the tail regeneration in a lizard and a newt species by the Gompertz growth law [7]
(4) R, is the length of the regenerating tail at the “day of initiation,” which is defined as the “day post-amputation on which the proliferation of cells first follows a Gompertz pattern” [3]. R, is the final tail length, t, the time after initiation, b a constant between 0 and 1. Baranowitz et al. [3] claim that the Gompertz fit is “almost perfect.” Unfortunately, they included neither very early nor very late length measurements of the regenerating tail, so the Gompertz function only fits the rather unspecific middle part of the sigmoid regenerative growth curve, and not the transitions to the plateau phases at the beginning and at the end of the regeneration process.
256
E. 0. VOIT, H. J. ANTON,
Savageau [20] introduced
the generalized
R(t)
=
AND J. BLECKER
hyperbola
R,.(t+a)’ b+(t+u)”
which is capable of representing hyperbolic as well as sigmoid curves. R, is the final value to which R(t) converges if t increases. u, b, c are nonnegative parameters. None of these equations considers that the regenerating organism, as a whole, may be growing simultaneously. Therefore, we will replace the constant R, by a function of time that accounts for the superimposed normal growth. Our generalized regenerative growth function is
R(t)
(t+a)=
=R,(t).
b+(t+u)’
a, 6, and c are nonnegative parameters. We will call R,(t) the “normal (growth) component” and (t + a)‘/[ b + (t + u)~ ] the “regenerative (growth) component.” The normal component, R,(t), is the current regeneration target; that is, at each time point, the regenerating part of the organism tries to regain that size, R,(t), that it would have had during normal development at that time point. Depending on the parameter values b and c, the regenerative component can exhibit different S-shaped curves (Figure 2); the parameter b is responsible for the steepness; the parameter c, for the curvature between the lag phase and the growth phase, and between the growth phase and the final plateau.
10
FIG. 2. zomponent:
20
The values of the parameters b and c determine the shape of the regenerative ab = 1, c = 4; @/I =100,000, c = 6; 06 = 50, c = 2. (See text for details.)
REGENERATIVE GROWTH
257
The parameter a in Equations (5) and (6) shifts the time scale of the regenerative component. Because at time t = 0 regeneration just begins, R (0) should be equal to zero, and hence, u is equal to zero by definition. That implies in the biological interpretation that R(t) does not describe, for instance, the whole length of a regenerating limb or the whole weight of a regenerating organ, but only the tissue that actually is regenerated at time t. A stump or the rest of an organ at time t = 0 is not included in R(t) and has to be added if Equation (6) is applied to experimental data for an entire limb or organ. In the analysis of experimental data on limb regeneration in newts, the normal component R,(t) was chosen to be a linear or a logarithmic function. The linear function includes R,(t) = constant, which is identical with Equation (5). The logarithmic function was selected from among the nonlinear functions because it reflects the decelerating growth phase that is often observed in animal growth after the initial exponential growth phase. A logistic equation or a general growth equation [20] would have been an alternative choice. With a linear normal component R,(t), the regenerative growth function can be written as R(t)
= R&l+&+
(7)
,with nonnegative parameters R,, g,,, b, and c. In comparison with Equation (5), the current regeneration target at time t = 0, R,, grows linearly with the normal-growth parameter g,. In logarithmic form, the regenerative growth function reads
R(r)=[s+pln(tcq)].&.
In contrast to the time-shift parameter a in Equation (6), the time-shift parameter q refers to the normal growth and therefore cannot be set equal to zero by definition. The normal component in Equations (5), (7), or (8) uses one (R,), two ( R,, g,), or three (s, p, q) parameters, respectively. These parameters may or may not be observable. For instance, in the limb regeneration of newts after amputation, R, is the length of the amputated part. In Equation (8), the length I, of the amputated appendage can be used to express the parameter q in terms of I,, S, and p by setting 1, = s + p ln(O+ q), q = exp[(I, - s)/p]. The parameters g,, p, and q that reflect the normal growth of the organism can be estimated separately from control experiments or, in limb regeneration, from the unaffected contralateral limb. In contrast, the parameters b
25%
E. 0. VOIT, H. J. ANTON,
AND J. BLECKER
and c cannot be observed in that sense. In a data analysis, all values of parameters that cannot be observed or obtained separately must be determined by mathematical methods. In the present analysis, this was done with a computer search procedure which estimates that set of parameter values yielding the least sum of squared differences (SSQ) between data and the mathematical function whose parameter values are being optimized. The search procedure used here was the Levenberg-Marquardt algorithm [12, 141 in the implementation of the software library IMSL [5]. Most of the data were fitted with a constant, a linear, and a logarithmic normal component. In most cases, one of the three functions clearly yielded the best fit as seen from the least SSQ, but in some cases, Equation (8) produced slightly better results than Equations (5) and (7). The question whether Equation (8) was only better because it contained one or two more parameters was answered with the MSE, test [16], which objectively decides if an improved fit is worth the higher number of parameters. If in the analyses of the following sections one fit is said to be superior to another, that refers to the result of the MSE, test.
ANALYSIS
OF LIMB REGENERATION
IN SALAMANDERS
In the following, we will use Equations (5), (7), and (8) to analyze increase in length of regenerating forelimbs of salamanders (Triturus pestris, T. vulgaris, Salamandra salamandra). Salamandra larvae were collected in the
the al-
vicinity of Wuppertal, W. Germany; the newts were obtained from laboratory spawnings. The animals were maintained in the dark throughout the experimental period at a constant temperature of 22f0.5”C and only brought into daylight for operations and observations. In all specimens, the right forelimb was amputated through the proximal third of the stylopodium. We amputated larvae (4 T. alp., 9 T. vulg., 3 S. salam.) between two and four weeks before metamorphosis, 3 S. salam. one month after metamorphosis, and 15 T. alp. and 5 T. v&g. one year and 10 T. alp. two years after metamorphosis. We also studied 5 T. alp. and 3 T. vu/g. larvae that were one or two years old but postponed metamorphosis. Such abnormalities are mostly due to a defect in the pituitary gland caused by incorrect invagination and induction during early development. The regenerating and the contralateral limbs were observed and drawn with a stereomicroscope having a Leitz drawing prism attached. Observations were made every other day during the first three weeks and subsequently every 3 to 7 days, until the regenerative growth had reached the final plateau phase. Here we will only mathematically analyze the increase in length of the regenerating limb. More experimental details and further results are or will be published elsewhere ([4]; Anton, Voit, and Blecker, in preparation).
REGENERATIVE
1.
259
GROWTH
THE NORMAL
GROWTH
COMPONENT
Depending on an animal’s age, the shape of the regenerative growth curve varies from a steeply inclining curve in larvae to a shallower curve in adults with a prominent lag phase (Figure 3). The longer lag phase in postmetamorphic animals is caused by the extension of the regression phase due to advanced ossification and could also be affected by the darkness (cf. [13]). The mathematical analysis of about 50 regenerative growth curves showed that Equation (8) was superior to Equations (5) and (7) when data came from larvae, whereas Equations (5) and (7) were optimal in describing postmetamorphic regeneration. This finding agrees with our experimental observations showing that normal growth of larval limbs is about logarithmic [corresponding to the logarithmic term in Equation (S)], whereas postmetamorphic limb growth is nearly linear with a very slight slope or none at all [corresponding to the linear and constant numerators in Equations (5) and (7), respectively]. When the growth of the unaffected contralateral limb was recorded, the parameter values for the optimal fit to the data could be determined in two ways: either all parameter values of the normal and the regenerative component were optimized simultaneously with the computer search algorithm mentioned in the last section, or the growth of the contralateral limb was approximated by a logarithmic function in larvae or by a linear function in postmetamorphic animals and these functions were used as normal compo-
5
c 0
U(t) R(t)
0
0
th
days of regeneration
Regenerative growth curves of a larval @ and a one-year-old @ newt (Triturtu ufpestris). The symbols represent the measured data; R(t) was calculated with Equation (8) in @ and Equation (7) in @; U(f) is the normal growth of the unaffected contralateral limb; th = half-time; S = length of the stump at amputation. (See text for further details.)
E. 0. VOIT, H. J. ANTON,
260
AND J. BLECKER
nents. Both methods yielded fits of about the same quality and even similar parameter values for the regenerative component. This implies that the parameter values of the normal and the regenerative components can be determined independently of each other. It can be concluded that neither component affects the other much, and therefore the mathematical separation of the two components is also biologically meaningful. This conclusion is further supported by the analysis of the regeneration of postmetamorphic forelimbs: for some of these cases the constant, linear, and logarithmic normal component yielded fits with about the same residual error SSQ. Again, about the same parameter values were found for the regenerative component. For instance, the best fits to the data for the forelimb regeneration of a two-year-old T. alpestris were obtained with the following parameter values: Equation Equation
(5) : (7):
Equation
(8):
SSQ =1.4386, b =1796, c =1.980, R, = 5,444; SSQ=1.4316, b=1557, c=1.943, R,= 5.461, g, = 0.000074; SSQ= 1.4343, b = 1452, c = 1.919, s = 0.0408, p=1.02242, q=1.15x1023.
(That the parameter values for b and c are in fact very similar will be shown in the next section, in particular, in Figure 4; three encircled symbols, VVV). In all three cases the residual error is about the same; the MSE, test favored Equation (5) over Equation (7) over Equation (8). It should be mentioned that the slope g, in Equation (7) is very small, and hence the linear function is about constant. The parameter values s, p, and q of the normal component in Equation (8) shift the logarithmic function in such a way that during the regeneration period the normal component is also approximately constant. This indicates that Equation (5) in this case is the appropriate description, as the MSE, test also shows. 2.
THE REGENERATIVE
COMPONENT
It is well known that the regeneration rate and capacity are age dependent and decrease if an animal grows older (e.g. [S, 17, 271). However, it is difficult to describe or measure this fact quantitatively, because nonadult animals regenerate and grow at the same time. As demonstrated above, the normal and the regenerative component of a regenerative growth curve can be separated mathematically. It is hence possible to study the regeneration rate and capacity of an animal while the superimposed normal growth is excluded. The regenerative component is characterized by the two parameters b and c which determine its shape [cf. Equations (5), (7), (8) and Figure 21. In order to compare the regenerative components of different animals, the values of b and c were considered as coordinates of points in a semilogarithmic plot, each point representing one regenerate (Figure 4).
REGENERATIVE
261
GROWTH
b . 108 5x105
10” 5x10’
IO’ 5x10=
lo3 500
100 50
10
1 FIG. 4.
Representation
2
1.5
of the regenerative
3
2.5 component
3.5 by the parameters
4
c h and
c’.
Each symbol represents one (h. c) pair and hence one regenerate. The three encircled symbols T correspond to best fits with Equations 5.7 and 8 to the same regenerate (cf. last section). T.
---I
ulp.
Larvae
1 year old 2 years old Abnormal larvae
: . .
T.
vulg.
s. sulun1.
0
0
0
0
A
262
E. 0. VOIT, H. J. ANTON, @
lo
so
0
@
10
10
@
100
120 days
FIG.
Trirurus
5.
Some
alpesrris.
AND J. BLECKER
of
HO regeneration
regenerative components of larvae and postmetamorphic The letters refer to the letters in Figure 4.
animals
of
Clearly, the points of larvae and of postmetamorphic animals are clustered in different parts of the plot, as indicated by the dotted “clouds.” Interpreted biologically, this means that the mode of regeneration is age dependent even if normal growth is excluded. It is striking that the clouds are elongated in directions like those marked by t, = 10,20,. These lines contain the (h, c) pairs of all regenerative components with the half-time t,;’ that is, each regenerate whose b, c-pair lies on the line t, has at time t, reached half of its “regeneration due”. That means, t, is the time when R(t), which is equal to the length of the regenerating limb diminished by the length S of the remaining stump, reaches half the length U(t) of the unaffected contralateral limb diminished by S: that is, R (t,,) = [U( t,,- S]/2 (cf. Figure 3).’ Along the t,-lines, the b, c pairs refer to differently shaped curves with more or less distinct lag phases (the points marked by letters A through F correspond to the curves in Figure 5). Interpreted biologically, the b, c pairs represent the animals’ individuality within the same developmental stage. The half-time t, allows us to compare the regenerative components of different animals with a single parameter. Although th does not consider the shape of the regenerative component, which somehow reflects the individual
‘The half-time is t,, = h”’ as calculated by setting 0.5 = tl;/(h + th) (c > 0). ‘With the given definition of the half-time, th is an observable quantity. For instance, limb regeneration after amputation is studied, R(f), U(r), and S are at hand.
if
REGENERATIVE
263
GROWTH
S salamandra
i ‘I:vulgaris
. .
* . .
.
l
.AA . :
T alpestris
l* :T , 10
1
I 30
* . .
. .
.
.
l
I
I 50
1 60
I 70
FIG. 6. Half-times fh of animals of different species and ages. 0 larvae, v 2 years old, A abnormal larvae. In all species, larvae and postmetamorphic strictly separated.
I80 th
+ 1 year old, animals are
among larvae and postmetamorphic animals of different ages, the half-time t, was found to be appropriate in distinguishing between larvae and postmetamorphic animals. In agreement with the known fact that the regeneration rate decreases in older animals, t, is much shorter in larvae than in postmetamorphic animals: In larvae of T. alpestris, I,, was found to be between 6.5 and 17; in one-year-old postmetamorphic T. afpestris, between 31 and 42; and in two-year-olds, between 34 and 72.3 Similar results were obtained for the other species analyzed (see Figure 6). Our data are not yet sufficient to determine the mathematical form for the fading of the regeneration rate. A plot of the t,-values versus the animals’ age (Figure 7) may suggest a nonlinear, perhaps logarithmic function. It also is possible that t, increases linearly with a steep slope before metamorphosis and later is again linear but with shallow incline. It could even be that t, is about constant in the larval stage and also constant, with a higher level, after metamorphosis. In any event, the mathematical analysis suggests what kind of experiments have to be done in order to elucidate the age dependence of the regeneration rate. Besides the normally developed newts, five T. alpestris and three T. vulgaris were studied that were one or two years old but still in the larval stage. The t,-values of these abnormal larvae were found among the larval as well as the postmetamorphic half-times (see Figure 6), which shows their variation
3t,, is a measure for the regeneration rufe. If r,, is high, it takes a long time to regenerate half of the lost part; it may also mean that the animal will not regain a full-sized part, an effect known from observations (e.g. [27]). th can then be considered as a measure for the regenerative ccrpacit~~.
264
E. 0. VOIT, H. J. ANTON,
AND J. BLECKER
th
.
70 60
50
8 .
40
h
30
0
20 s 6
10 LL
-2
0-Y
2
4
6
8
10
12
14
16
18
20
Age (months after metamorphosis) FIG. 7.
Dependence
of the half-times
th on the animals’
ages. 0 = T. alpesrris,
0 = T.
uulguris.
ambivalent status also on a mathematical level. The b, c pairs of these animals appeared within the larval and the postmetamorphic clouds and between the clouds of the (b, c) plot (Figure 4). DISCUSSION 1.
METHOD
OF ANALYSIS
Many regenerative growth curves show a sigmoid shape that can be described mathematically by very different functions. The choice of a particular function should therefore be based on reasons derived from the process that is being analyzed. Otherwise, the mathematical formulation may fit the experimental data nicely but might not yield any further insight in the biological process. For instance, an arctangent function has a sigmoid graph and might fit some data from regeneration. However, it seems difficult to connect the parameters of that mathematical function with mechanisms involved in the regeneration process. This example shows the crux of many mathematical models and analyses, which is that the agreement between experimental result and mathematical model does not prove the validity of the model. Experimental data may prove a mathematical model to be wrong, but because two models with different structures can yield the same output, it can always happen that the model works differently than the process being studied but still shows the same reaction to a given input. The risk of using a wrong model becomes lower if (i) the structure of the model can be derived
REGENERATIVE
GROWTH
265
from experimental knowledge, and (ii) many details, results, conclusions and predictions from the model agree with the process being analyzed. The mathematical model used in this paper was chosen for two reasons: (i) The model includes as special cases the regenerative growth functions by Spencer and Coulombe [24] and by Savageau [20] that were successfully applied earlier to experimental data; and (ii) because the model [Equations (7) and (S)] can be written as power-law system. A power-law system is a set of nonlinear differential equations that are derived from very general, nonrestricitive assumptions on the underlying mechanisms of biological or chemical processes (e.g. [21]). More specifically, the power-law system can be shown to be a “natural” description for any growth processes [18-211. The model (6) contains two components that determine the regenerative growth: the normal and the regenerative component. The normal component was kept very general in Equation (6), but was restricted in Equations (7) and (8) to linear and logarithmic functions. Both functions may only be approximations to the “real” growth process, but seem to be reasonable in limb regeneration of newts as seen from the observed normal growth of unaffected limbs [4]. However, other functions R,(t) in Equation (6) may be more appropriate for the analysis of other regeneration processes. For the regenerative component we chose the generalized hyperbolic function proposed by Savageau [20], which also includes the regenerative function used by Spencer and Coulombe [24]. As an alternative, we could have tried other growth laws, for instance the Gompertz law, which Baranowitz et al. [3] applied to data from regenerating tails in a newt and a lizard species, or the general power-law equation [20], which includes all laws mentioned as special cases. However, since we felt that other growth laws could hardly improve the fit over the simple two-parameter hyperbolic function [cf. Equation (7) and Figures 1 and 31, we only used the latter as regenerative component. Due to its two parameters, the regenerative component can mimic a variety of different sigmoid curves (Figure 2). That the S-shape of a regenerative growth curve is mainly due to its regenerative component rather than to the normal component can be concluded from adult animals that do not grow any more but whose regenerative growth curves are still sigmoid. A crucial property of the model (6) is that normal and regenerative components are separable. There is no proof of the validity of this assumption, and one could imagine that the two components interacted in such a way that they could not be separated mathematically. However, there are indications that the two components are independent and superimposed: (i) In old animals, constant, linear, and logarithmic normal components sometimes yielded similarly good fits to the data. In all three cases, the parameter values for the regenerative component were about the same, which indicates that the regenerative component does not balance out different normal components.
266
E. 0. VOIT, H. J. ANTON,
AND J. BLECKER
(ii) In larvae, the values for the normal growth parameters could be obtained from records on the growth of the unaffected contralateral limb of the same animal. The parameter values for the regenerative component were very similar, whether those parameter values were used for the normal component or the parameter values of both components were optimized simultaneously. (iii) Besides the conclusions from this mathematical analysis, it has long been known that in tadpoles regeneration occurs faster when more has to be regenerated and, more precisely, that independent of the amputation level, the same percentage of the lost part is always being regenerated at a given time [6, 91. The same phenomenon was later found in lizard and newt tails and in fish fins (for references see [ll, 31). This strongly suggests that in different animals and different amputation levels the same regenerative component is merely multiplied by a constant factor that reflects the species, the size of the lost part, and the animal’s individual metabolism. In Equation (7), this factor is R,.
These findings imply that the separation of the regeneration process into two independent, superimposed components for normal and regenerative growth is not only mathematically possible but also biologically meaningful. 2.
BIOLOGICAL
RESULTS
It is well known that in newts the rate and capacity of limb regeneration highly depend on an animal’s age [8; 9, p. 146; 17; 23; 27; 26, p. 71. The mathematical analysis presented here yields the same result but in much more detail. Besides the description of the regeneration process with Equations (7) and (8), some insight was obtained by studying the crucial parameters h and c of the regenerative component and the half-time t,. As a simple, one-parameter measure, t, represents the regeneration rate and its age-dependent fading. The value of t, was found to increase with age from about 12 days in larvae to about 35 or 45 days in adult newts; similar half-times were found in different newt species and in the salamander. This agrees with observations that larvae regenerate twice as fast as adults, if not more (e.g. [23, 261). The functional dependence of t, on age could not be derived, due to a shortage of sufficient data. In any event, the mathematical analysis, and in particular the introduction of t,, as a measure for regeneration rate and capacity, show which experiments would have to be done and how they could be evaluated in order to determine how regeneration changes with age. From the existing data, t, can be thought to increase rapidly during the first months of life and slower afterwards, according to a logarithmic or S-shaped function. This would agree with the often observed decelerating growth in animals after an initial exponential growth phase. The function could also consist of two
REGENERATIVE
GROWTH
267
linear phases with different slopes and/or different additive terms before and after metamorphosis. It would not seem unreasonable to explain the discontinuity by metamorphosis. The half-time is a characteristic of regenerative growth curves but does not represent the curve completely, e.g. very different curve shapes can reach half of the final value at the same time. In contrast, the parameter-value pair (b, c) contains all information about the regenerative component and so implies its shape. The mathematical analysis of our data showed that the (b, c) pairs varied considerably among all animals studied and even within groups of “comparable” animals, for instance, among larvae of the same species and age. However, the groups of larvae and of postmetamorphic animals turned out to be strictly separate in the (b, c) plot (Figure 4). The clouds of different species overlap somewhat but might be separable by statistical means. The separation in clouds shows that the representation of the regenerative growth component by its (b, c) pair reflects the developmental age as well as the individual variability among the animals of the same group. The latter could hardly be seen from the half-times and so demonstrates the cost of simplifying the regenerative component from two parameters to one. The analysis of some one- and two-year-old larvae, which normally would have already gone through metamorphosis much earlier, showed that their (b, c) pairs lie in the larval or in the postmetamorphic cloud or between the clouds (Figure 4). The exact conditions of these abnormal larvae are not known, but it could be that the different (b, c) pairs reflect different degrees of defects of their pituitary glands. In any event, the example demonstrates how from the (b, c) pair analysis animals of uncertain origin or with unusual properties can be classified. This way, one could also quantitatively study the influences of temperature (cf. [22]), light-dark regime (cf. [13]), or other experimental or environmental conditions that could be relevant for limb regeneration. As an example of experimentally affected regeneration, we will analyze elsewhere the regeneration of chimeric newts (Anton and Voit, in preparation). In our analysis, we have only studied the regeneration of amputated limbs. It would now be interesting to apply the same methods to other regeneration processes, particularly to the regeneration of the tail, which is also being intensively studied (for references see [9, 3, 261). One could then compare our quantitative results on the limb with those obtained by other authors on the tail (e.g. [3]), and also the (b, c) parameter values and the t,-values for different limbs and the tail within the same animal. This would show whether similar qarameter values were found in all regenerating parts of the same organism or whether their regenerative growth satisfied the allometric growth law [cf. Equation (l)]. Such a similarity or relationship would lead to the assumption of a central control instance goveming the regeneration process.
268
E. 0. VOIT, H. J. ANTON,
Otherwise,
AND J. BLECKER
if the parameter values showed no similarities, one perhaps would
rather surmise that the regenerating
stump would react more autonomously
and that its cells would possess the necessary information the regeneration process.
for the control of
REFERENCES 1
H. J. Anton, Die Regeneration von chimarischen Extremitlten bei den Urodelen unter besonderer Beriicksichtigung der Frage nach der Herkunft und den Differenzierungsleistungen des Regenerationsblastems, Inaugural-Dissertation, Koln, 1953.
2
H. J. Anton, Autoradiographische Untersuchungen liber den EiweiRstoffwechsel der Extremitltenregeneration der Urodelen. Wilh. Roux Arch. 161:49-88 (1968). S. A. Baranowitz, P. F. A. Maderson, and T. G. Connelly, Lizard and newt regeneration: A quantitative study, J. Exp. Zool. 210:17-37 (1979).
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REGENERATIVE 20
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