Physiological time and time-invariance

J. theor, Biol. (1983) 104, 349-351

Physiological Time and Time-invariance NICO M. VAN STRAALEN Biologisch Laboratorium, Vtije Universiteit, De Boelelaan 1087, Postbus 7 161, 1007 MC Amsterdam, The Netherlands (Received 5 January 1983) An operational definition of physiologicaltime is suggestedthat involves a non-linear transformation of a physical time-scalesuch that a certain temperature-dependentprocessbecomestime-invariant. The definition expressesthat physiologicaltime has the dimensionof time, and can be measuredin units suchasdaysor weeks.It is shownthat the quantification of a physiological time-scale requires the specification of an arbitrary constant, which equalsthe rate of changeof the processin physiological time. When the process-rate also depends on someinternal variable, the interaction of this variable with temperature must be multiplicative, if a singlephysiologicaltime-scaleis to be used.The relationshipwith conventional usesof physiological time is discussed.The time-scale,as defined here, reducesto degree-daysummationor development accumulationif certain multiplicative constantsare deleted.

1. Introduction The concept of physiological time is intuitively obvious to some extent, but has nevertheless been explicated in various ways in biological literature. In applied botany and entomology, it refers to the “heat unit approach” (Wang, 1960), and physiological time is measured in degree-days (Gilbert & Gutierrez, 1973). More generally, Stinner, Gutierrez & Butler (1974) and Taylor (1981) use “development accumulation” as the basis for a physiological time-scale. VanSickle (1977), on the other hand, denotes by “physiological age” any suitable variable (e.g. body size) that is related to age by a monotonic function and that serves as an indicator of life-history events. Lindstedt & Calder (1981) discuss physiological time of homeotherms in the context of the basal rates of cyclic events, such as heartbeat,

which show a power relationship with body size. Moreover, the term “biological time” is used in the field of biological rhythmicity, where the period of biological clocks is usually temperature-independent (Winfree, 1980). It is remarkable that in some uses of the concept, physiological “time” does not have the dimension of time. To denote by “time” quantities having 349

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no dimension (development accumulation), or dimension temperature x time (degree-days) seems rather inexpedient. It is the purpose of the present paper to suggest a definition of physiological time that avoids this confusion of dimensions and expresses physiological time in units of time, such as days or weeks. Not all of the uses of physiological time cited above are covered in this paper; the definition proposed is restricted to the context of the temperature-dependence of biological processes in poikilotherms. An application of this approach, involving the demographics of Collembola populations, will be given in a separate paper. 2. Definition The notion of time-invariance is central to many population models in ecology, because time-invariant models are most amenable to analysis. For instance, in an age-structured population model, no analytical solution for the age-structure can be obtained if the age-specific demographic functions depend also on time (Charlesworth, 1980). A model is designated as time-invariant if the defining relations (structure) of the model do not change with time (see, e.g. Kalman, Falb & Arbib, 1969, section 1.1 for a formal definition). A time-invariant model is insensitive to positive linear transformations of the time, i.e. to transformations of the type

where r denotes the original, and i the transformed time-scale, and cy and p are constants ((u must be positive). This expresses that shifts in the zero-point of time, and changes of time-units do not affect the outcome of the model. The application of time-invariant models to the population ecology of poikilotherms poses a serious problem, since in poikilotherms all life-history processes depend on temperature, which changes with time. A shift of the time-scale generally produces another temperature schedule and the dynamics of the population cannot be described by the same model. Although it is possible to include temperature as an additional variable in the model, the problem might also be circumvented by defining a physiological time-scale as follows: A physiological time-scale for a specified biological process is a time-scale obtained by transforming a physical time-scale so that the rate of change of the process becomes time-invariant in physiological time. This allows a temperature-independent has been transformed appropriately.

model to be used, after the time

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In this paper “physical time” denotes a set of time-scales implied by physical processes; these time-scales can be obtained from each other by positive linear transformations. For example, the Julian time-scale, which numbers the days in one sequence from 4713 B.C., may be taken as a reference and any positive linear transformation there from, such as time in weeks after 1 January 1979, is one of the physical time-scales. Likewise, “physiological time” denotes a set of time-scales, related to each other by linear transformations, but related to a physical time-scale by a non-linear transformation. As pointed out by Whitrow (1980), two time-scales can only be considered as effectively the same if they are linearly related. Conversely, only by applying a non-linear transformation, we pass to an essentially different set of time-scales. The definition of physiological time given here is implicit in many ecological papers which use the concept; e.g. Lee et al. (1976) noted that the use of degree-days considerably simplified the analysis of an age-structured population model. I will show below, by simple calculus, that a consistent development of the definition leads to a small, but important, modification of the conventional notions of degree-days and development accumulation. 3. Consequences At first we will consider a very simple biological process which would have constant rate of change if the temperature were constant. However, in physical time its rate of change is not constant, because the temperature varies. Let t denote physical, T physiological time and P the function that maps t into r. Furthermore, let F denote the function that describes the temperature dependence of the process and T the function that gives the temperature schedule in physical time. The rate of change of the process (d.s/dt) can then be given as: $=F(T(t)).

(1)

From the definition of physiological time given above we require the rate of change in physiological time to be constant. Denote this constant by k, then P must be such that

Using the chain rule for differentiation, (2) into: $=$F(T(t))

we may combine equations (1) and

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which is satisfied if P is written

as

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(3) where x is a dummy variable of integration and f. is an arbitrary constant. Because equation (3) defines a scale transformation, P must be a monotonic function, so that a one-to-one correspondence of physical and physiological time is achieved. This requirement will be satisfied if the rate of change of the process is positive (or negative) for all temperatures, as is usually the case in practice. The following points emerge from this exposition: (i) Physiological time can be defined only with reference to a specified biological process, which must have non-negative (or non-positive) rate of change in physical time. Since different processes, even in the same animal, may respond differently to temperature (unless they are coupled in the way discussed below), a universal physiological time-scale, even for a single temperature schedule, does not exist. This contrasts with the situation in physics, where the time-scales implied by various physical processes seem to be identical in practice (although this cannot be guaranteed a priori), which gives rise to the concept of universal physical time (Whitrow, 1980). In other words, the set of physical time-scales can in itself be considered asan interval-scale, which is unique up to the positive linear transformations (Pfanzagl, 1968, section 1.8). In physiological time, one can imagine several interval-scales which are not mutually exchangeable. (ii) Defining physiological time requires specification of two arbitrary constants, corresponding to the desired rate of change of the process in physiological time (k) and the zero point of physiological time in physical time (to). Although its value is arbitrary, the use of the constant k is not trivial. In the first place, it ensures a balance of dimensions: deleting k causesthe confusion of dimensions referred to above. In the second place, it allows a comparison of different physiological time-scales; there are situations where it is convenient to assign different k-values to the physiological time-scales of different species (van Straalen, unpublished); if k is deleted, this cannot be achieved. The formalism given here can be extended easily to those caseswhere the process under study does not only depend on temperature, but also on some internal variable. For example, the rate of body-growth may depend on temperature, but usually also depends on the body-size. In this case, equation (1) can be written as ds -=F(T(t),s). (4) dt

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It will be shown that a physiological time-scale makes sense only if the interaction between temperature and the internal variable is multiplicative, i.e. if equation (4) can be written as

where F, denotes the temperature response at some reference value of s and f(s) is a dimensionless function which expresses that the temperature response at other values of s can be obtained by multiplying F, by an amount which depends only on S. The process will now be time-invariant in physiological time if

where k, denotes the rate of change in physiological time of the process at the reference value of s. It is easy to see that this leads to the same formulation of P (equation (3)). To illustrate the criterium of multiplicative interaction, consider the case of temperature-dependent logistic body-growth. A physiological time-scale for growth as a whole can be specified only if the growth rate can be written as $= F,(T(t))(as

-bs2).

This implies that the temperature should have an equal effect on both a and 6. Consequently, it should not affect the quotient a/b, which equals the asymptotic maximal size; it should affect only the rate at which animals grow towards this maximal size. This constraint may severely limit the application of physiological time-scales, since many animals grow larger when exposed to persistent low temperatures. The principle of multiplicative interaction can also be generalized to two or more different processes, such as body-growth and moulting, or bodygrowth of different species. A single physiological time-scale makes sense only if these different processes have the same temperature-response, up to a multiplicative constant. Additionally, it must be noted that equation (3) defines a fixed time-scale, which does not move along with an individual animal as it passes through its various life-history stages (or body sizes). In fact, as was also argued by Curry, Feldman & Sharpe (1978), equation (3) can only be regarded as such if the process under study depends on temperature only. The general expression for physiological age (not shown here) is more complicated.

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with Conventional

Uses of Physiological

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At first, it is clear that development accumulation, as used by Taylor (1981), is equivalent to the integral in equation (3). Since developmental rate usually has dimension (time)-‘, the integral is a dimensionless quantity and l/k has the dimension of time. Thus, physiological time has the time-dimension. As argued above, k ensures a balance of dimensions and cannot be simply deleted. Secondly, if F is linear, with slope c and threshold temperature h (the rate of change is zero for all temperatures below h ), then equation (3) may be written

provided that T(t) 2 h for all t. (To allow for below-threshold temperatures, the temperature schedule must be truncated at h.) The integral in equation (5) corresponds to the familiar degree-day summation (Kitching, 1977). In our case, this is related to physiological time through the multiplicative constant c/k, which has dimension (temperature)-‘. The appearance of c in equation (5) acknowledges the fact that the progress of physiological time does not only depend on the threshold temperature, but also on the slope of the temperature response. The conventional degree-day summation does not account for this. In the third place, let us consider processes that are only observed at certain discrete points in time. For example, in the development of eggs, only the time of laying (stage entry times) and time of hatching (stage exit times) are recorded, and the progress of the process cannot be observed in between. These data may be summarized by a function R, which specifies

q/k {

FIG. 1. Possible shape of the function P, which maps physical time (t) into physiological time (7). q/k equals the time-interval needed to complete a single stage in physiological time.

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for each physical time t at which an egg is laid (or could have been laid), the corresponding physical time U, at which this egg will hatch (or would hatch if it were laid at time t). Now let 4 denote an arbitrary constant equalling the part of the process elapsing between stage entry and stage exit (e.g. in the case of body-growth, q equals the size-increment pertaining to the stage; in the case of instar-development, q may be dimensionless and valued at unity or 100). The relationship between R and the definition of physiological time is as follows (Figs 1 and 2). An egg laid at physical

FIG. 2. Shape of the function R, which maps (physical) stage entry times (t) into (physical) stage exit times (u). R was constructed graphically from P in Fig. 1, with q/k as indicated. I, and u1 are the same for both Figures.

time tl is laid at physiological time 71 =P(ti). Since in physiological time it develops at constant rate k, it will hatch q/k time units later in physiological time. This corresponds to the physical time-point u1 = R (tl). In other words, for each time t the following relation holds: P(t)+;=P(R(t)).

(6)

I do not see how this equation can be solved in general for P(r) if R(t) is given. (This would enable one to determine the temperature response under any variable temperature regime from observations on stage entry and stage exit times.) Since two points of P determine one point of R, this may be intrinsically impossible. On the other hand, if P is given, R can be constructed graphically (Figs 1 and 2). From P being monotonic, it follows that R cannot go below the line u = f : because physiological time is not permitted to move backwards, an

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egg cannot hatch before it is laid. R can also be expressed directly in terms of F and T; to see this, substitute equation (3) into equation (6) and rearrange to obtain: R(I)

If

F(T(x))dx

=q.

(7)

A similar equation is used in computing the development of insect stages (cf. Allen, 1976; Logan et af., 1976). Finally, a quantity known as the thermal constant may be expressed in the terminology used here. If the response function F is linear, equation (7) reduces to R(l)

[T(x)-h]dx

--F

where h and c are as in equation (5). This expresses that the area under the time-temperature graph above the threshold h, when evaluated for the duration of a process with marked beginning and end, is constant, i.e. it does not depend on the actual duration of the process. This is an empirical fact, known for a long time (Wang, 19601, but the underlying principles on which it is based are seldom indicated. The thermal constant (here q/c) has dimension temperature x time and has been used effectively for predicting the seasonal occurrence of insects (e.g. Harcourt, 1981). As Laudien (1973) has pointed out, it is not correct to refer to this constant as a heat quantity. 5. Conclusions

Although some of the results arrived at in this paper are intuitively obvious and have been used in practice for some time, the analysis given may be useful since it provides a common base for various practices by deriving these from the notion of time-invariance. On the other hand, some new points have been presented which do require the analysis given. The definition proposed is expedient in that physiological time has the dimension of time. Thus, if a population model is applied in physiological time, its variables referring to a rate of change in physical time still do so in physiological time, and retain their biological interpretation. I feel indebted to Dr E. N. G. Joossefor encouragement and comments. Some ideas,expressedin this paper, originated from discussions with Dr P. G. Doucet, and I alsobenefited by commentsand hints from Dr J. A. J. Metz; to both of them I am very grateful. Furthermore, I wish to thank Miss D. Hoonhout for typing the

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manuscript. The investigations were supported by the Foundation for Fundamental Biological Research (BION), which is subsidized by the Netherlands Organization for the Advancement of Pure Research (ZWO). REFERENCES ALLEN, J. C. (1976). Environ. Entomol. 5,388. CHARLESWORTH, B. (1980). Evolution in Age-structured Populations. Cambridge: Cambridge University Press. CURRY,G.L.,FELDMAN,R.M.,& SHARPE,P.J.H.(~~~~).J. theor.Biol.74,397. GILBERT,N.& GUTIERREZ, A. P.(1973). J. AnimalEcol. 42,323. HARCOURT,D. G. (1981). Can. Ent. 113,601. KALMAN,R.E.,FALB,P.L.&ARBIB,M. A.(1969). TopicsinMathematicalSystem Theory. New York: McGraw-Hill. KITCHING,R.L.(~~~~). Aust.J. Ecol. 2,31. LAUDIEN, H. (1973). In: Temperature and Life (Precht, H., Christophersen, J., Hensel, H. & Larcher, W. eds) pp. 355-377. Berlin: Springer-Verlag. LEE, K. Y., BARR, R. O., GAGE,!% H. & KHARKAR, A. N.(1976). J. theor. Biol. 59, 33. LINDSTEDT, S. L. & CALDER III, W. A. (1981). Quart. Rev. Biol. 56, 1. LOGAN, J. A., WOLLKIND, D. J., HOYT, S. C. & TANIGOSHI, L. K. (1976). Environ. Entomol. 5, 1133. PFANZAGL, J. (1968). Theory of Measurement. Wiirzburg-Wien: Physica-Verlag. STINNER,R.E.,GUTIERREZ, A.P.& BUTLERJR, G. D.(1974). Can.Ent. 106,519. TAYLOR,F.(~~~~). Am.Nat. 117,l. VANSICKLE,J. (1977). I. theor. Biol. 64,571. WANG, J. Y. (1960). Ecology, 41,785. WHITROW, G. J. (1980). The Natural Philosophy of Time. Oxford: Clarendon Press. WINFREE, A. T. (1980). The Geometry of Biological Time. New York: Springer-Verlag.