Energy and time optimization during exit from torpor in vertebrate endotherms

Abstract

Torpor is used in small sized birds and mammals as an energy conservation trait. Considerable effort has been put towards elucidating the mechanisms underlying its entry and maintenance, but little attention has been paid regarding the exit. Firstly, we demonstrate that the arousal phase has a stereotyped dynamic: there is a sharp increase in metabolic rate followed by an increase in body temperature and, then, a damped oscillation in body temperature and metabolism. Moreover, the metabolic peak is around two-fold greater than the corresponding euthermic resting metabolic rate. We then hypothesized that either time or energy could be crucial variables to this event and constructed a model from a collection of first principles of physiology, control engineering and thermodynamics. From the model, we show that the stereotyped pattern of the arousal is a solution to save both time and energy. We extended the analysis to the scaling of the use of torpor by endotherms and show that variables related to the control system of body temperature emerge as relevant to the arousal dynamics. In this sense, the stereotyped dynamics of the arousal phase necessitates a certain profile of these variables which is not maintained as body size increases.

Appendix

Upregulation of T _s

The graphical visualization of the dynamics of M and T_B, determined by Eqs. 29 through 34, requires that numerical values are assigned to all constants. The organism is considered to have body mass m = 50 g, specific heat capacity c = 4.12 J/^oC·gram (Haemmerich et al. 2005), euthermic body temperature T_E = 38 °C and initial temperature T₀ = 12 °C. Considering Eq. 3 and a mass of 50 g, total thermal conductance (h) has a value of 0.001310 watts/^oC (0.22 mL O₂ h⁻¹ °C⁻¹). The environment is considered to be at constant temperature T_A = 10 °C. These values result in an euthermic resting metabolic rate M_E = 1.16 watts and an initial metabolic rate M₀ = 0.083 watts, independently of the gain k. For the sake of comparison, the elephant shrew Elephantulus rozeti with body mass of 45 g at T_A of 10 °C has an euthermic oxygen consumption rate of 5 mL O₂ g⁻¹ h⁻¹ which results in 1.26 watts (Lovegrove et al. 2001).

The time course of metabolic rate and body temperature described by Eqs. 34–39 are plotted in Fig. 

Fig. 11

Examples of dynamics of metabolic rate (blue lines) and body temperature (red lines) along the time (in hours) accordingly to equations from 29 to 34. Darker colours represent low gain (r = 0.5), intermediate colours represent critical gain (r = 1) and lighter colours represent high gain (r = 10)

11. Note that the brighter lines, which represent the high gain case, have dynamics that resemble the stereotyped behaviour of exit from torpor shown in Fig. 1. Therefore, in the remaining of the analysis, we focus only on values of r greater than 1.

As explained in the main text, we have the mathematical framework for the general behaviour of the exit from torpor, nevertheless, without the 2·M_E peak. This can be accomplished by an upregulation of T_S. To implement the change in the setpoint during the process of exit, Eq. 7 was changed to:

$$\frac{d{T}_{S}}{dt}=2\cdot \gamma \cdot \left({T}_{F}(t)-{T}_{S}\right)$$

(35)

where T_F(t) is given by:

$${T}_{F}(t)={T}_{G}+\left({T}_{G}-{T}_{E}\right)\cdot \left({e}^{-2\cdot \gamma \cdot (t-\tau )}-1\right)\cdot \theta (t-\tau )$$

(36)

T_G is a value which the setpoint would try to achieve before aiming for T_E, τ is the time at which the setpoint dynamic would change and $\theta$(t) is the Heaviside Step Function. Equation 36 describes a function which has constant value T_G until time $\tau$, after which there is a step change to T_F(t > $\tau$) = T_E. Considering the same 50 g organism described before, we have T_G = T_E + 7 °C (in this case, 45 °C) and $\tau$ = 5,400 s (1.5 h). New equations for T_S, T_B and M were obtained through the following system:

$$\frac{d{T}_{B}}{dt}=-\frac{h}{m\cdot c}\cdot {T}_{B}+\frac{1}{m\cdot c}\cdot M+\frac{h}{m\cdot c}\cdot {T}_{A}$$

(37)

$$\frac{dM}{dt}=k\cdot ({T}_{S}-{T}_{B})$$

(38)

$$\frac{d{T}_{S}}{dt}=2\cdot \gamma \cdot \left({T}_{F}(t)-{T}_{S}\right)$$

(39)

Numerical integrations were done for T_S, M and T_B—the unique dynamic for T_S is shown in Fig. 

Fig. 12

Dynamic for T_S (in °C) as a function of time (in hours), according to Eq. 35. Notice that at 1.5 h there is a change in the behaviour of the dynamic due to the change in T_F (Eq. 36). It is important to highlight that this dynamic does not depend on the values of T_B, M or r

12, while some dynamics for M and T_B, which depend on r, are shown in Fig. 5 of the main text.

Calculations of the total time and energy spent during the process were done for different values of r. Total time was defined as the time at which T_B reaches and crosses T_E for the first time (after that, it starts to oscillate around T_E). The total energy spent was the integral of M(t) from time zero to that total time. Figures 6 and 7 show these results (see main text).

For different values of r, T_B will cross T_E at different phases of the oscillation, giving rise to the discontinuity seen in Figs. 6 and 7, at around r = 85. The insets in Fig. 5 illustrate this.