Macromolecular rate theory explains the temperature dependence of membrane conductance kinetics

Abstract

The factor Q₁₀ is used in neuroscience to adjust reaction rates of voltage-activated membrane conductances to different temperatures and is widely assumed to be constant. By performing an analysis of published data of the reaction rates of sodium, potassium, and calcium membrane conductances, we demonstrate that 1) Q₁₀ is temperature dependent, 2) this relationship is similar across conductances, and 3) there is a strong effect at low temperatures (<15°C). We show that macromolecular rate theory (MMRT) explains this temperature dependency. MMRT predicts the existence of optimal temperatures at which reaction rates decrease as temperature increases, a phenomenon that we also found in the published data sets. We tested the consequences of using MMRT-adjusted reaction rates in the Hodgkin-Huxley model of the squid’s giant axon. The MMRT-adjusted model reproduces the temperature dependence of the rising and falling times of the action potential. Furthermore, the model also reproduces these properties for different squid species that live in different climates. In a second example, we compare spiking patterns of biophysical models based on human pyramidal neurons from the Allen Cell Types database at room and physiological temperatures. The original models, calibrated at 34°C, failed to generate realistic spikes at room temperature in more than half of the tested models, while the MMRT produces realistic spiking in all conditions. In another example, we show that using the MMRT correction in hippocampal pyramidal cell models results in 100% differences in voltage responses. Finally, we show that the shape of the Q₁₀ function results in systematic errors in predicting reaction rates. We propose that the optimal temperature could be a thermodynamical barrier to avoid over excitation in neurons. While this study is centered on membrane conductances, our results have important consequences for all biochemical reactions involved in cell signaling.

Significance

Determining the temperature dependence of biochemical reaction rates is essential for theoretical and computational studies to understand metabolic processes. This is particularly important in neuroscience. We show that the value of the temperature adjustment factor, Q₁₀, is temperature dependent for sodium, potassium, and calcium neuronal membrane conductances. Macromolecular rate theory, originally developed to study this dependence in enzymatic processes, provides the best explanation for the temperature dependence of Q₁₀ and predicts the existence of optimal temperatures that could affect neuronal excitability. Since macromolecular rate theory is based on universal thermodynamical properties of macromolecules, this theory could be used to study temperature dependence in all biochemical processes associated with neuronal function and plasticity.

Introduction

The function of the nervous system, from synapses to behavior, is temperature dependent. Changes in neuronal temperature are due to circadian (1), environmental (2,3,4), and experimental conditions (5,6). There is increasing interest in developing therapeutic techniques to deliver thermal stimulation to deep regions of the nervous system and a corresponding need to understand their effect on neuronal excitability (7,8,9,10,11,12,13,14). Presently, temperature adjustments to the voltage (V)- and temperature (T)-dependent reaction rate ($r\left(V,T\right)$) of neuronal membrane conductances use the phenomenological $Q_{10}$ factor. The value of $Q_{10}$ is defined as the ratio of a reaction rate at a difference of 10 K ($Q_{10}=r\left(V,T+10\right)/\mathrm{r}\left(V,T\right)$). This factor has been used since the seminal work of Hodgkin and Katz (15) and is widely assumed to be fixed or to have a negligible temperature dependence (16,17). However, increasing experimental evidence across the biosciences shows that the value of $Q_{10}$ cannot be assumed to be fixed but instead is part of a function that depends on temperature (18,19,20,21,22,23,24,25,26,27,28). Measurements on enzymatic reactions, bacterial processes, and plant respiration, among others, all suggest a similar quantitative temperature dependence of $Q_{10}$ and question its usefulness (29,30,31,32,33,34,35,36,37). More importantly, across biology, a wide range of reaction rates show a negative curvature as a function of temperature (33,38,39). The recently developed macromolecular rate theory (MMRT) provides a thermodynamical-based foundation to explain and predict the negative curvature in reactions rates and, as a consequence, strong changes in the value of $Q_{10}$ with respect to temperature. Since thermodynamical properties apply to all macromolecules (40), including those that mediate neuronal membrane conductances, then it is of importance to determine if the traditional $Q_{10}$ approach used in neuroscience should be replaced by MMRT.

Hodgkin-Huxley equations are used to model neuronal voltage-gated ion conductances (41). The dynamics of each conductance is described with a time activation function, $\tau {\left(V,T_{\mathit{exp}}\right)}_{T_{\mathit{exp}}}=1/r\left(V,T_{\mathit{exp}}\right)$, measured at an experimental temperature ($T_{\mathit{exp}}$). We use a rate coefficient function, $\kappa \left(T\right)$, to modify the rate to a different temperature, $T$, so $\mathrm{r}\left(V,T\right)=\kappa \left(T\right)r\left(V,T_{\mathit{exp}}\right)$, where $\kappa \left(T_{\mathit{exp}}\right)=1$. The factor $Q_{10}$ is used to calculate rate coefficients at other temperatures, $\kappa \left(T\right)={Q_{10}}^{\left(T-T_{\mathit{exp}}\right)/10}\kappa \left(T_{\mathit{exp}}\right)$. From a biophysical standpoint, any biochemical reaction has an activation barrier (42). The activation barrier is the energy needed to pass from the ground state (e.g., closed channel) to the transition state (peak of the activation barrier). In the phenomenological Arrhenius function or in its statistical mechanics version, known as transition rate theory, the energy barrier is assumed to be temperature independent (43,44). This assumption is appropriate for small molecules but not for macroproteins (30,31). Temperature changes can cause conformational transitions in macroproteins due to formation or destruction of noncovalent bonds, such as hydrogen bonds and van der Waals interactions (45). In turn, conformational changes affect the heat capacity of the energy barrier ($\Delta {Cp}^{‡}$) (46). The MMRT considers the thermodynamical effects of temperature changes on $\Delta {Cp}^{‡}$ (31,45). In doing so, MMRT reproduces the temperature dependence of $Q_{10}$, the widely observed high value of $Q_{10}$ at low temperatures, and the existence of an optimal temperature, $T_{opt}$, at which reaction rates start to decrease as a function of increasing temperature (29,30,31,32,47,48,49). The value of $T_{opt}$ is not necessarily related to the denaturation temperature (50,51,52,53,54,55,56,57,58), but to changes in ${\Delta \mathrm{C}}_p^{‡}$ (31), a feature not explained by any Arrhenius associated theory (30).

In this study, we show that the values of the factor $Q_{10}$ for sodium, potassium, and calcium voltage-gated ion conductances change as a function of temperature. We demonstrate that MMRT explains this temperature dependence, including the existence of a $T_{opt}$, for the three types of conductances. When we applied MMRT reaction rate correction to the Hodgkin-Huxley model, we reproduced the temperature dependence of the rising and falling phases of the action potential. Furthermore, the same model is capable of reproducing the values of $Q_{10}$ reported for different squid species. We quantified the error of using a fixed $Q_{10}$ versus MMRT to evaluate the Hodgkin-Huxley model and different types of biophysical realistic models of pyramidal cells. Taken together, our results show that the MMRT provides a thermodynamical based theory to understand how temperature affects neuronal conductances. Our results are of importance to neuroscience given the large-scale efforts to model neuronal function (59).

Theory

Based on Arrhenius (60), the rate coefficient is

$$k^{Arr}\left(T\right)=A{e}^{-E_a/RT}\text{,}$$ (1)

where $A$ is a constant, $E_a$ is the temperature-independent activation energy, and $R$ is the universal gas constant. Using transition-state theory (61), the relationship is

$$k^{trt}\left(T\right)=\left(k_{B}T/h\right)e^{\raisebox{1ex}{$-\Delta G^{‡}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.}\text{,}$$ (2)

where $k_B$ and $h$ are the Boltzmann’s and Planck’s constants, respectively, and $\Delta G^{‡}$ is the free energy of the activation barrier that is related to the enthalpy ($\Delta H^{‡}$) and the entropy ($\Delta S^{‡}$):

$$\Delta G^{‡}=\Delta H^{‡}-T\Delta S^{‡}\text{.}$$ (3)

Substituting Eq. 3 in Eq. 2, we get

$$k^{trt}\left(T\right)=\left(\left(k_{B}T/h\right)e^{\raisebox{1ex}{$\Delta S^{‡}$}\!\left/ \!\raisebox{-1ex}{$R$}\right.}\right)e^{\raisebox{1ex}{$-\Delta H^{‡}$}\!\left/ \!\raisebox{-1ex}{$RT$}\right..}$$ (4)

The $Q_{10}$ factor using the Arrhenius equation is

$$Q_{10}^{Arrh}=e^{\frac{10E_a}{RT\left(T+10\right)},}$$ (5)

or with transition rate theory, we get the Arrhenius-$Q_{10}$ function

$$Q_{10}^{trt}=e^{\frac{10\Delta G^{‡}}{RT\left(T+10\right)}.}$$ (6)

When assuming that $\Delta H^{‡}$ and $\Delta S^{‡}$ are temperature independent,

$$Q_{10}^{trt}=e^{\frac{10\Delta H^{‡}}{RT\left(T+10\right)}.}$$ (7)

The pre-exponential factor ($(\left(T+10\right)/T$) in $Q_{10}^{Arrh}$ or $Q_{10}^{trt}$ varies less than 0.25% between 5°C and 25°C. The value of $\Delta H^{‡}$ is assumed to be positive (45). Fitting Eq. 7 to a constant value of $Q_{10}=3$ over the 0°C–40°C range results in $\Delta H^{‡}=296$ kJ/mol. The values of $Q_{10}$ over this range vary less than 8% from the desired value of 3 (17).

Heat capacity is the amount of heat added to the system to change the temperate by $\partial T$. At constant pressure, this is

$${\Delta \mathrm{C}}_p^{‡}\partial T=\partial \Delta Q$$ (8)

using the definition of entropy

$$\partial \Delta Q=T\partial \Delta S^{‡}$$ (9)

and then using Eqs. 8 and 9 and re-arranging

$$\frac{{\Delta \mathrm{C}}_p^{‡}\partial T}{T}=\partial \Delta S^{‡}$$ (10)

and integrating and rearranging

$${\Delta S^{‡}=\Delta \mathrm{C}}_p^{‡}\mathit{Ln}\left(\frac{T}{T_0}\right)+{\Delta S}_{T_o}^{‡}\text{,}$$ (11)

where $T_0$ is a reference temperature. Similarly, the relationship with enthalpy is

$$\partial \Delta Q=\partial \Delta H^{‡}\text{,}$$ (12)

then

$${\Delta \mathrm{C}}_p^{‡}\partial T=\partial \Delta H^{‡}\text{,}$$ (13)

and the integration results in

$$\Delta H^{‡}={\Delta \mathrm{C}}_p^{‡}\left(T-T_o\right)+{\Delta H}_{T_o}^{‡}\text{.}$$ (14)

Combining Eqs. 11 and 14 with Eq. 2, we get the MMRT-$k\left(T\right)$ function

$$k_{MMRT}=\frac{k_{b}T}{h}{e}^{\frac{-\left({\Delta \mathrm{C}}_p^{‡}\left(T-T_o\right)+{\Delta H}_{T_o}^{‡}\right)}{RT}+\frac{\left({\Delta \mathrm{C}}_p^{‡}\mathit{Ln}\left(\frac{T}{T_0}\right)+{\Delta S}_{T_o}^{‡}\right)}{R}.}$$ (15)

To obtain the MMRT-$Q_{10}$ function, we perform a Taylor expansion of first order on the Logarithm and simplify

$$Q_{10}^{MMRT}=e^{\frac{10T\left({\Delta \mathrm{C}}_p^{‡}\left(T-T_o\right)+{\Delta H}_{T_o}^{‡}\right)+50T{\Delta \mathrm{C}}_p^{‡}-500{\Delta \mathrm{C}}_p^{‡}}{R{T}^2\left(T+10\right)}.}$$ (16)

It is easy to show that the MMRT predicts a $T_{opt}$, at which the reaction rate starts decreasing as a function of increasing temperature (29,31,51).

$$\frac{{\partial k}_{MMRT}}{\partial T}=\frac{\partial }{\partial T}\frac{k_{b}T}{h}{e}^{\frac{-\left({\Delta \mathrm{C}}_p^{‡}\left(T-T_o\right)+{\Delta H}_{T_o}^{‡}\right)}{RT}+\frac{\left({\Delta \mathrm{C}}_p^{‡}\mathit{Ln}\left(\frac{T}{T_0}\right)+{\Delta S}_{T_o}^{‡}\right)}{R}}=0$$ (17)

$$T_{opt}=\frac{{\Delta \mathrm{C}}_p^{‡}{T}_o-{\Delta H}_{T_o}^{‡}}{{\Delta \mathrm{C}}_p^{‡}+R}$$ (18)

Materials and methods

The Hodgkin-Huxley model is (62,63)

$$C_{\mathit{m}}\frac{\mathit{d}\mathit{V}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}\mathit{=}\mathit{〖}\mathit{-}{\mathit{[}\overline{\mathit{g}}\mathit{〗}}_{\mathit{m}}\left(\mathit{V}\left(\mathit{t}\right)\mathit{-}{\mathit{V}}_{\mathit{l}}\right)\mathit{+}{\overline{\mathit{g}}}_{\mathit{k}}{\mathit{n}}^{\mathit{4}}\left(\mathit{t}\right)\left(\mathit{V}\left(\mathit{t}\right)\mathit{-}{\mathit{E}}_{\mathit{k}}\right)\mathit{+}{\overline{\mathit{g}}}_{\mathit{N}\mathit{a}}{\mathit{m}}^{\mathit{3}}\left(\mathit{t}\right)\mathit{h}\left(\mathit{t}\right)\left(\mathit{V}\left(\mathit{t}\right)\mathit{-}{\mathit{E}}_{\mathit{N}\mathit{a}}\right)\mathit{]}\mathit{+}I\left(\mathit{t}\right)\mathit{,}$$ (19)

where $C_m$ is the membrane capacitance; ${\overline{g}}_m$ is the leak conductance; ${\overline{g}}_k$ is the maximum potassium conductance; ${\overline{g}}_{Na}$ is the maximum sodium conductance; $V_l$, $E_k$, and $E_{Na}$ are the reversal potential for the leak, potassium, and sodium currents, respectively; and $I\left(t\right)$ is the input current. The gating variables $x=\left[n\left(t,V,T\right),m\left(t,V,T\right),h\left(t,T,V\right)\right]$ follow a first-order reaction equation:

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\mathrm{x}\left(\mathrm{t},\mathrm{V}\right)={\alpha }_{\mathrm{x}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)\left(1-\mathrm{x}\left(\mathrm{t},\mathrm{V}\right)\right)-{\beta }_{\mathrm{x}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)\mathrm{x}\left(\mathrm{t},\mathrm{V}\right)\text{.}$$ (20)

The functions ${\alpha }_x\left(V\left(t\right)\right)$ and ${\beta }_x\left(V\left(t\right)\right)$ are the forward and backward rates, respectively. These functions were experimentally determined at $T_{\mathit{exp}}=6.3°$C:

$${\alpha }_{\mathrm{n}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)=\frac{0.1-0.01\left(\mathrm{V}\left(\mathrm{t}\right)-{\mathrm{V}}_0\right)}{{\mathrm{e}}^{\left(1-0.1\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)\right)}-1}\text{,}$$ (21)

$${\beta }_{\mathrm{n}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)={0.125\mathrm{e}}^{-\frac{\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)}{80}}\text{,}$$

$${\alpha }_{\mathrm{m}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)=\frac{2.5-0.1\left(\mathrm{V}\left(\mathrm{t}\right)-{\mathrm{V}}_0\right)}{{\mathrm{e}}^{\left(2.5-0.1\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)\right)}-1}\text{,}$$

$${\beta }_{\mathrm{m}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)={4\mathrm{e}}^{-\frac{\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)}{18}}\text{,}$$

$${\alpha }_{\mathrm{h}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)={0.07\mathrm{e}}^{-\frac{\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)}{20}},\mathrm{a}\mathrm{n}\mathrm{d}$$

$${\beta }_{\mathrm{h}}\left(\mathrm{V}\left(\mathrm{t}\right)\right)=\frac{1}{{1+\mathrm{e}}^{\left(3-0.1\left(\mathrm{V}\left(\mathrm{t}\right)+{\mathrm{V}}_0\right)\right)}}\text{.}$$

The reaction of each conductance is defined as

$$r_x\left(V\right)=\frac{1}{{\tau }_x\left(V\right)}={\alpha }_x\left(V\right)+{\beta }_x\left(V\right)\text{,}$$ (22)

and the temperature adjusted reaction is

$$r_x\left(V,T_2\right)={Q_{10}}^{\raisebox{1ex}{$\left(T_2-T_{\mathit{exp}}\right)$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}\left({\alpha }_x\left(V\right)+{\beta }_x\left(V\right)\right)\text{.}$$ (23)

We used two types of multicompartmental neuronal models to test the effect of using MMRT on mammalian cells. We downloaded 12 models constructed based on patch-clamp recordings of human cortical pyramidal cells from the Allen Brain Cell Types database (64). Each one of the selected models corresponded to different donors and consisted on a reconstrucuted neuron and associated passive and active conductances. The passive conductances were identical throughout the model and the active conductances included sodium, potassium, and calcium and were only present at the soma. A second model corresponded to a multicompartmental model based on electrophysiological and anatomical data from a murine hippocampal pyramidal neuron (65). This model was downloaded from modelDB (66) and contained sodium, potassium, and calcium conductances and NMDA receptors distributed across the morphology. Both models used NEURON (67) for the simulations. We developed custom Python and MATLAB (MathWorks, Natick, MA, USA) (68) code for running and analysis of the simulations.

In the Hodgkin-Huxley model, we began by characterizing the rheobase (minimum stimulus to generate a spike). In all cases, the stimulation consisted of injecting constant current for $0.2\mathrm{m}\mathrm{s}$. We classified the response as an action potential by the presence of an after-hyperpolarization and a minimum amplitude of 45 mV from resting potential. We then measured the rising and falling phases of action potentials. The duration of the rising and falling phase was measured using three reference points: when the voltage derivative crosses a threshold (5–50 mV/ms linearly increased from 0°C–20°C) to the action potential peak and when the voltage crosses back the resting potential after the peak. We calculated $Q_{10}$ as the ratio of the reciprocal of the duration of the rising and falling phases at two temperatures 10°C apart.

The literature search was conducted using Goggle Scholar (scholar.google.com). We used search terms like “temperature dependence neuronal conductance,” ‘temperature dependent $Q_{10}$ neuron,” and variations to look for sodium, potassium, and calcium conductances. We collected data from voltage clamp experiments that reported or included in their figures information about the reaction rate of membrane conductances. When the numerical values were not available, we used the GRABIT (69) routine to extract the data points from figure plots. In total, we obtained 10 papers for sodium (21,26,27,70,71,72,73,74,75,76), six papers for potassium (19,21,22,77,78), and nine papers for calcium (18,20,24,25,79,80,81,82,83). We also obtained the data for the rising and falling phases of the action potential in the squid’s giant axon (15,84). We programmed custom code in MATLAB to fit Eqs. 2, 6, 15, and 16 to the experimental data collected from the literature. The goodness of each fit was evaluated with the mean squared error and r-squared values. All models and analysis files are available in our GitHub repository (www.github.com/SantamariaLab).

If $Q_{10}$ is temperature dependent, then using the traditional interpolation to calculate the values of reaction rates needs to be adjusted. Based on the definition (see introduction), it is easy to then derive an iterative process in which the reaction rate changes at small temperature increments of $\Delta T$ to go from the $T_{\mathit{exp}}$ to a final temperature ($T_f$) along all the intermediate temperatures:

$$Q_{10,adjusted}\left(T_{\mathit{exp}}to{T}_f\right)=\prod _{i=0}^{\left|\frac{T_f-T_{\mathit{exp}}}{\Delta T}\right|-1}{\left(Q_{10}\left(T_{\mathit{exp}}\pm i\Delta T\right)\right)}^{\frac{\Delta T}{10}}\text{.}$$ (24)

Results

In this project, we wanted to ask two questions: is the factor $Q_{10}$ constant for neuronal membrane conductances? And, if not, what are the consequences on the predictions from biophysical-based neuronal models?

The factor Q₁₀ is temperature dependent and is described by MMRT

We performed a search of published voltage clamp studies that characterized the temperature dependence responses of sodium, potassium, or calcium voltage-gated ionic conductances (18,19,20,21,22,24,25,26,27,70,71,72,73,74,75,76,77,78,79,80,81,82,83). We plotted $Q_{10}$ data for each conductance without further discrimination on the ion channel subtypes or if the reaction time was for the activation or inactivation components. This analysis evidently shows that $Q_{10}$ depends on temperature for all three conductances (Fig. 1 A–C). For each data set, we fitted the MMRT-$Q_{10}$ (Eq. 16) and Arrhenius-$Q_{10}$ (Eq. 6) functions. As expected from the plots, the mean squared errors of the fits were larger for the Arrhenius models than for the MMRT (data not shown). The values of ${\Delta \mathrm{C}}_p^{‡}$ for each conductance are within the range of −1 to −12 kJ mol⁻¹ K⁻¹ determined for various other enzymes (29,31). We used ${\Delta \mathrm{C}}_p^{‡}$ and ${\Delta H}_{T_o}^{‡}$ to predict the values of $T_{opt}$ for each conductance (Eq. 18), which resulted in a range of 46°C to 57°C, consistent with changes in ${\Delta \mathrm{C}}_p^{‡}$ below denaturation values (85) (Table 1). Thus, this analysis shows that the factor $Q_{10}$ is temperature dependent and is described by the MMRT-$Q_{10}$ function.

Figure 1.

Macromolecular rate theory (MMRT) explains the temperature dependence of factor $Q_{10}$ for neuronal conductances. (A–C) MMRT-$Q_{10}$ (solid) and Arrhenius-$Q_{10}$ (dashed) fits to $Q_{10}$ versus temperature data for sodium (A), potassium (B), and calcium (C) gating kinetics. (D) MMRT-$k\left(T\right)$ fits to rate coefficient versus temperature data for sodium (empty squares), potassium (filled circles), and calcium (empty circles) conductances.

Table 1.

Summary of coefficients from the fits of macromolecular rate theory (MMRT) $Q_{10}$, $k\left(T\right)$, and Arrhenius-$Q_{10}$ functions to sodium, potassium, and calcium conductance data sets

	MMRT-$Q_{10}$			Arrhenius-$Q_{10}$	Average MMRT rate
	${\Delta \mathrm{C}}_p^{‡}$	${\Delta H}_{T_o}^{‡}$	$T_{opt}$	$\Delta G^{‡}$	${\Delta \mathrm{C}}_p^{‡}$	${\Delta H}_{T_o}^{‡}$	${\Delta \mathrm{S}}_o^{‡}$	$T_{opt}$
GNa	−2.49 ± 1.54	76.72 ± 14.23	52.15	86.26 ± 4.42	−2.70 ± 0.92	33.90 ± 1.40	−113 ± 38.90	49 ± 13.84
GK	−4.15 ± 1.36	86.51 ± 1.83	46.63	97.96 ± 7.13	−1.70 ± 0.59	31.80 ± 2.06	−130 ± 7.77	54.52 ± 13.67
GCa	−3.61 ± 1.86	94.78 ± 1.69	57.06	98.18 ± 3.31	−5.07 ± 3.58	70.80 ± 1.11	−2.30 ± 138	40 ± 1.67

The coefficients were ${\Delta \mathrm{C}}_p^{‡}$ (kJ mol^-1 K^-1), heat capacity; ${\Delta H}_{T_o}^{‡}$ (kJ mol⁻¹), enthalpy of reference temperature; ${\Delta \mathrm{S}}_o^{‡}$ (J mol⁻¹ K⁻¹), entropy of reference temperature; $\Delta G^{‡}$ (kJ mol⁻¹), free energy of energy barrier; $T_{opt}$ (Celsius), optimal temperature. For MMRT-$Q_{10}$ and Arrhenius-$Q_{10}$, values are reported with ±95% confidence intervals. The average MMRT-$k\left(T\right)$ were the average values from all the fits to rate coefficient plonts in Fig. 1D. In these columns the data are reported as mean ± SE. G_Na, sodium; G_K, potassium; G_Ca, calcium.

Instead of fitting the values of $Q_{10}$, we fitted the MMRT-rate function (Eq. 15) to 12 data sets that reported rate changes as a function of temperature (18,26,72,73,74,75,76,77,78,79,81,82). For each data set, we obtained $k\left(T\right)$ by normalizing the reported reaction rates to the lowest temperature data point in each set. For all data sets, the log-linear plots show a negative curvature relationship (Fig. 1 D). From these fits, we extracted values for ${\Delta \mathrm{C}}_p^{‡}$, ${\Delta H}_{T_o}^{‡}$, and ${\Delta S}_{T_o}^{‡}$. The average values of ${\Delta \mathrm{C}}_p^{‡}$ and ${\Delta H}_{T_o}^{‡}$ were in agreement with those obtained using MMRT-$Q_{10}$ (Table 1). The values of $T_{opt}$ were also in agreement with their corresponding predicted values using the MMRT-$Q_{10}$ approach. Interestingly, the median value of $T_{opt}$ was 41.38°C for all conductances, suggesting that it could be present within physiological temperatures (Fig. 2). Taken together, our analyses show that the values of ${\Delta \mathrm{C}}_p^{‡}$ and ${\Delta H}_{T_o}^{‡}$ might be similar among conductance types, that neuronal conductances have a $T_{opt}$ that limits their reaction rate as a function of temperature, and that this value is below denaturation temperatures.

Figure 2.

The optimal temperature ($T_{opt}$) of sodium (Na), potassium (K), and calcium (Ca) neuronal conductances is similar. The values of $T_{opt}$ were obtained from fits using the MMRT-$k\left(T\right)$ function. Error bars report mean ± SE. The larger symbols next to the scatter plots are the predicted $T_{opt}$ for each conductance species derived from the MMRT-$Q_{10}$ fits.

In order to apply the MMRT rate function to adjust membrane conductances, it is necessary that the combination of ${\Delta \mathrm{C}}_p^{‡}$, ${\Delta H}_{T_o}^{‡}$, and ${\Delta S}_{T_o}^{‡}$ result in $k\left(T_{\mathit{exp}}\right)=1$. We chose to adjust the values of ${\Delta S}_{T_o}^{‡}$ because its value does not affect the rate of change within the MMRT rate equation and represent the difference in entropy between the experimental and reference temperatures. As an example, we parametrized the MMRT-$k\left(T\right)$ functions using the values of ${\Delta \mathrm{C}}_p^{‡}$ and ${\Delta H}_{T_o}^{‡}$ from the MMRT-$Q_{10}$ fits at a $T_{\mathit{exp}}$ = 20°C. This shows the full structure of the MMRT-$k\left(T\right)$ functions with the lowest $T_{opt}$ corresponding to potassium conductances and the highest to sodium conductances (Fig. 3).

Figure 3.

Plots of the average MMRT-$k\left(T\right)$ function for Na, K, and Ca conductances referenced at $20°\mathrm{C}$$\left(k\left(T=20\right)=1\right)$. The value of ${\Delta S}_{T_o}^{‡}$ was adjusted to 17.40 J mol⁻¹ K⁻¹ for Na, 51.00 J mol⁻¹ K⁻¹ for K, and 79.20 J mol⁻¹ K⁻¹ for Ca.

Effects of using thermodynamical-based temperature adjustment in the Hodgkin-Huxley model

From the original work of Hodgkin and Katz in the squid’s giant axon, the $Q_{10}$ of the rising and falling rates of the action potential are used to study membrane excitability (15). The fit of MMRT-$k\left(T\right)$ to the original Hodgkin and Katz data reproduces precisely their nonlinear temperature dependence (Fig. 4 A; Table 2). We used these fits to modify the kinetics of the sodium and potassium gates of the Hodgkin-Huxley model and tested whether the modified model reproduced the Hodgkin and Katz data. Since the falling phase of the action potential is composed of the inactivation of the sodium conductance and the activation of the potassium conductance, we modified the inactivation gate using the average rate between the rise and fall fits (Fig. 4 B). The measured rising and falling times of the MMRT-modified Hodgkin-Huxley simulations show great consistency with the experimental results (Fig. 4 C). Thus, the MMRT fits to the rise and fall times of the action potential can be used to reproduce the temperature dependence of the sodium and potassium conductances in the Hodgkin-Huxley model.

Figure 4.

MMRT applied to the Hodgkin-Huxley model. (A) Rise and fall rates and fits of the original data from Hodgkin and Katz using the MMRT-$k\left(T\right)$ function (15). (B) The rate coefficients derived from fitting the rise and fall rates of the action potential can be used to adjust the gating kinetics of the Hodgkin-Huxley model. (C) Reproducing the rise and fall times from the Hodgkin and Katz data using the simulated data from (B). (D) Predicting the dependence of $Q_{10}$ for rising and falling phase of the action potential for different species of squid that live in cold and temperate climates (84).

Table 2.

Coefficients from the fits of MMRT-$k\left(T\right)$ function to the rise and fall rates of the action potential in the squid’s giant axon reported by Hodgkin and Katz

	MMRT-HK
	${\Delta \mathrm{C}}_p^{‡}$ (kJ mol−1 K−1)	${\Delta H}_{T_o}^{‡}$ (kJ mol−1)	${\Delta S}_{T_o}^{‡}$ (J mol−1 K−1)	$r^2$
Rise	−0.89 ± 0.31	33.05 ± 1.82	−122.20 ± 6.05	1
Fall	−2.82 ± 0.53	51.51 ± 1.81	−51.44 ± 5.975	1

Values reported with ±95% confidence intervals. ${\Delta \mathrm{C}}_p^{‡}\text{,}$ heat capacity; ${\Delta H}_{T_o}^{‡}$, reference temperature enthalpy; ${\Delta S}_{T_o}^{‡}$, entropy.

The factor $Q_{10}$ of the rise and fall times of the action potential has been determined for different squid species that live in tropical and temperate climates (84). We used the MMRT rate fits to the Hodgkin and Katz data to predict the values of $Q_{10}$ at different temperatures and compared them with those found in different species. Remarkably, this shows that $Q_{10}$ extracted from a single species of squid reproduces the relative temperature independence of the rising time and the high values of $Q_{10}$ at low temperatures for the falling time (Fig. 4 D). We replicated these results using the $Q_{10}$ calculated from the rising and falling times measurements from the MMRT-adjusted models. This analysis suggests that the sodium and potassium conductances found in different animals are the same or have similar heat-dependent kinetics.

Nonlinear effects in the excitability of models at room and body temperatures

We determined the effects of temperature dependence on biophysical models of pyramidal cells. As a first example, we obtained 12 multicompartmental models built based on anatomical reconstruction and electrophysiological measurements of human donors from the Allen Institute Cell Types database. All the models contained sodium, potassium, and calcium conductances. In all cases, the models used $Q_{10}=2.3$. We modified the models using the MMRT-$k\left(T\right)$ coefficients described in Fig. 3. The values of ${\Delta S}_{T_o}^{‡}$ were adjusted to the reported experimental parametrization temperature of each conductance to have $k\left(T_{\mathit{exp}}\right)=1$. We injected constant current (range 0.1–0.5 nA) to get a spike train at 34°C and compared the response to the same stimulus at 21°C (Fig. 5 A). When using MMRT adjustments, all models generated realistic spikes at both temperatures. In contrast, seven out of 12 original models resulted in spurious spikes with no after-hyperpolarization (Fig. 5 B). In only one case did the two approaches generate the same type of spikes and spike trains.

Figure 5.

Effect of temperature correction in the excitability of biophysical models of human cortical pyramidal cells. We compared the models, originally parametrized at 34°C with $Q_{10}=2.3$ to corrections using MMRT-$k\left(T\right)$ fits. (A) Spike trains of models at 21°C and 34°C stimulated with the same constant current. (B) Isolated spikes from (A) at 21°C. Models were obtained from the Cell Types database from the Allen Institute.

As a second example, we used a multicompartmental model of a hippocampal pyramidal neuron (65). This model was developed to reproduce the response of the apical tuft to synaptic theta-burst stimulation at 35°C. As in the previous example, we adjusted the rates according to the reported T_exp of each conductance. We then compared the response of the models at 21°C and 35°C. In this case, we found that at 21°C, the two models responded almost identically. However, at 35°C, the MMRT response was more than 100% than in the original model (Fig. 6). Overall, our examples suggest that MMRT allows biophysical neuronal models to have a robust response to changes in temperature and that using fixed values of $Q_{10}$ or MMRT result in large differences in model behavior.

Figure 6.

Effects of temperature changes to synaptic stimulation voltage responses in the apical tuft of a multicompartmental hippocampal pyramidal cell model. Plots of the original model with fixed $Q_{10}$ values and MMRT at 21°C and 34°C.

Using Q₁₀ to predict reaction rates results in accumulating errors

The computational models we presented show the quantitative consequences in the shape and properties of action potential generation in biophysical neuronal models. However, the $Q_{10}$ and MMRT assumptions have fundamental differences on the dependency of reaction rates to temperature.

The $Q_{10}$ function is concave (positive curvature). However, the experimental evidence and MMRT show that the reaction rates are convex. We illustrate the error introduced by the $Q_{10}$ function by plotting the reaction rates of each conductance at a $T_{\mathit{exp}}=20°C$ and predict the rate up to $35°C$ using both approaches (Fig. 7 A). This shows that the interpolated values (20°C–30°C) have a difference of up to 6%, while the extrapolated (30°C to 35°C) values exponentially increase to 25% (Fig. 7 B). Such deviations can be seen at any temperature (data not shown).

Figure 7.

The errors in predicting MMRT rate coefficients by using the $Q_{10}$ approximation. (A) Plots of MMRT-$k\left(T\right)$ function adjusted at $T_{\mathit{exp}}=20°C$ for the Na, K, and Ca conductances and the corresponding $Q_{10}$ function. (B) Percentage difference between the MMRT-$k\left(T\right)$ and $Q_{10}$ function over the same range of temperatures. (C) Cumulative error when using an iterative method to calculate $k\left(T\right)$ based on $Q_{10}$ and a temperature step $\Delta T$.

The concavity of the $Q_{10}$ function introduces a systematic error when trying to predict the reaction rates of MMRT-type conductances. To demonstrate this, we developed a recurrent method that calculates the values of MMRT rates based on the $Q_{10}$ function at increments of temperature $\Delta T$ (Eq. 24) and applied it to the sodium conductance. To start the iterative process, we used MMRT-$Q_{10}\left(T_{\mathit{exp}}=20°C\right)$ and $\Delta T=10°\mathrm{C}\text{.}$ As expected from the definition of $Q_{10}$, the prediction using Eq. 24 matches the value of the MMRT-$k\left(T\right)$ function at $30°C$. However, starting from the same point but at $\Delta T=5°\mathrm{C}$, the first iteration lands at $25°\mathrm{C}$, which has an error of 6% from the MMRT value. This error is then compounded with the next step, and so on. Thus, when using other values than $\Delta T=10°\mathrm{C}\text{,}$the prediction generates an error that linearly accumulates (Fig. 7 C). Therefore, while Eq. 24 is mathematically correct, it is the wrong physics to model the phenomenon.

Discussion

In the present work, we showed that, as opposed to what is generally assumed in neuroscience, the factor $Q_{10}$ used to modify the gating variables of neuronal membrane conductances is temperature dependent. We showed that MMRT explains this temperature dependence and reproduces experimental data. Furthermore, the theory provided testable predictions on the existence of optimal temperatures. Optimal temperatures have not been reported for these types of molecules but can be observed in the experimental data. MMRT is a thermodynamics-based theory that has been applied to different biophysical systems, such as reaction rates on individual enzyme kinetics, soil respiration, methanogenesis, and nitrification (30,31). While we concentrated on the effects on Hodgkin-Huxley-type models of membrane conductances, MMRT could also be applied to all biochemical reactions involved in neuronal function. Our results go beyond their applicability to biophysical models of neurons because they could contribute to the understanding of the effects of thermal neuronal stimulation.

The MMRT is based on the fundamental relationships of $\Delta G^{‡}$, $\Delta H^{‡}$, $\Delta S^{‡}$, and ${\Delta \mathrm{C}}_p^{‡}$. Originally, it was developed to explain the intermediate conformational changes of biomolecules involved in biochemical processes (30). These changes could be associated with hydrogen bonds being formed or broken within proteins, which would change the ${\Delta \mathrm{C}}_p^{‡}$ of the system and, as a consequence, cause the temperature dependence of $\Delta H^{‡}$ and $\Delta S^{‡}$ (30). MMRT is increasingly being used to explain temperature effects across the biosciences (30,34,35,37,47,49).

An important consequence of the interplay of ${\Delta \mathrm{C}}_p^{‡}$ and $\Delta H_{T_0}^{‡}$ is the existence of $T_{opt}$, which corresponds to the maximum rate of a reaction, in contrast to the ever-increasing rates in the Arrhenius approach. Our analysis suggests that $T_{opt}$ is relatively high with a median value of $41°C$; however, it is important to note that measurements report lower values than predicted (30). In any case, the value of $T_{opt}$ is below denaturation temperatures for many enzymes as also suggested by our analysis.

Is the existence of $T_{opt}$ of physiological or computational importance to neurons? Most animals are poikilotherms. It has been suggested that the nervous systems of these organisms have compensatory mechanisms to operate efficiently under large thermal fluctuations (86,87,88,89). In homeotherms, changes in temperature affect the activity of sleep-active neurons, and localized changes in temperature can control sleep cycles (for a review, see (90)). There is also evidence that heat-activated channels control the excitability of hippocampal neurons (91). The effect of fever in the activity of the nervous system is of great importance because hyperthermia can result in seizures. However, the mechanisms are not clear (for a review, see (92) and (93)). From an evolutionary point of view, organisms and their enzymes evolved to be efficient at the temperatures they experience as reflected in the evolution of ion channels (94,95,96). This has resulted in psychrophilic, mesophilic, and thermophilic enzymes having different optimal temperatures (for a general reference, see (37)). In this context, the presence of $T_{opt}$ could be a thermodynamical barrier to reduce the excitability of neurons as temperatures increase from the natural environment. Computationally, the combination of conductances with different $T_{opt}$ in the same neuron could result in differential excitability properties, even for small changes in temperature.

In the original work of Hodgkin and Katz, the value of $Q_{10}$ was, indeed, not constant (see their Table 2A (15)). However, we could not find models in modelDB, a database of neuronal models, that did not have fixed $Q_{10}$. The idea that $Q_{10}$ is fixed is embedded in widely used simulation environments (67). We used two models as examples of the quantitative and qualitative changes in cellular electrophysiology. We selected these models because they focused on the cerebral cortex and the hippocampus, areas of interest in computational neuroscience. These models could yield a 100% difference depending on if a fixed or MMRT-adjusted $Q_{10}$ was used. Importantly, there are large-scale programs that aim to build realistic models of the nervous system (97,98). Paramount to these efforts is generating models with coefficients that can be used not only to replicate the data in which they are based but to predict behavior under different circumstances. As is well known, this problem is not well defined in neuronal modeling with multiple combinations of membrane conductances replicating the same set of spike trains (99). Our results suggest that performing experiments at different temperatures and using physics-based correction approaches could contribute to better parametrize neuronal models.

While the original motivation for this study was on adjusting simulation parameters, our results have wider applicability. Effects on synaptic strength as a function of temperature are well documented (100). Changes in brain temperature could happen due to environmental or circadian conditions (1). Furthermore, there is increasing interest and technological development to deliver thermal stimulation to deep regions of the nervous system (7,8,9,10,11,12,13,14). These technologies could stimulate regions that are technically impossible for current optogenetic tools. Finally, MMRT potentially applies to all enzymes involved in neuronal function, which could affect our understanding on cellular mechanism of learning and memory.

Author contributions

Conceptual design and manuscript drafting, F.S.; simulation and manuscript drafting, B.P.; and simulation, N.B.

Editor: Gabriela Popescu.