https://orcid.org/0000-0003-2301-8791
Figures
Abstract
Temperature and pressure affect neuronal gating kinetics. We recently used thermodynamic macro-molecular rate theory to describe the effects of temperature on the activation rate function of sodium, potassium, and calcium voltage activated conductances. Here, we extend the theory to include the effects of both, temperature and pressure. The theory includes transition changes in heat capacity, entropy, enthalpy, activation volume, expansivity, and compressibility during protein conformation. The complete model replicates experimental results from the literature. We used the expanded model to study how temperature and pressure affect the generation of action potentials in the Hodgkin-Huxley model and in detailed biophysical and morphological models of human cortical neurons. In particular, our results show how pressure can affect the optimal temperature of reaction rates and how small changes in pressure could affect spike timing and correlations across neurons. Our work provides a physics-based approach to adjust reaction rates of neuronal conductances to study cellular function in evolution and under extreme heat and pressure conditions such as those found in blast waves or electro-mechanical neuronal couplings.
Citation: Miller JA, Pahlavan B, Gamboa B, Santamaria F (2025) Incorporating the thermodynamic effects of temperature and pressure on modeling neuronal gating kinetics. PLoS One 20(10): e0333592. https://doi.org/10.1371/journal.pone.0333592
Editor: Ghanim Ullah, University of South Florida, UNITED STATES OF AMERICA
Received: March 11, 2025; Accepted: September 16, 2025; Published: October 9, 2025
This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.
Data Availability: All simulation files, analysis scripts, and data are available in https://github.com/SantamariaLab or by request. They are also in ModelDB (https://modeldb.science/2019887) data base Model Number 2019887.
Funding: BG AFOSR LRIR 23RHCOR012 FS NIH-NINDS 1R01NS130759. AFOSR LRIR: Air Force office of Scientific Research - Laboratory Research Initiation Request. NIH-NINDS: National Institutes of Health-National Institute of Neurological Disorders. The funders had no role in study design, data collection and analysis, or preparation of the manuscript. The views expressed are those of the author and do not reflect the official guidance or position of the United States (U.S.) Government, Department of Defense, United States Air Force or United States Space Force. Distribution A: Approved for public release; distribution unlimited (P.A. #AFRL-2024-0248, 16 JAN 2024).
Competing interests: The authors have declared that no competing interests exist.
Introduction
Practically all studies on the effects of temperature [1–6] and pressure [7–9] on the activation rate of voltage-gated conductances use an exponential function: either Q10, the Arrhenius function [10], or its related transition state theory [11]. This approach assumes that temperature only affects the free energy () of the energy barrier of the activation gating mechanism through constant entropy (
) and enthalpy (
). However, macroproteins show a universal non-Arrhenius temperature-dependent behavior characterized by a decrease in reaction rate after an optimal temperature (Topt) not due to denaturation [12]. We recently used Macromolecular Rate Theory (MMRT) to demonstrate that sodium (Na), potassium (K), and calcium (Ca) membrane conductances all have Topt within physiological ranges not associated with denaturation and that the Arrhenius equation produces additive errors in predicting temperature effects [13]. The MMRT assumes that temperature affects both the enthalpy and entropy of the energy barrier through changes in the heat capacity (
) of enzymes.
There is a great deal of interest in understanding how thermodynamic variables affect enzymatic function [14,15]. For example, temperature [16,17], pressure [18–22], and osmotic flow [23,24]. In particular, there are multiple studies to understand how temperature [1,25–28] and pressure [29–32] affect neuronal function. A unified model could be useful for a wide range of applications, such as the effects on cellular function in extremophilic bacteria [33] and deep-sea marine organisms [20,21,34]. This would also be of interest in understanding neuronal function. For example, the heat and pressure waves of a concussive blast [35,36] or electro-mechanical neuronal couplings affecting anesthesia [37,38].
In this study, we extend our MMRT-based approach [13] to integrate the effects of pressure [39], providing a quantitative thermodynamic description of voltage-gated membrane conductances behavior. We first develop the theory and then we evaluate the validity of its parameters through data fitting. We then use the model to understand how the effects of temperature and pressure on voltage gated conductances impact action potential generation, firing frequency, and precise timing. We conclude by exploring physical interpretations of the parameters in the model.
Theory
Macromolecular rate theory.
The value of between the ground and transition state of an ion channel is:
Changes in heat, , are related to
and
via
when
:
Integrating assuming constant [17], to get
and
and substitute in Eq 1, we get the basis of MMRT.
where , and
correspond to a reference temperature To.
The rate coefficient function is based on the Eyring equation, Eq 4.
where kB and h, are Boltzmann’s and Planck’s constants, respectively, and R is the universal gas constant. The value of Topt is where [13,16,40].
Incorporating the effects of pressure.
The effect of pressure, P, on a reaction rate is mediated by the activation volume, [41,41–44]:
Several studies suggest a positive change in for ion channel opening conformations [29–32,45–48]. Based on [39] and [19,21,22]:
Entropy depends on temperature and pressure
Using Eq 2 for and the Maxwell relation
we get:
The isobaric thermal volume is with α the expansivity coefficient:
Thus,
Similarly for volume:
Using the isothermal volume compressibility, , with the compressiblity coefficient,
, we get
Assuming that Cp, , and
are temperature and pressure independent, the integrals are:
Where To, is a reference temperature with associated reference values Po, So and . We substitute in Eq 6.
We can convert this into values for the change between the ground and the transition state of the reaction. Because MMRT uses , we can make the substitution
and ultimately produce:
Where and
. Eq 15 incorporates the terms from Eq 3 and Eq 5. To obtain the kinetics, we plug Eq 15 into Eq 4.
The activation volume as a function of pressure and temperature is found in Eq 13b as:
The value of Topt is:
Materials and methods
We used our previous parametrization of MMRT for the values of and
[13]. As we did before, the values of
were adjusted so k = 1 at
C. The reference temperature and pressure were
C and Po = 1 atm, respectively.
We conducted a literature search for experimental data on the effects of pressure on the kinetics of voltage-gated channels[29–32,45,47–49]. We used the results from these papers to determine the other parameters, see Table 1.
The values for ,
, and
were obtained from our previous publication [13].
Studies on the squid’s giant axon [30–32] reported values of , Table 1. We assumed a linear relation between reference activation volume and temperature, which resulted in
at To. The values for
and
were very close to each other [30,31], so we used their average for the simulations.
From the values of and
at different temperatures and using Eq 16 we calculated an average value of
, which is consistent with an earlier report [47]. The change in α across an increase in temperature was found to be negative [50,51] but positive for unfolding [51]. So, we might also estimate
at a lower
. Therefore, we estimate a high magnitude value of
and a low value of 10−2.
From approximate
. Since the scale of
is 102, and assuming both terms are approximately equal or the first dominates, the scale of
. This is near reported values of α in the
range [50–53] and will be fine for analyzing the impact of this parameter.
Dreydroppel et al. [39] provides a value for of 1.8 GPa−1 with
of
. Others reported compressibilities on the scale of
[50,52,54]. Assuming the same
as in [39] suggests a high value of
.
Rapid changes in pressure could result in increases in temperature, known as adiabatic heating. Based on previous reports [31,46,47,49] we assume an adiabatic heating of per
(
).
We tested the thermodynamic effects of pressure and temperature on membrane conductance gating on neuronal models of action potential generation. First, we used a Hodgkin-Huxley system of equations. We multiplied the reaction rate of each conductance by the rate coefficient k normalized by modifying to the control experimental condition of
, Eq 4. In another set of simulations, we chose four models from the Allen Brain Cell Types database. We used an identical approach as in our previous publication [13]. All simulation files, analysis scripts, and data are available in github.com/SantamariaLab or by request. They are also in ModelDB data base Model Number 2019887.
Results
Pressure effects on rate coefficient function
To gain intuition on how pressure affects the reaction rate coefficient of membrane conductances we plotted k (Eq 4 with Eq 15) as a function of temperature and pressure. We used our previously calculated averaged temperature thermodynamic parameters for Na conductances (,
,
) and our estimated value of
. For the expansivity and compressibility we used
, and
because this combination resulted in decreasing values of the reaction rate as a function of pressure, consistent with experimental reports, see Table 1 and Methods. We plotted the value of k at three representative pressures (atmospheric pressure, 1atm; average ocean depth,
; and bottom of the Mariana Trench,
), Fig 1A. With this combination of parameters the value of Topt varied over a small range, from
at 1 atm to
at 1,000 atm, Fig 1B.
(A) Reaction rate coefficient vs temperature at 1 atm (), average ocean depth
atm
, and Mariana trench
atm
. (B) The rate coefficient as a function of pressure and temperature. Parameters were:
,
,
, and
,
and
.
We wanted to calculate the value of , based on electrophysiology recording and compare to our estimate. To do this we used the model to fit values of k extracted from experiments on different neurons and conductances [30–32,48,55], Fig 2. Depending on the source of experiments, we used the average temperature parameters we previously calculated for Na and K conductances. As we did in the past, we also fit the value of
because this varies as a function of the experimental temperature and does not affect the rate of change of the MMRT function. In all fits we assumed
and
. The fits had a mean R2 of
. This analysis shows very accurate values of both parameters. The average value of
was −133.97 ± 3.88 SD
, with an average 95% confidence interval of
SD
. The values of
were very close to those that we reported for potassium channels in our previous study. For
the average value was 40.93 ± 11.60 SD
and a 95% confidence interval of 9.20 ± 6.47, which are in the range of values reported of Na and K conductances, see Table 1.
Adiabatic heating effects on pressure measurements
We studied the effects of incorporating adiabatic heating in the model, Fig 3. We plotted four isotherms of the rate coefficient function using and
. We applied a one-degree increase for every
(
) from rapid pressure change. Depending on the starting temperature, adiabatic heating temperature change can have a significant impact on the shape of k. The effect of rapid heating is minimal when the pressure change is at Topt. At suboptimal temperatures, the heating increases rate whereas at supraoptimal temperatures heating is adverse. This behavior reflects the temperature optimum over rate and is a significant quantitative result that should be considered for rapid or transient pressure changes.
The pressure dependence of rate is shown at multiple temperatures. Because increasing pressure can cause a temperature increase, we show that effect with the dashed line. The optimal temperature with the parameters used was and the rates are referenced at
.
Sensitivity to expansivity, compressibility, and activation volume
In the previous sections we first calculated the value of to avoid over-parametrization and the numerical effects on the fitting procedure of parameters with large differences in their orders of magnitude and quadratic effects of temperature and pressure. Here we perform a sensitivity analysis of the model by varying the values of
and
and
, Fig 4.
(A) The optimal temperature (Topt), and rate coefficient at Topt, and at when using the combination of the extreme values of
and
.(B-D) Sensitivity of the model to individual variations of pressure parameters. The reference model is plotted with a dash blue line in each panel. The range of values were:
from –1 to 0 in
increments;
from 100 to 500 in
increments; and
from 5 to 70 in
increments. The range of parameters is plotted from low to high value as green to cyan. We used as a reference the model with:
,
,
,
,
,
,
and
.
We first studied how the model behaved when using the extreme values of and
, Fig 4A. We plotted the value of Topt, k at Topt, and at an experimental temperature, which we selected to be
, all as a function of pressure. When using the low value of
we obtained a linear relationship between Topt with pressure independently of the value of
. When using
and
, which are found in soluble proteins [39,50–54], there is a minimum effect of pressure on Topt. In contrast, when using the high value of
there is a non-linear behavior of Topt. The analysis of the rate coefficient at Topt or at
suggest that there are interactions between the values
and
that result in decreasing reaction coefficient behavior as a function of pressure.
To estimate the relative effect of varying the pressure parameters we took as a reference our model parametrized with ,
and
, Fig 4B–4D. This shows that the behavior of Topt is highly sensitive to the values of
. The behavior of the rate coefficient is sensitive at Topt but is less at our designated experimental temperature. In contrast, the behavior of Topt is not sensitive to the values of
but could have a strong effect at experimental temperatures on the value of the rate coefficient. A similar effect is seen with the values of
. Taken together, this analysis provides a methodology to distinguish between the effects of each of these parameters on how the reaction coefficient function is affected by pressure.
Effects of temperature and pressure on action potential generation and timing
We performed an analysis of the spiking and firing rate of the Hodgkin-Huxley equations under different temperature and pressure conditions, Fig 5. Increasing pressure resulted in a broadening of the action potential and a lengthening of the inter-spike interval. However, at higher temperatures, pressure had a stabilizing effect on the shape of the action potential. In all cases, we used the and
, Fig 5A. The summary data, Fig 5B, shows a continuous decrease in firing rate due to pressure (top), a temperature dependence that peaks at
, followed by a failure to generate action potentials past
, note that we required a minimum amplitude of 20 mV to detect an action potential) (center). We also see a similar behavior of firing rate as a function of input current (bottom).
(A) left - single action potentials generated rheobase for different pressures and temperatures. Right - Spike trains under the same conditions. (B) Firing rate vs pressure, temperature, and input current. For each panel the pressure and temperature panels the input current was . For the bottom panel the temperature was
.
The plots in Fig 5 suggest a weak pressure effect on action potential generation and average firing rate. However, in those plots, we noticed an effect on spike timing. To study the possibility that pressure could affect spike timing, but not firing rate, we performed a series of simulations in which the Hodgkin-Huxley model was stimulated with random current plus a constant component. We selected the random amplitude and DC offset to generate variable spike trains, Fig 6A. We then used the same random sequence to stimulate an identical model while varying only the pressure. We decided to study lower pressures, including one in the range of intracranial values, 0.02 atm, [35]. These simulations showed that even at very low pressures, the spiking activity could be different from the control simulation, see 1.02 vs 1.00 atm in Fig 6A. As the pressure increased, the spike trains became more different. However, the average firing rate of the entire simulation, 15sec, remained basically the same, Fig 6B. To evaluate changes in spiking activity, we calculated the difference in spike time from the control simulation. This pairwise calculation could be constant, corresponding to a shift in the spiking activity. However, the difference in spike time showed variability that seemed to correlate with pressure, Fig 6C. Indeed, when we calculated the standard deviation of the spike differences, there was a pressure effect. These results could be because the simulation at the higher pressure could be slowing down with respect to the control simulation with a dependence on the random noise. However, these differences remained even after averaging multiple simulations (10) using different random number sequences, Fig 6D. In order to test further the idea that the spike trains became decorrelated, we calculated the correlation coefficient of the instantaneous inter-spike interval sequences. This shows that even at the lowest pressure, the spike trains had a low and non-statistically significant correlation coefficient. We binned the ISI sequences in chunks of 10 to test if this correlation could become significant by averaging the noise. Even with this filter, there was only a significant correlation at 1.02 and 2.00 atm, Fig 6E. Together, our results suggest that small pressure changes can affect precise spike timing and correlation of spike trains across neurons.
(A) Examples of spike traces of the Hodgkin-Huxley model at different pressure receiving identical sequences of input random currents. For each value of pressure we repeated the simulations 10 times with different input current random sequences. (B) Average firing rate vs pressure. Error bars are for the standard deviation calculated on the 10 different runs. (C) Examples of spike time differences for simulations that had the same random input sequences of stimulation but different pressures (colors correspond to pressures in A). (D) Standard deviation of the spike time differences vs pressure. (E) Average correlation coefficients of the inter-spike intervals (ISI) with respect to the simulation at 1 atm (black). We recalculated the correlation coefficients after averaging 10 ISIs, showing two pressures in which the correlations were statistically significant.
Finally, we applied the extended theory to biophysical models of human cortical pyramidal cells, see Methods. We analyzed changes in spiking over pressure ranges in blast conditions (10 atm), Fig 7. At these relatively low values of pressure changes the effects were notable in pace-making type neurons. In these and the Hodgkin-Huxley simulation we assumed steady-state temperature and pressure, and so additional temperature change from adiabatic heating was not considered. Overall, our results show that while temperature generally increases the firing rate up to Topt, or up to the point of over-saturation, pressure inhibits or delays action potential generation. While these effects could be mild at the single-cell level, both burst desynchronization between multiple neurons and error in very precise temporal codes could arise from pressure effects.
Rows: four different cortical models. Columns: Three different temperatures. Each model was run with normal and blast-type pressure (10 atm). The models were obtained from the Cell Types database from the Allen Institute, see text for details.
Discussion
In this work, we combined concepts from MMRT and transition state theory to integrate the effects of temperature, volume, and pressure on the activation energy of voltage-gated membrane conductances. While the values of ,
, and
are well understood, the extended theory uses other variables and parameters that require further understanding.
The activation volume of voltage-gated ion channels
The length of a voltage-gated ion channel (VGIC) is around 45 Å [56–58]. The VGICs are roughly cylindrical [56,58,59] with diameters around 10 Å [56,58,60]. During opening or closing the volume of a pore may change [61–64]. The physical process of activating a channel requires dewetting that can happen by a 1-2 Å decrease in pore radius [61–63]. This simple cylindrical model is an approximation, where real channels often undergo multiple complex conformational changes which all may contribute to . Nevertheless, to gain intuition on the physical meaning of
we will assume a VGIC of height 50 Å and diameter 10 Å. We will also assume that the channel is described by a two-state process, open and closed, Fig 8. Assuming a 1 Å radial decrease when changing states, we can calculate a physical volume difference of
which is on the same scale as values found in multiple experiments [29–32,45–47]. This similarity between geometrically and experimentally determined values of activation volume lead us to propose that these properties could be physically modeled based on protein structure, instead of fitting and estimating them, and could be an interesting future direction of research.
In the closed state, the radius shrinks by 1 Å which allows the fully hydrophobic part of the pore to dewet. The blue represents regions occupied by water molecules.
Expansivity, compressibility, and further development
Just as allows suggestions of underlying mechanisms such as pore constriction, expansivity and compressibility may offer insights into the greater complexity behind pressure and volume effects on ion channels. For instance, compressibility is relevant to aromatic ring flip conformations of proteins [39]. Positive transition compressibility suggests that the ion channel volume is more susceptible to pressure when open. Expansivity is an important parameter as it sets the temperature dependence of activation volume. If transition expansivity is positive, one would suspect channel volume to be less partial to temperature when closed. Intuitively, this seems to contradict the concept of a closed, evacuated pore as liquid water in the open state should be less expansive. That interpretation makes negative transition values seem more plausible.
Conceptually, is the difference in the partial temperature derivative of volume between the transition and ground states of a reaction. The derivatives can be large, but if we take the high value of the parameter
and Eq 13b with our reported
we can see that just a
increase would set
. Negative activation volumes are possible for processes such as unfolding [43], but are likely nonphysical for gate opening. That value also implies an extreme 100% change in activation volume. The same applies for a high compressibility of
which suggests activation volume would be zero after about
of pressure which we know cannot be true from the experimental data. So, the Topt is not a significant function of pressure.
Finally, it should be noted that ,
, and
may all depend on the pressure and temperature themselves. We mentioned that
is assumed constant, but it could be given a linear temperature dependence [17], some parameter-based function of temperature and pressure [65], and it has been suggested to increase with pressure [66]. If
had a significant pressure dependence, it could also explain the apparent temperature dependence of
, rather than or in addition to
. For simplicity, we did not include adiabatic heating in our simulations. Any of these effects, if significant at a biological temperature range or at high pressures, would limit the model at its current level of detail, but not necessarily invalidate the modeling strategy. Though, outside of biological ranges, where there is protein denaturation or cell death, the model regardless of parameters would certainly no longer apply.
Conclusion
We presented thermodynamic theory that can be integrated with studies of neuronal excitability [67,68]. More broadly, a unified temperature and pressure theory can be used to compare the enzymatic kinetics of diving creatures and extremophilic bacteria in hundreds of atmospheres in the ocean [20,21,34] with land creatures or their ancestors. Our work provides a platform to study the evolution of preferred body temperatures [33] with the optimal temperature of enzymes. Pressure has mechanical consequences for neuronal function and structure [69,70] due to membrane mechanics. In the context of human health, changes in intracranial pressure (ICP) can arise from many phenomena such as intracerebral hemorrhage [71], plateau waves [72], microgravity [73], and impacts and blasts [35,36]. For example, acceleration effects can change the ICP and have been shown to have negative effects on cognitive performance [74,75]. Pressure changes caused by ICP are around 10-100 mmHg (0.05atm) which we showed have little effect on the firing rate [76], but that could affect precise spike timing. It remains to evaluate in a similar way if the small pressure changes from action potential propagation [9] could have a significant effect. In any case, our work suggests that pressure affects precise spike timing and we suggest that cumulative effects could modify network dynamics and performance. The mechanisms we describe could combine with other cellular communication mechanisms important for network activity, such as the effects of pressure on synaptic release [47]. Other areas to consider are how membrane thermodynamics affect anesthesia [37,38] and how membrane changes could alter channel function [7]. While in this study we focused on developing theory for applications on neuronal intrinsic excitability driven by voltage activated ion channels there could be other emergent network-level effects or intracellular metabolic pathways that together could enrich the thermodynamical effects on neuronal and network function. As such, our work promotes the return to experimentation and discussion of pressure, especially since pressure experiments can reveal important biological properties [77].
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