Temperature-regulated circadian clock model without spatial heterogeneity

We first considered a temperature-regulated circadian clock model without spatial heterogeneity. This formulation couples the temperature signaling cascade to the core circadian clock gene network and provides the common modeling framework used in both the homogeneous and compartmentalized settings. In the absence of regional subdivision, the model can also serve as an effective description of peripheral circadian clocks. The corresponding dynamics are described by a system of coupled ordinary differential equations for temperature-responsive signaling and clock gene regulation.

In the framework, we first specify the temperature-responsive signaling pathway that drives thermal input to the circadian clock (Fig 1A). Cellular responses to sudden temperature fluctuations can be mediated by HSPs. In unstressed cells, HSPs bind HSF1, forming an inactive HSF1:HSP complex. Upon exposure to heat stress, denatured proteins accumulate, prompting HSPs to dissociate from HSF1 and bind these misfolded proteins instead. The released HSF1 then trimerizes into its active form and translocates to the nucleus, where it binds to heat shock elements (HSEs) and activates transcription of heat shock genes [33]. As levels of HSPs rise, they in turn facilitate the conversion of active HSF1 trimers back to inactive monomers, decreasing transcriptional activity and returning HSF1 to its basal distribution. This negative feedback loop allows HSF1, under the regulation of HSPs, to sense and respond to cellular stress, thereby maintaining proteostasis. Meanwhile, once activated HSF1 trimers are formed during this process, they can directly promote the transcription of Per2 mRNA by binding to HSEs within the Per2 gene. Also, HSF1 can interact with the BMAL1:CLOCK complex, thereby attenuating E-box-mediated transcription [20]. Together, these pathways enable the circadian clock to sense and respond to temperature signals. To describe the temperature-dependent heat shock response, we adopted a mathematical model developed by Szymańska and Zylicz for heat shock protein synthesis under varying thermal conditions [33]. We then coupled this module to an established circadian clock model [34] by modifying the Per2 mRNA equation. In particular, we added an HSF1-trimer-dependent activation term to represent HSF1-mediated induction of Per2 transcription. Therefore, the dynamics of Per2 mRNA included the basal transcription, BMAL1:CLOCK-dependent transcriptional activation, repression by PER:CRY complexes, HSF1-trimer-mediated induction, and first-order degradation. To simplify notation, all activating and inhibitory regulatory terms were written using common Hill functions and , respectively. The resulting equation is

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Schematic illustration of the two model architectures.

A: Homogeneous temperature-regulated clock model. A single compartment model integrates the heat shock signaling cascade with the core circadian clock network, representing a spatially uniform peripheral oscillator entrained by thermal input. B: Compartmentalized SCN clock model. The SCN consists of ventral and dorsal compartments, each with a local clock network. The two regions are connected by peptide signaling: ventral VIP regulates both regions, and dorsal AVP acts locally. These signals modulate Per expression and interregional phase relationships, allowing analysis of how spatial organization influences SCN thermal responses.

https://doi.org/10.1371/journal.pcbi.1014831.g001

where .

Here, denotes the BMAL1:CLOCK complex, denotes HSF1 trimers, and , , , and denote the inhibitory PER:CRY complexes PER1:CRY1, PER1:CRY2, PER2:CRY1, and PER2:CRY2, respectively.

HSF1 can also regulate the circadian clock through interaction with the BMAL1:CLOCK complex [20]. To represent this process, we introduced a new variable, , denoting the complex formed between HSF1 and BMAL1:CLOCK.

The HSF1 and BMAL1:CLOCK equations were modified accordingly to include the formation and dissociation of the HBC complex. Specifically, the terms and represent the reversible association and dissociation of the BMAL1:CLOCK:HSF1 complex. All other terms in these equations describe synthesis, degradation, chaperone interactions, and oligomerization dynamics inherited from the original model [33].

The effect of temperature was incorporated into the heat shock signaling module using a temperature-dependent function that has been approximated by Peper et al. [35]. Here, T denotes the external temperature. This function was used to modulate HSF1 activation in response to temperature changes.

In the present model, the newly introduced HSF1–clock coupling parameters were treated as effective phenomenological terms, since these interaction strengths are difficult to estimate directly from currently available experimental data. In this formulation, heat stimulation regulates Per2 expression through two competing routes: direct activation by HSF1 trimers and indirect attenuation of BMAL1:CLOCK-dependent transcription through formation of the HSF1:BMAL1:CLOCK complex. Because experimental studies show a rapid increase in Per2 expression following heat stimulation, the reference parameter set was specified so that the direct HSF1-trimer-mediated activation dominates the indirect inhibitory effect of BMAL1:CLOCK sequestration. The experimentally observed acute induction of Per2 served as the main biological constraint in selecting these heat-response coupling parameters. To further assess the robustness of this parameter selection, all newly introduced parameters were evaluated jointly, as described below and in Section B of S1 Text.

Except for the additional coupling terms described above, the remaining equations follow the original circadian clock model [34] and the HSP:HSF1 interaction framework [33]. The complete equations and parameter values are given in Section A of S1 Text. Representative autonomous oscillations of the peripheral clock model at a constant 37°C are shown in S1 Fig, including the mRNA time courses of the core clock genes Bmal1, Per1, Per2, Cry1, Cry2, and Rev-erb.

A compartmentalized model of the SCN’s circadian clock with temperature regulation

We next extended the framework described above to a compartmentalized model of the SCN, in which the ventral and dorsal regions are represented as distinct but coupled compartments. Each compartment contains the same core circadian clock network and a local temperature-responsive signaling pathway, allowing thermal input to regulate the ventral and dorsal clocks separately while preserving explicit regional phase differences. Interregional communication was introduced through peptide-mediated signaling. Specifically, the ventral SCN predominantly expresses VIP, whereas the dorsal SCN is characterized by AVP expression [10]. In the model, VIP release is regulated by PER:CRY complexes in the ventral compartment, while AVP release is regulated by PER:CRY complexes in the dorsal compartment [36,37]. AVP promotes Per1 and Per2 transcription only in the dorsal compartment, whereas VIP promotes Per1 and Per2 transcription in both ventral and dorsal compartments through CREB-mediated activation at CRE sites [10,38]. Thus, the coupling is asymmetric: VIP, produced in the ventral compartment, acts on both ventral and dorsal clocks, whereas AVP, produced in the dorsal compartment, acts only within the dorsal clock (Fig 1B). To represent these interactions, we introduced two variables corresponding to VIP and AVP concentrations. The dynamics of VIP and AVP are given by the following equations.

The resulting peptide signals were then incorporated into the Per1 and Per2 equations of the ventral and dorsal compartments to represent peptide-mediated coupling between SCN subregions. Using the activating and inhibitory Hill functions defined above, the clock gene equations can be written in the following generic form for , where V stands for ventral SCN and D for dorsal SCN:

Here, and denote the peptide-mediated coupling terms. In the ventral compartment, only VIP-dependent regulation is included, whereas in the dorsal compartment both VIP and AVP-dependent regulation are present. Therefore,

and have a similar form.

The VIP- and AVP-related coupling parameters in the compartmentalized SCN model were also introduced as effective phenomenological terms to represent neuropeptidergic regulation at the model level, rather than as directly measured kinetic constants. In this formulation, VIP provides the dominant coupling signal from the ventral compartment to the SCN network, whereas AVP primarily captures local regulation within the dorsal compartment. Under a constant temperature of 37°C, the model produces a 23.6-h autonomous oscillation and an 8.6-hour phase difference between the ventral and dorsal circadian oscillators, consistent with experimental observations showing that PER2 peak fluorescence occurs earlier in the dorsal SCN than in the ventral SCN [39]. This phase offset is used to represent the intrinsic regional timing difference within the SCN. To compare simulations with experimental measurements, which typically report clock gene expression at the level of the global SCN, we defined the global SCN signal as a weighted average of the ventral and dorsal oscillations. Accordingly, the global SCN signal was taken as one third of the ventral expression plus two thirds of the dorsal expression, based on the approximate 2:1 ratio of AVP to VIP neurons [10,40]. S2 Fig shows representative autonomous dynamics of the compartmentalized SCN model at a constant 37°C, including the mRNA time courses of the core clock genes Bmal1, Per1, Per2, Cry1, Cry2, and Rev-erb. Additional details of the compartmentalized SCN model, including the full equations, parameter values, and initial conditions, are provided in Section A of S1 Text. To examine the sensitivity of the model behavior to the newly introduced coupling parameters, we performed a Latin hypercube sampling analysis over all such parameters, including those governing both HSF1–clock coupling and VIP/AVP-mediated coupling (Section B of S1 Text). Specifically, 10,000 parameter sets were generated by varying each parameter within of its nominal value.

The model captures distinct thermal responses in peripheral clocks and the SCN

We first tested whether the model could recapitulate the experimentally observed differences in thermal responses between peripheral clocks and the SCN [21]. We began by evaluating the response of the homogeneous model, which serves as an effective representation of peripheral clocks, to acute thermal stimulation through comparison with experimental observations [41]. Consistent with experimental measurements [34], the peak of simulated Per2 expression was assigned to CT9. We then simulated a 2-hour heat pulse at CT4, during which the temperature was elevated from 37°C to 41°C (gray shading), as shown in Fig 2A. The heat pulse induces a rapid increase in HSF1, which affects Per2 expression through two opposing pathways. On the one hand, the model includes an inhibitory effect of HBC formation on BMAL1:CLOCK-mediated E-box activity, which counteracts Per2 transcription. On the other hand, elevated HSF1 trimer levels (red) enhance HSE-mediated activation of the Per2 gene and promote its transcription. The net effect of these competing processes is an acute increase in Per2 mRNA (dark blue). Thus, the model captures the transient induction of Per2 in response to temperature elevation. We then compared the magnitude of simulated Per2 induction with the experimental values reported by Ohnishi et al. [41] to assess whether the model response is quantitatively reasonable. Experimentally, mice were exposed to a 2-hour heat treatment at 41°C starting at CT4. Per2 mRNA levels in kidney, liver, and submandibular gland were measured relative to the 37°C control condition, which was normalized to 1, allowing the percentage increase induced by heat to be compared across tissues. Under this normalization, the magnitude of acute Per2 induction in the model falls within the experimentally observed range (Fig 2B), supporting the quantitative reasonableness of the short-term thermal response of peripheral clocks.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Model and experimental validation of acute and cyclic temperature responses in peripheral clocks.

A: Model simulation of a 2-hour heat pulse (41°C, shaded gray) applied at CT4, showing Per2 mRNA (solid line), Per2 mRNA without heat pulse (dashed line), HSF1 trimers, and HSF1. The heat pulse at CT4 induces a marked acute increase in Per2 mRNA compared to the no-pulse condition. B: Quantitative comparison between experimental values reported by Ohnishi et al. [41] for different peripheral tissues and the simulated fold-change in Per2 mRNA expression following a 2-hour temperature elevation (41°C vs. 37°C). C: Entrainment of peripheral clock to opposing (inverted) temperature cycles. The top shows simulated Per2 oscillations synchronized to antiphase temperature cycles (36°C/38.5°C). The bottom shows an author-generated heatmap visualization of the experimental data reported by Buhr et al. [21]. D: Polar plot of the simulated Per2 peak phases on days 2, 3, and 4 after entrainment under opposite temperature cycles, shown as three concentric circles from inner to outer. Each dot marks a Per2 peak phase; blue and red dots correspond to the peak phases of the blue and red curves in Fig 2C, respectively.

https://doi.org/10.1371/journal.pcbi.1014831.g002

Having established that the homogeneous model captures the acute response of peripheral clocks to heat pulses, we next examined whether it could also reproduce entrainment by temperature cycles. Ethan D. Buhr et al. applied the same temperature cycle protocol separately to pituitary and lung explant cultures [21]. In each tissue type, cultures were first synchronized under temperature cycles with the same phase, after which one group was switched to a fully inverted 12-hour temperature cycle between 36°C and 38.5°C while the other remained on the original schedule. After 4 days, Per2 bioluminescence in the two groups became nearly 180° out of phase in both tissues. For comparison with the model output, the experimental data reported by Buhr et al. [21] were independently visualized in heatmap format. The resulting heatmaps, shown in the bottom of Fig 2C, reveal clear antiphase expression under the two temperature schedules. We then performed corresponding simulations using identical initial conditions and imposed opposite 4-day temperature cycles through the temperature input function I. After 4 days of entrainment, the simulated Per2 oscillations also became entrained to the respective thermal schedules and ultimately evolved into opposite phases (blue and red curves in Fig 2C). Fig 2D further summarizes the simulated Per2 peak phases after entrainment under opposite temperature cycles. The two entrained states were approximately anti-phasic, consistent with experimental observations in lung and pituitary cultures exposed to inverted schedules [21]. This agreement indicates that the homogeneous model captures the main qualitative outcome of temperature-driven entrainment in peripheral clocks.

Using the same protocol, we next applied a 4-day inverted temperature cycle to the compartmentalized SCN model. In contrast to the homogeneous model, the SCN exhibited only minor phase modulation under opposite temperature schedules and failed to develop antiphase Per2 rhythms. As shown in Fig 3A, the phase difference between the two conditions remained far smaller than the near 180° separation observed in peripheral clocks. This result is consistent with experimental evidence that the intact SCN is resistant to temperature entrainment [21]; the lower panel of Fig 3A shows the corresponding experimental data reported by Buhr et al. [21], independently visualized here in heatmap format. A natural question is whether this resistance results from weak thermal responsiveness of the SCN. To test this possibility, we examined the model response to a single heat pulse at 38.5°C. As shown in Fig 3B, the global SCN Per2 level still exhibited a marked acute increase following thermal stimulation. Thus, the failure of the SCN to undergo phase inversion under temperature cycles cannot be explained by an absence of acute heat responsiveness.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Responses of the SCN model to temperature cycles and heat pulse.

A: Per2 rhythms (blue and red lines) do not exhibit phase reversal in response to opposite temperature schedules. The lower heatmaps are author-generated visualizations of the experimental data reported by Buhr et al. [21]. B: Per2 expression in the SCN model in response to a 2-hour heat pulse (38.5°C, gray shaded area) at CT12 after 12-hour low temperature (36°C). The solid and dashed lines represent the control and heat pulse conditions, respectively.

https://doi.org/10.1371/journal.pcbi.1014831.g003

These results establish that the model captures the key experimental contrast between peripheral clocks and the intact SCN under thermal stimulation, providing the basis for subsequent analysis of the mechanism underlying SCN temperature resistance.

Regional phase differences underlie SCN resistance to thermal entrainment

Because the temperature signaling cascade and the core circadian regulatory network are conserved between the SCN and peripheral tissues, the difference in thermal entrainment is likely to arise from how these signals are integrated within the SCN network. We therefore turned to the spatial organization of the SCN to investigate how internal phase heterogeneity shapes the collective response to thermal input. To dissect this mechanism, we first considered the response of the SCN to a single heat pulse.

The SCN Per2 phase response was quantified as the shift in the first trough occurring 12 hours after the end of the heat pulse. When a 2-hour temperature pulse at T = 39.1°C was applied at CT6, the global SCN Per2 exhibited a delay of approximately 1 hour (upper panel, Fig 4A). Because the global SCN signal reflects the combined activity of ventral and dorsal SCN subregions, we next analyzed the responses of these two compartments separately. As shown in the lower panel, the ventral SCN exhibited an approximately 4-hour phase advance in Per2 (red curve), whereas the dorsal SCN showed an approximately 2.4-hour phase delay (blue curve). Thus, although each subregion displayed a relatively large phase shift in response to the same heat pulse, the two shifts occurred in opposite directions. Since the global SCN Per2 signal is a weighted integration of ventral and dorsal oscillations, these opposing responses partially cancel at the tissue level, resulting in only a modest net delay of the overall SCN. A similar pattern was observed when the heat pulse was applied at CT18. In this case, the global SCN rhythm exhibited a net phase delay of only approximately 0.3 hours (Fig 4B).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Phase response of Per2 in the SCN model to temperature pulses.

A–C: Per2 expression in the SCN model following a 2-hour temperature pulse (39.1°C, gray shaded) applied at CT6 (A), CT18 (B), or CT14 (C). Upper: Overall SCN Per2 with and without pulse. Lower: Ventral and dorsal SCN compartments under pulse (solid) and control (dashed). Arrows indicate phase shift measurement. D: Phase response curves (PRCs) for ventral, dorsal, and overall SCN to a 2-hour temperature pulse (39.1°C) at different circadian times. Black arrows mark pulse application times at CT6, CT14, and CT18.

https://doi.org/10.1371/journal.pcbi.1014831.g004

At the subregional level, however, the ventral SCN showed a phase delay of about 3.3 hours, whereas the dorsal SCN exhibited a phase advance of about 1.6 hours. Thus, as at CT6, the two compartments responded in opposite directions, and their effects partially canceled in the global SCN signal. Therefore, the responses at CT6 and CT18 illustrate a compensatory mechanism that stabilizes the SCN phase against thermal perturbation. At both stimulation times, the same heat pulse drives opposite phase shifts in the ventral and dorsal SCN, with one region advancing while the other delays. Because the global SCN rhythm integrates signals from both compartments, these opposing responses reduce the net phase shift of the intact SCN.

Although opposing phase responses in the two subregions do not occur at all stimulation times, attenuation of the global SCN phase response remains a general feature across circadian time. For example, when a temperature pulse was applied at CT14 (Fig 4C), both the ventral and dorsal SCN exhibited phase delays of approximately 2.1 and 1.2 hours, respectively, rather than shifting in opposite directions. The global SCN Per2 therefore showed a net phase delay of about 1.5 hours. However, over the full 24-hour cycle, comparison of the ventral, dorsal, and global SCN Per2 PRCs shows that the global SCN typically exhibits attenuated phase shifts relative to the subregional responses (Fig 4D). This indicates that interregional compensation dampens net SCN phase resetting even when the two subregions do not respond in strictly opposite directions, thereby contributing to the thermal robustness of the intact SCN.

Since the global SCN signal is constructed as a weighted average of the ventral and dorsal signals, the effect of the assumed ventral:dorsal weighting ratio on this attenuation was examined next. We tested ventral:dorsal weighting ratios of 1:3, 1:1, and 3:2 around the biologically motivated baseline. The resulting global SCN phase shifts were then compared with the corresponding phase shifts of the dorsal and ventral SCN subregions. Across all tested weighting ratios, the above conclusion remained unchanged: the phase shift of the combined global SCN signal lay between the individual dorsal and ventral phase shifts, consistent with a reduced net phase response arising from interregional phase heterogeneity (S3A Fig). Thus, the proposed mechanism does not depend on the specific baseline averaging assumption, although the weighting ratio does influence the extent of this reduction in the global phase response to some degree. As shown in S3A Fig, when the relative contribution of the ventral SCN subregion increases, the phase-cancellation effect becomes more pronounced. S3B and S3C Fig support this trend more directly at two representative stimulus times, CT8 and CT20, where the dorsal and ventral SCN subregions shift in opposite directions. At CT8, increasing the relative ventral contribution progressively reduced the global phase delay, whereas at CT20 it progressively reduced the global phase advance. Thus, despite quantitative differences, the qualitative inference that interregional phase heterogeneity reduces the net global phase shift remains unchanged across weighting ratios.

Having established that the effect was not specific to the weighting ratio, we next asked whether the same qualitative result remained robust to perturbations of the newly introduced coupling parameters. Using Latin hypercube sampling, 10,000 parameter sets were generated by varying all newly introduced parameters within of their nominal values (see Section B of S1 Text). Among these, 3523 parameter sets produced stable global oscillations with a period near 23.6 h, although the interregional phase difference varied substantially, as shown in S4 Fig. In the majority of synchronized cases, the dorsal SCN led the ventral SCN by up to 11.8 h (n = 3201), with most phase differences clustered near 8.5–9 h, whereas a smaller subset showed ventral-leading phase relationships of approximately 6–11.8 h (n = 322). Even under these perturbations, the same qualitative result was observed in both lead-lag configurations: whenever a phase difference existed between the two subregions, the global SCN phase shift generally lay between the corresponding ventral and dorsal shifts across stimulus times, consistent with attenuation at the level of the integrated SCN signal (S5 Fig). The same qualitative result was reproduced under a narrower perturbation range (S6 Fig). Therefore, we conclude that the reduced global SCN phase response emerges as a robust consequence of heterogeneous subregional phase shifts rather than a byproduct of the averaging assumption or a narrowly tuned parameter set.

SCN thermal resistance decreases as the dorsal–ventral phase difference narrows

Given the attenuation of the global SCN PRC under the natural 8.6-hour phase difference between ventral and dorsal subregions, we next asked whether the magnitude of this phase difference affects the thermal robustness of the SCN. To address this, we systematically varied the phase difference between the two compartments and quantified the resulting global phase shift under thermal stimulation.

To focus specifically on the role of phase difference, we decoupled the ventral and dorsal SCN compartments and adjusted their initial conditions to produce different phase differences between the two subregions. This setup allowed us to directly evaluate how the magnitude of the dorsal–ventral phase difference affects the net phase shift of the SCN in response to heat stimulation. As shown in Fig 5A, when a 2-hour heat pulse was applied at CT6.5 under a dorsal–ventral phase difference of 1 hour, the Per2 rhythms in the ventral and dorsal SCN were delayed by approximately 2.1 and 2.8 hours, respectively. The resulting global SCN phase delay was about 2.6 hours, which is close to the shifts observed in the individual subregions. In contrast to the natural 8.6-hour phase difference, a 1-hour phase offset produces much less cancellation between regional responses. Thus, when the interregional phase difference is small, the global SCN no longer exhibits strong attenuation of the thermal phase shift. This pattern is also evident in the phase response curves. As shown in Fig 5B, the PRCs of the global SCN (yellow), ventral SCN (red), and dorsal SCN (blue) are highly similar across the tested stimulation times.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Phase response of the SCN model without VIP coupling for different dorsal–ventral phase differences.

A,B: Per2 expression and phase response after a 2-hour, 39.1°C temperature pulse at CT6.5, with a 1-hour phase difference between dorsal and ventral SCN compartments. (A) Upper: overall SCN Per2 expression with and without pulse. Bottom: Per2 in ventral and dorsal compartments. (B) Corresponding PRCs for overall SCN (yellow), ventral (red), and dorsal (blue); the black line indicates the pulse time. The net phase shift of the overall SCN is 2.6 hours. C,D: Same as (A, B), but with a 4-hour dorsal–ventral phase difference. The net phase shift of the overall SCN is 1.9 hours. E,F: Same as above, but with an 8-hour phase difference. The net phase shift of the overall SCN is 1.1 hours. Solid and dashed lines represent simulations with and without the heat pulse. Arrows indicate measurement of phase shifts.

https://doi.org/10.1371/journal.pcbi.1014831.g005

When the phase difference between ventral and dorsal SCN subregions was increased to 4 hours, the Per2 rhythm in the ventral SCN advanced by approximately 0.8 hours, whereas the dorsal SCN delayed by approximately 2.8 hours (Fig 5C). Under this condition, the net phase delay of the global SCN was reduced to about 1.9 hours, compared with 2.6 hours when the interregional phase difference was 1 hour. This reduction was also reflected in the global SCN PRC, whose amplitude decreased as the phase difference widened (Fig 5D). A similar trend was observed as the interregional phase difference increased further. When the phase difference was further increased to 8 hours (Fig 5E and 5F), the offset between ventral and dorsal responses became even more pronounced, and the net phase shift of the global SCN decreased to only about 1.1 hours. The amplitude of the global SCN PRC was further reduced under this condition. Thus, increasing the phase difference between ventral and dorsal subregions progressively attenuated the net phase resetting of the SCN.

To quantify this relationship systematically, we varied the phase difference between ventral and dorsal SCN subregions from 0 to 8 h in 1-h increments and measured the resulting global Per2 phase shift in response to a 2-hour temperature pulse applied at CT6.5 (Fig 6A). The global phase shift decreased as the interregional phase difference increased and reached a minimum at approximately 8 hours. Notably, this minimum occurs close to the natural phase difference used in the SCN model. These results show that larger dorsal–ventral phase differences produce stronger compensation between subregional responses and thereby enhance thermal resistance of the SCN. This dependence on phase difference extends across the circadian cycle. Fig 6B shows a heatmap of the global SCN phase shift over a range of interregional phase and pulse timings. At any given stimulation time, the SCN phase shift becomes smaller as the phase difference between ventral and dorsal subregions increases. In addition, the maximum phase shift observed across all stimulation times is progressively reduced at larger phase differences. These results indicate that the phase difference between ventral and dorsal SCN subregions is a key determinant of SCN thermal resistance, with larger phase differences producing stronger attenuation of temperature-induced phase shifts.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. Phase shifts in the SCN decrease as dorsal–ventral phase difference increases.

A: Absolute value of the Per2 phase shift in the overall SCN following a 2-hour, 39.1°C temperature pulse applied at CT6.5, shown as a function of dorsal–ventral phase difference. B: Heatmap showing the absolute value of Per2 phase shifts in the SCN after a 2-hour, 39.1°C temperature pulse, plotted across a range of dorsal–ventral phase differences (0–8 hours) and pulse times (CT0–24).

https://doi.org/10.1371/journal.pcbi.1014831.g006

Consistent with this trend, we further analyzed the synchronized parameter sets obtained under the perturbation range. Cases with relatively small dorsal–ventral phase differences (0.1–4 h; n = 70) exhibited larger global SCN phase shifts than most other synchronized cases (S7 Fig), reinforcing the conclusion that when the dorsal and ventral subregions undergo more similar phase shifts, attenuation at the global level is weakened. Thus, both the decoupled phase-difference analysis and the parameter-perturbation analysis support a negative relationship between the magnitude of the dorsal–ventral phase difference and the phase-shift response to thermal stimulation.

The spatial pattern of thermal input may contribute to SCN thermal resistance

The analyses above showed that the phase difference between the dorsal and ventral SCN can cause the two regions to exhibit different phase shifts in response to the same thermal perturbation, thereby partially canceling the net phase shift of the global SCN rhythm. To further demonstrate that this effect arises primarily from interregional phase heterogeneity, we introduced a reduced two-oscillator Poincaré model. This simplified model retains only the basic dynamical ingredients relevant to the present question, including a stable phase difference between the two oscillators, interregional coupling, and external perturbation. Despite its simplicity, it reproduced the core mechanism identified above: two oscillators at different phases can respond differently to the same perturbation, thereby attenuating the net phase shift of their combined signal (S8 Fig).

Building on this result, we next asked whether the spatial pattern of stimulation also affects how the local phase responses combine into a net phase shift. In the Poincaré model, the phase response curves of the combined signal differed between dual stimulation, in which both oscillators were perturbed simultaneously, and ventral-only stimulation, in which only the ventral-like oscillator was perturbed (Fig 7A). Over approximately half of the circadian cycle, the global phase shift was smaller under dual stimulation than under ventral-only stimulation, as indicated by the shaded region. Notably, this reduced global phase shift occurred under a stronger input condition, because both oscillators received the same perturbation under dual stimulation. Instead, the shaded interval corresponded to phases at which the dorsal-like and ventral-like oscillators shifted in opposite directions in response to the same perturbation, thereby reducing the net phase shift of the combined signal. Representative examples further illustrated this relationship (Fig 7B): under dual stimulation (solid bars), the two oscillators reset in opposite directions, resulting in a reduced net phase shift of the combined signal, whereas ventral-only stimulation (hatched bars) produced a larger global phase shift. The examples include both configurations, with the dorsal-like oscillator delaying while the ventral-like oscillator advanced, and vice versa. We next examined whether the molecular compartmentalized SCN model showed the same pattern as the reduced model. Phase-shift summaries at CT4 and CT18 showed that, under dual stimulation, the dorsal and ventral regions again shifted in opposite directions, with a smaller resulting global phase shift than under ventral-only stimulation (Fig 7B, solid versus hatched bars). The phase response curves of the compartmentalized SCN model were also consistent with this interpretation (S9 Fig). In particular, the phase interval over which dual stimulation produced an attenuated global phase shift fell within the range in which the dorsal and ventral regions shifted in opposite directions, although this interval was relatively narrow. This likely reflects the simplified structure of the present compartmentalized molecular SCN model, in which the global output is represented as a weighted sum of the oscillatory signals from the two subregions.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Effect of stimulation pattern on SCN phase shift and Per2 induction.

A: Phase response curves of the combined signal in the reduced Poincaré model for ventral-only stimulation (green) and simultaneous stimulation of both oscillators (yellow). Gray shading indicates phases where simultaneous stimulation produces a smaller phase shift than ventral-only stimulation. B: Representative phase-shift decompositions are shown under four conditions, corresponding from left to right to Poincaré CT9, Poincaré CT21, molecular CT18, and molecular CT4. Within each condition, solid bars represent dual stimulation and hatched bars represent ventral-only stimulation. Yellow, blue, and red bars indicate the phase shifts of the global SCN signal, dorsal, and ventral components, respectively. C: Net Per2 phase shift after a 2-hour temperature pulse at CT4, as a function of pulse temperature, for three conditions: dual SCN stimulation (yellow), ventral-only stimulation (green), and homogeneous model (blue). D: Fold increase in Per2 expression after stimulation under the same conditions. Fold increase was calculated as the ratio of the maximum Per2 level after stimulation to the level immediately before stimulation.

https://doi.org/10.1371/journal.pcbi.1014831.g007

We next increased the thermal stimulus at CT4 and compared the responses of the molecular model under dual and ventral-only stimulation together with those of the homogeneous single-oscillator model (Fig 7C and 7D). Even under the stronger perturbation, dual stimulation still produced the smallest phase shift (yellow), whereas the homogeneous model (blue) showed a substantially larger shift. Notably, under dual stimulation, increasing the stimulus strength led to a larger acute increase in Per2 without producing a larger phase shift, which instead became slightly smaller at 41°C (Fig 7C and 7D). By contrast, under ventral-only stimulation (green) and in the homogeneous model (blue), stronger thermal input increased both the acute Per2 response and the phase shift, and the phase shifts remained larger than those under dual stimulation. The same qualitative conclusion was also obtained when the acute response was quantified using the absolute Per2 increase rather than fold induction (S10 Fig).

Thus, in the presence of interregional phase heterogeneity, dual stimulation can produce a substantial acute Per2 response while limiting the shift of the global circadian phase at certain circadian phases, and this effect can persist even when the thermal stimulus is strengthened.