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ORIGINAL ARTICLE
Mathematical model of the interaction between the dorsal
and ventral regions of the suprachiasmatic nucleus of rats
Modelo matemático da interação das regiões dorsal e
ventral do núcleo supraquiasmático de ratos
Bruno da Silva Brandão Gonçalves1, Breno Tercio Santos Carneiro1,
Crhistiane Andressa da Silva1, Diego Alexandre da Cunha Fernandes1,
Fabiano Santos Fortes1, João Miguel Gonçalves Ribeiro1, Rafaela Cobuci Cerqueira1,
Sergio Arthuro Mota Rolim2, John Fontenele Araújo1
ABSTRACT
Background and objective: In mammals, the main biological clock
that is synchronized by light is located in the suprachiasmatic nucleus
of the hypothalamus, which can be divided into two distinct regions:
the ventrolateral and the dorsomedial. Both behave as separate oscillators that interact with each other to form the circadian rhythm.
Methods: Our objective was to develop a mathematical model to
understand how these regions of the suprachiasmatic nucleus coordinate the circadian rhythm of motor activity in rats. To accomplish
this, we performed simulations with light-dark cycles of 24 (T24)
and 22 hours (T22) and simulations with constant darkness (CD). In
the model, we developed equations to describe the circadian rhythm
of a clock protein. Results: For the two light-dark and constant darkness cycles, the model was able to reproduce the synchronization with
T24, the dissociation with T22, and the free-running rhythm with
constant darkness. The results show that the intensity of coupling
between the two oscillators and their periods define the output of the
rhythm. Conclusions: The proposed model is consistent with data in
the literature and suggests new experimental approaches. This model
will contribute to a better understanding of the interaction between
the two regions of the suprachiasmatic nucleus.
Keywords: Mathematical models; Circadian rhythms; Motor activity; Suprachiasmatic nucleus; CLOCK proteins; Biological clocks/
physiology; Animal; Rats
RESUMO
Introdução e objetivos: No hipotálamo se encontra o principal relógio biológico sincronizado pela luz, que pode ser dividido em duas
regiões distintas: uma chamada ventrolateral e outra dorsomedial.
Ambas se comportam como osciladores distintos que se relacionam
para formar os ritmos circadianos. Métodos: Desenvolver um modelo
matemático para entender como essas regiões do núcleo supraquias-
mático coordenam o ritmo circadiano da atividade motora em ratos.
Para isso, foram realizadas simulações com ciclo claro-escuro de 24
(T24) e de 22 horas (T22), e em condição de escuridão constante (EE).
No modelo desenvolvido, foram utilizadas equações que descrevem o
ritmo circadiano dos níveis de uma proteína-relógio fictícia. Resultados: Para os diferentes ciclos de claro-escuro e escuro constante, o modelo foi capaz de reproduzir a sincronização ao T24, a dissociação em
T22 e o curso livre em EE. Os resultados apontaram que a intensidade
do acoplamento entre os dois osciladores e seus períodos define a saída
do ritmo. Conclusões: O modelo proposto foi capaz de reproduzir
dados da literatura e sugerir novas abordagens experimentais. Essas
novas manipulações podem contribuir para uma melhor compreensão
de como ocorre a interação entre as duas regiões do núcleo supraquiasmático.
Descritores: Modelos matemáticos; Ritmo circadiano; Atividade
motora; Núcleo supraquiasmático; Proteínas CLOCK; Relógios biológicos/fisiologia; Animais; Ratos
INTRODUCTION
In mammals, the suprachiasmatic nucleus (SCN) is responsible for generating the circadian expression of several physiological and behavioral variables such as locomotor activity,
sleep-wake cycle and body temperature (1). Electrophysiological data suggest that each cell of the SCN should be considered as self-sustained (2) and might have different periods,
albeit within a circadian limit (3). Because the SCN is necessary and sufficient to generate the circadian rhythm, it is
known as the principal circadian oscillator in mammals. The
SCN can be divided into morphologically and functionally
distinct dorsal and ventral regions (4). The dorsal, or dorsomedial (dm), region, contains a large population of neurons
Study carried out at Universidade Federal do Rio Grande do Norte – UFRN, Natal (RN), Brazil.
1
Center for Research on Rhythm, Sleep, Memory and Emotion, Department of Physiology, Biosciences Center, Universidade Federal do Rio Grande do Norte – UFRN,
Natal (RN), Brazil.
2
Center for Research on Rhythm, Sleep, Memory and Emotion, Department of Physiology, Biosciences Center, Universidade Federal do Rio Grande do Norte – UFRN,
Natal (RN), Brazil; Edmond and Lily Safra International Institute of Neuroscience of Natal, Natal (RN), Brazil.
Corresponding author: Bruno da Silva Brandão Gonçalves – Departamento de Fisiologia, Centro de Biociências, Universidade Federal do Rio Grande do Norte – Campus
Universitário, Lagoa Nova – Caixa Postal 1506 – CEP 59078-970 – Natal (RN), Brazil – Phone: (84) 3215-3409 (Ext 218); Fax: (84) 3211-9206
Received: December 12, 2009; Accepted: March 10, 2010
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Gonçalves BSB, Carneiro BTS, Silva CA, Fernandes DAC, Fortes FS, Ribeiro JMG, Cerqueira RC, Rolim SAM, Araújo JF
that produce vasopressin, while the ventral, or ventrolateral
(vl), region is mainly comprised of neurons that produce vasoactive intestinal peptide (4). Generally, the two regions are
coupled oscillators, but this condition may change when the
light-dark cycle is delayed or when the period of illumination is reduced (5).
When rats are kept in a 22-hour (11:11) symmetrical light-dark cycle, there is a stable oscillation that uncouples these regions (6). During in vivo dissociation, the
rhythm of locomotor activity has two distinct components; one component is synchronized with the light, and
the other free-runs with a period longer than 24 hours
(6)
. By analyzing the expression of clock genes (Per and
Bmal1) at specific phases, de la Iglesia et al. (7) showed
that the component synchronized with the 22-hour lightdark cycle is associated with activity of the vl region,
while the free-running rhythm depends on variations in
the dm.
In the present study, we developed a mathematical model of the SCN to study how vl and dm regions work together
to create the circadian rhythm of motor activity in rats. The
model’s equations describe the circadian rhythm of a clock
protein.
METHODS
Mathematical model
Figure 1A shows the two oscillators that form the SCN. Following the system proposed by Schwartz et al. (8), light acts
directly on the vl-SCN region, and its output is projected to
the oscillator that represents the dm-SCN region.
(A)
(B)
Figure 1: (A) Scheme proposed for the SCN. The light-dark cycle directly
affects the vl-SCN oscillator, and this output synchronizes the dm region.
(B) Mathematical equations that describe the dynamics of the vl-SCN and
dm-SCN oscillators.
To simulate circadian oscillation, we used the Goodwin
model (9) with three variables (Figure 1B). X is the concentration of mRNA of a particular clock gene, Y is the concentration of the protein produced by gene X, and Z represents
an inhibitor of X synthesis in the cell nucleus. This model
was used to describe the vl-SCN and dm-SCN oscillators,
with changes in some constants.
The constants ax,y,z and bx,y,z correspond to the production
and degradation of the X, Y, and Z variables. The constant
‘c’ modulates the output of the oscillator and influences both
vl-SCN and dm-SCN. For the vl-SCN oscillator, a constant
‘j’ was associated with light (j=0 for dark and j=0.4 for
light), and for the dm-SCN oscillator, ‘j’ was associated with
the vl-SCN output (j=c * vl-SCN output).
The oscillators were adjusted so that the period of the
v1-SCN oscillator were equal to 24.25 hours and the period
of the dm-SCN oscillator were equal to 24.4 hours, as proposed by Schwartz et al. (8). In order to do that, the constant
‘bx’ was set to 0.33 and 0.325 for vl-SCN and dm-SCN,
respectively. The other constants were set as follows: ax=ay
=az =0.7; by=bz=0.35.
To calculate the output of the oscillators, we considered
the value of the variable Y. When the value of Y remains below a threshold, the output is equal to 0. When the value
exceeds this threshold, the output has the same value as Y.
The threshold was defined as 2/3 of the maximum value of Y.
Data analysis was performed using the program “El
temps” (A. Díez-Noguera, Universitat de Barcelona, 1999).
With this tool, we discovered existing periods in the rhythm
that are consistent with the periodogram described by Sokolove and Bushell (10). Graphical representations of rhythm
(actograms) were constructed from the output of the dmSCN and vl-SCN oscillators separately.
RESULTS
The simulations were performed with a light-dark cycle
of 24 (T24) and 22 (T22) hours and in conditions of constant darkness (CD). In each condition, we used three values for the constant ‘c’. This enabled the analysis of weak
(c=0.0094), medium (c=0.15), and strong (c=0.4) interactions between the oscillators. For the weak interaction, the
value of ‘c’ was adjusted so that, in all conditions, the oscillators presented different periods. For the medium interaction, the constant was chosen so that the dm-SCN oscillator showed two periods in T22. For the strong interaction,
we chose a value of ‘c’ so that both regions presented the
same period in all cycles.
T24
For all types of interactions, the period of the vl-SCN region
was of 24 hours (Figure 2A, B, and C). For strong interSleep Sci. 2010;3(1):�����
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Mathematical model of the interaction between the dorsal and ventral regions
actions, the dm-SCN region had an average period of 24
hours (Figure 2A and B) and a phase difference of 7.4 and
7.9 hours in comparison to the vl-SCN region. The dm-SCN
oscillator had a period of 24.6 hours when the interaction
was weak (Figure 2).
(A)
(B)
(C)
(A)
(B)
(C)
Figure 3: The vl-SCN oscillator (gray) and dm-SCN oscillator (black)
were submitted to constant darkness. Under this condition, strong (A),
medium (B) and weak (C) interactions between the oscillators were simulated.
(A)
Figure 2: Simulation of the output of the vl-SCN (gray) and dm-SCN
(black) oscillators with a photoperiod of 24 hours. The simulations were
made with strong (A), medium (B) and weak (C) interactions between
the oscillators.
(B)
Constant darkness
Only the weak interaction between the vl-SCN and dm-SCN
oscillator caused their periods to be uncoupled, at 24.33 and
24.5 hours, respectively (Figure 3C). For the medium and
strong interactions, the oscillators have the same period,
24.33 hours (Figure 3A and B). The lag of the oscillators
was of 7.2 hours for strong interactions and 7.5 hours for
medium interactions.
T22
As with the T24 cycle, the vl-SCN oscillator had the same
period as the external stimulus, in this case, 22 hours (Figure 4A, B and C). When the interaction between regions
is strong, the dm-SCN and vl-SCN regions have the same
period and a phase difference of 7.6 hours (Figure 4A). Two
periods (22 and 24.67 hours) were found in the dm-SCN osSleep Sci. 2010;3(1):�����
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(C)
Figure 4: Vl-SCN (gray) and dm-SCN (black) oscillators during a photoperiod of 22 hours. The simulations were made with strong (A), medium
(B) and weak (C) interactions between the oscillators.
Gonçalves BSB, Carneiro BTS, Silva CA, Fernandes DAC, Fortes FS, Ribeiro JMG, Cerqueira RC, Rolim SAM, Araújo JF
cillator with a medium interaction (Figure 4B). The dm-SCN
oscillator had a period of 24.75 hours when there was a weak
interaction (Figure 4C).
We reduced the value of the constant ax to 0.5 to study
how t (the endogenous period) interferes with dissociation.
After this reduction, the vl-SCN and dm-SCN oscillators
had periods of 23.5 and 23.8 hours, respectively. We then
reduced ax to 0.5 for the T22 cycle for a medium interaction
between the regions. In this case, both oscillators showed the
same period as the external stimulus, 22 hours (Figure 5A).
When the period of simulation was changed to 20 hours, the
dm oscillator presented a free-running rhythm.
(A)
(B)
Figure 5: Variation of t in the vl (in gray) and dm (black) regions. For a
22 hours photoperiod (A), the oscillators have the same period of stimulation. By submitting the oscillator to a 20-hour photoperiod (B), the dm
region enters a free-running rhythm while the vl region adjusts to the
new stimulus.
DISCUSSION
In this work, we simulated the two main oscillators that
make up the SCN. The first oscillator received stimulus
from the light-dark cycle, and the second had as its input
the output of the first oscillator. Anatomically, the vl region,
represented by the first oscillator, receives a dense projection
from the retina (11) (Figure 1A). Thus, we assumed that the
intensity of incoming light synchronizes the vl oscillator in
all of the simulated cycles.
The connection between the two oscillators caused us to
consider data showing a large synaptic projection from the
vl-SCN region to the dm-SCN region with little evidence of
reciprocity (12). In our model, the dm-SCN oscillator receives
the output of the vl-SCN oscillator as input.
At the cellular level, circadian oscillation depends on
the sequential activation of clock genes (13). This complex
mechanism also depends on proteins that regulate transcription, binding, and entry of the core clock proteins (14). To
simplify these steps and maintain the behavior of molecular
oscillation, we used a model that simulates the production
of a hypothetical clock gene.
Experimental data shows that light leads to increased
transcription of genes in neurons of the SCN (15). We modified our equation from the original model, increasing the
effect of light for the vl-SCN oscillator. Synaptic activation,
which increases the production of neuronal genes (16,17), was
simulated by adding the output of the vl-SCN oscillator in
the production of the gene to the dm-SCN oscillator.
The behavior of the dm-SCN oscillator is related to the
coupling constant ‘c’. In other words, the intensity of the
output of the photo-responsive oscillator modulates the output of the dm-SCN oscillator. There is a total separation
between the regions only when the interaction is weak for
all levels of illumination. For constant darkness, Kunz et al.
(18)
showed that a reduction in interaction between the two
groups of oscillators led to results similar to ours (Figure
3C).
With a strong interaction, there is a synchronization of
the two oscillators for all photoperiods simulated. When
the intensity is increased, Granada and Herzel (19) showed
that the oscillators synchronized only with stimuli of different frequencies. Therefore, when we changed the photoperiod from T24 to T22, a greater interaction was required to
achieve synchronization between the oscillators.
Despite being composed of multiple oscillators, the vl
and dm regions show a unique period under stable conditions (20). Herzog et al. (21) showed that neurons in culture
have periods ranging from 21.5 to 26 hours (21). As the coupling between neurons increases, the variability decreases
(21)
. For this reason, we chose to represent each region as a
single oscillator.
According to simulations carried out by Schwartz et al.
(8)
, synchronization of oscillators depends on the relationship
between the photoperiod and t. By reducing the oscillators
to T22, there was synchronization with medium interaction.
A 20-hour simulation period was required to dissociate the
oscillators.
The model uses some simplifications, such as a single hypothetical clock gene and only two oscillators, but it is able
to show how the interaction between two anatomically and
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Mathematical model of the interaction between the dorsal and ventral regions
functionally distinct regions interact to render the output
of the SCN.
Although simplified, the model presented reproduces
experimental (1,6-8) and computational (5,18,19) results that are
similar to those described in the literature. Additionally,
this study suggests that the intensity of the interaction between the oscillators and their periods define the output of
the simulated rate. Thus, this work suggests new experimental approaches for the study of circadian rhythm in the
dorsomedial and ventrolateral regions of the SCN.
REFERENCES
1. Anglès-Pujolràs M, Chiesa JJ, Díez-Noguera A, Cambras T. Motor activity rhythms of forced desynchronized rats subjected to
restricted feeding. Physiol Behav. 2006;88(1-2):30-8.
2. Moore RY, Spesh JC, Leak RK. Suprachiasmatic nucleus organization. Cell Tissue Res. 2002;309(1):89-98.
3. Welsh DK, Logothetis DE, Meister M, Reppert SM. Individual
neurons dissociated from rat suprachiasmatic nucleus express independently phased circadian firing rhythms. Neuron. 1995;14(4):
697-706.
4. Diez-Noguera A. A functional model of the circadian system
based on the degree of intercommunication in a complex system.
Am J Physiol. 1994; 67(4 Pt 2):R1118-35.
5. Schwartz WJ. Circadian rhythms: a tale of two nuclei. Curr Biol.
2009;19(11):R460-2. Comment in Curr Biol. 2009/19(10)84852.
6. Campuzano A, Vilaplana J, Cambras T, Dıez-Noguera A. Dissociation of the rat motor activity rhythm under T cycles shorter
than 24 hours. Physiol Behav. 1998;63(2):171-6.
7. de la Iglesia HO, Cambras T, Schwartz WJ, Diez-Noguera A.
Forced desynchronization of dual circadian oscillators within the
rat suprachiasmatic nucleus. Curr Biol. 2004;14(9):796-800.
8. Schwartz MD, Wotus C, Liu T, Friesen WO, Borjigin J, Oda GA,
et al. Dissociation of circadian and light inhibition of melatonin
release through forced desynchronization in the rat. Proc Natl
Acad Sci USA. 2009;106(41):17540-5.
9. Goodwin BC. Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul. 1965;3:425-38.
10.Sokolove PG, Bushell WN. The chi-square periodogram: its utility for analysis of circadian rhythms. J Theor Biol. 1978;72(1):13160.
11.Morin LP, Allen CN. The circadian visual system, 2005. Brain
Res Rev. 2006;51(1):1-60.
12.Leak RK, Card JP, Moore RY. Suprachiasmatic pacemaker organization analyzed by viral transynaptic transport. Brain Res.
1999;819(1-2):23-32.
13.Schibler U. The daily timing of gene expression and physiology in
mammals. Dialogues Clin Neurosci. 2007;9(3):257-72.
14.Baggs JE, Price TS, DiTacchio L, Panda S, Fitzgerald GA, Hogenesch JB. Network features of the mammalian circadian clock.
PLoS Biol. 2009;7(3):e52
15.Yan L, Takekida S, Shigeyoshi Y, Okamura H. Per1 and Per2 gene
expression in the rat suprachiasmatic nucleus: circadian profile
and the compartment-specific response to light. Neuroscience.
1999;94(1):141-50.
Sleep Sci. 2010;3(1):�����
����
–44
16.Bito H, Deisseroth K, Tsien RW. Ca2-dependent regulation in
neuronal gene expression. Curr Opin Neurobiol. 1997;7(3):41929.
17.Hardingham GE, Arnold FJ, Bading H. Nuclear calcium signaling controls CREB-mediated gene expression triggered by synaptic activity. Nat Neurosci. 2001;4(3):261-7.
18.Kunz H, Achermann P. Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators. J Theor Biol. 2003;224(1):63-78.
19.Granada AE, Herzel H. How to achieve fast entrainment? The
timescale to synchronization. PLoS One. 2009;4(9):e7057.
20.Butler MP, Silver R. Basis of robustness and resilience in the suprachiasmatic nucleus: individual neurons form nodes in circuits
that cycle daily. J Biol Rhytms. 2009;24(5):340-52.
21.Herzog ED, Aton SJ, Numano R, Sakaki Y, Tei H. Temporal precision in the mammalian circadian system: a reliable clock from
less reliable neurons. J Biol Rhytms. 2004;19(1):35-46.