BACKGROUND
most organisms use circadian rhythms to keep the temporal order and anticipate daily variations in their environmental changes  <cit> .
circadian rhythms are the sustained oscillations occurring with a periodicity close to  <dig> h in almost all living organisms from the cyanobacteria to plants, insects and mammals which are innately generated by an internal timing mechanism, i.e., the so-called circadian clock.
in mammals, the master circadian clock is located in the suprachiasmatic nucleus  of the ventral hypothalamus  <cit> , where it orchestrates the diurnal changes in both the physiology and behavior.
as such, the scn is capable of generating self-sustained rhythmicity in its intrinsic biological processes.
the mammalian circadian clock regulates many bodily functions, such as the sleep-wake cycles, the neuroendocrine levels, mental alertness, physical strength, renal and liver activity, body temperature, blood pressure, blood viscosity and the release of appropriate hormones at different times  <cit> .
in addition to these widely well-known effects, circadian rhythms also play a role in the pathogenesis and guide the optimal treatment for certain diseases including the arthritis, asthma, cancer, cardiovascular disease, diabetes, duodenal ulcers, hypercholesterolemia, and seasonal affective disorders  <cit> .
at the molecular level, molecular and genetic studies indicate that a circadian period arises from a system of interconnected feedback loops which control the transcription of a small number of "clock" genes  <cit> .
it is intriguing that the mechanism of circadian rhythms relies exactly on the interaction of these negative or positive feedback loops, which has been proposed to be important for generation of the basic circadian rhythm.
in other words, every oscillator has both positive and negative elements to establish the feedback loop.
the positive elements of the loop activate the expression of the negative elements, in which way moving a system away from its equilibrium state and become more unstable.
whereas, the negative elements feed back to block their own activation , which usually makes a system go into some equilibrium state to become more stable.
importantly, the key aspects of these feedback loops  can be modulated by the changes in clock.
although the sustained circadian rhythms are produced by the central pacemaker located in the scn of the anterior hypothalamus  <cit> , peripheral tissues such as liver, kidney or skeletal muscle can also give rise to circadian rhythms  <cit> .
recent work also showed that the clock timing and entrainment could be affected at a post-transcriptional level  <cit> .
thus, it is important to investigate the molecular basis of the post-transcriptional, i.e., mirna-mediated regulation of the circadian timing both in the scn and peripheral tissues.
mirnas are small non-coding rnas, with approximately  <dig> nucleotides long that are involved in post-transcriptional regulation of the gene expressions in most living organisms.
they are well conserved in eukaryotic organisms and are thought to be vital and evolutionarily ancient components of the genetic regulation  <cit> .
mirna molecules are partially complementary to one or more mrna molecules, with a main function of down-regulating the gene expression in a variety of manners.
they play a crucial role in diverse biological processes during the normal development such as the developmental timing and patterning, apoptosis, cell proliferation, organ development, as well as the pathological responses like the tumorigenesis  <cit> .
however, up to now for most mirnas, very little is known regarding their functions and precise regulation mechanisms.
thus, great efforts have been made to discover novel mirnas and to elucidate their regulatory mechanism in recent years.
lately, a large fraction of mirnas were found exhibiting strict developmental stage and tissue-dependent expression manner which is critical for their appropriate activities  <cit> , and the clock-relevant mirnas are the same cases.
sempere et al.
proved that the canonical myomirs  mir- <dig>  -133a and - <dig> are highly expressed in both human and mouse heart and skeletal muscles  <cit> .
and mir- <dig> is, especially, specifically restricted to the skeletal muscle, but is absent or expressed at relatively low levels in other tissues, which is exactly the reason making itself a unique one among the myomir family  <cit> .
mir- <dig> reinforces the muscle differentiation program by reducing the levels of dna polymerase and the inhibitory hlh protein id, which functions as a negative regulator of myod  <cit> .
in addition, chromatin immunoprecipitation experiments also demonstrated that the expression of muscle-enriched mirnas was controlled by the myogenic regulatory factors myod <dig> and myogenin  <cit> , two key components of the core circadian clock.
indeed, mir- <dig> was recently found to be a direct transcriptional target of myod <dig>  <cit> , a muscle-specific regulator that can stimulate the expression of mir- <dig>  despite of the role as a clock-controlled gene, myod <dig> was also a constituent of the skeletal muscle circadian transcriptome  <cit> .
up to now, the clock gene is the only reported mir- <dig> target involved in the circadian mechanism  <cit> .
taken together, the unique exclusively and highly expressed muscle-specific mirna- <dig> might represent an intriguing tool to regulate the circadian rhythm process  <cit> .
it is known that circadian clock system involves many components and complex interactions in living organisms  <cit> .
clearly, this complexity renders it extremely difficult to intuitively understand the clock control mechanism or even a simple configuration of those core genes.
to explore the specific effects of various genes that control the circadian clocks, it is necessary to resort to computational systems biology methods, which have emerged as powerful tools to supplement the experimental work and provide insights into the operation of the gene networks.
traditionally, the modeling of chemical reactions can be achieved either by using differential equations built based on the law of mass action or by the use of their stochastic counterpart  <cit> .
the modeling of circadian rhythms has also begun a long time before as recently reviewed  <cit> .
as a result, many interesting properties of the rhythms have been uncovered by the mathematical models, like the discovery that there might exist multiple sources of periodic behavior in the genetic regulatory network controlling the circadian oscillations.
recently a new deterministic model for study of the mammalian circadian clock has been proposed  <cit> .
this model was developed based on incorporation of the intertwined positive and negative regulatory loops involving the per, cry, bmal <dig>  clock and rev-erbα genes, which account for the autonomous, sustained circadian oscillations in conditions corresponding to continuous darkness.
interestingly, the importance of the mirna regulation in circadian oscillators has also been demonstrated in two dynamics models recently reported  <cit> .
however, up to now, little is known about the contribution effects of the translational controls to the circadian rhythms system.
no further model for the mammalian circadian clock involved in the mirna-mediated regulation mechanism has been developed, particularly for the mir- <dig>  a critical fine-tuner of the core circadian clock.
to address this issue and in particular to explore the translational control by mirnas, in the present work we attempt to establish a detailed quantitative mathematical model for the circadian rhythms oscillators in mammalian clock, which involves a clock-myod1-mir- <dig> feedback loop controlling the timing of the circadian cycle.
the process includes several aspects.
firstly, the generation of the basic circadian rhythm in our model depends on two negative autoregulations of the gene expression.
one is the negative autoregulatory feedback exerted by the clock-bmal <dig> on the expression of bmal <dig> gene.
the other is the clock expression which is subjected to a negative autoregulation by the clock protein.
secondly, besides the negative autoregulatory feedback of the gene expression, the interplay of the negative and positive feedback loops is also taken into consideration.
actually, these interlocked feedback loops potentially allow for multiple inputs and outputs at different phases  <cit>  and help to dissect their differential roles in the system.
as a result, a theoretical model was developed in this work and the dynamic mechanism it uncovered is well supported by several recent experimental studies  <cit> .
the model and related discovery might be helpful for deep understanding of the mechanism of mirna-mediated circadian rhythms.
it points out the crucial role of the mirna-mediated control on the gene expression during the skeletal muscle development and disease, and considers the potential of mirnas as therapeutic targets.
RESULTS
dynamics of the pathway
recent studies showed that the molecular mechanism of circadian oscillations not only relies on the feedback loops of the gene expression, but also is affected by the interfering of mirnas in the circadian rhythm choreography  <cit> .
therefore, in this work, a mir-206-mediated mammalian circadian clock model was developed and the properties of the oscillations involved in the model were investigated which incorporated the main components that play roles in the circadian rhythms.
using a set of parameters with appropriate biological values, the modeling work was carried out presently  by numerical integration of eqs.
-.
the reasonability of this model can be demonstrated by the simulated periods of oscillations which are close to  <dig> h in continuous darkness for those key components like clock protein.
these oscillations are constant and self-sustained  since the conditions of the equations are constant in time.
however, because some parameter values remain to be determined experimentally, the occurrence of autonomous sustained oscillations accounted by the model might suffer from a semiarbitrary choice of parameter values.
despite of this, the model we developed still yields a period of oscillations in continuous darkness close to  <dig> h, as all these parameters are within a reasonable physiological range .
the observed dynamics of the clock system are well consistent with the general principle that the circadian rhythms in mammals can persist in continuous darkness or light  <cit> .
the sets  <dig>   <dig>   <dig> and  <dig> represent the cases for the risc formation rate c <dig> =  <dig>   <dig> ,  <dig>  and  <dig>  nm-1·h- <dig>  respectively.
the oscillation periods of the four parameter sets are  <dig> ,  <dig> ,  <dig> and  <dig>  h respectively.
the parameter set  <dig> was selected because the circadian oscillation period is consistent with the physiologic phenomenon.
the black curves of figure  <dig> representing those oscillations of the system in the absence of mir- <dig> disclose some other interesting information.
as seen from the curves, obviously without the mir- <dig> regulation only a subtle change is observed in the oscillatory pattern of the system compared to the mir-206-mediated case, indicating that the impact of mirna regulation on the oscillatory pattern formation in circadian rhythms may not be enormous.
however, the oscillatory appearance of some key components such as myod <dig> mrna and its protein changes significantly, with the amplitude increased by about 77% and 133%  respectively, but the increase in the amplitude of bmal <dig> mrna, clock mrna and their proteins is less pronounced as approximately 10% .
the same tendency observed in figure  <dig> is also proven by figure  <dig>  which accurately depicts the expression increase for those key components in the pathways with or without the mir- <dig> regulations.
clearly, for several crucial components like the clock mrna , bmal <dig> mrna , cytosolic and nuclear clock proteins  and the complex between the nuclear clock and bmal <dig> proteins , only small variations in the amplitude of the components are observed without the mir- <dig> regulation.
but for other components, like the cytosolic clock proteins , myod <dig> mrna  and cytosolic myod <dig> , their expression levels are largely enhanced in the absence of mirna.
this may be due to the fact that myod is a direct target of the circadian transcriptional activators in, which binds in a rhythmic manner to the core enhancer of the myod promoter  <cit> .
therefore, the complex in directly controls the mmy and myc expression, which is more important than the indirect mirna regulation.
these results may be related to the fact that the transcription and translation of key circadian core components, on one hand are tightly connected with each other, and on the other hand may buffer each other.
such a regulatory feature could also explain why a major increase in the transcription, like the one caused by the myod <dig> gene, results in only a modest increase in myod <dig> mrna abundance and probably an even more modest increase in the translated protein.
from the above results, three interesting findings are presented: 1) although the mir- <dig> regulation is crucial in controlling the dynamics of the circadian rhythms system, it is not strong and fits exactly some recent data demonstrating that a drosophila mirna can function as a buffering agent against the environmental perturbations during the development  <cit> .
2) sustained oscillations with a circadian period close to  <dig> h occur in our model in the continuous darkness, which is precisely regulated by the interlocked positive and negative feedback loops.
and these loops effectively maintain the oscillations and the stability of the system, as indicated by the small variation in the amplitude of several key components .
but the expression levels of cytosolic clock proteins, myod <dig> mrna and cytosolic myod <dig> are largely enhanced in the absence of mirna, reflecting the fact that the complex in's direct effects on the mmy and myc expression is more important than the indirect mirna regulation.
these observations may provide a plausible mechanism through which tissue-specific factors such as myod <dig> and mir- <dig> can convey unique tissue requirements to the circadian clock.
3) once again, it is confirmed that mir- <dig> is necessary for accurate circadian timekeeping, since in the absence of mirna, the oscillation period of the system as shown by the three representative components, bmal <dig>  myod <dig> and clock mrnas and their corresponding proteins is not very constant and changes with time.
at  <dig> h, the period is  <dig>  h. but when the simulation time increases to  <dig> h, the period has changed up to ~ <dig>  h, and even up to ~ <dig>  h at  <dig> h or much longer time.
however the system is more stable in the presence of mirna when the period of the oscillatory system is always  <dig>  h, which further indicates the important role of mir- <dig> plays in the accurate circadian timekeeping.
dynamic sensitivity analysis
having many successfully applications in the study of biochemical systems  <cit> , sensitivity analysis provides a systematic framework to investigate how changes of a parameter cause changes of the dynamic behavior of a pathway, and which parameters are the most crucial ones impacting the whole system.
in the present work, a sensitivity analysis for all parameters  of the mammalian circadian pathway mediated by mir- <dig> was conducted, when a total of  <dig>  local sensitivities were calculated and normalized with  <dig> scaled sensitivity absolute values .
as a result,  <dig> rate constants are found larger than  <dig> , where negative s indicates that the reaction output decreases with the increasing rate constant.
in figure 3a, only those reactions or parameters with significant effects as revealed by the sensitivity analysis on the system are shown, and others with weak or no influences  on the pathway were omitted for clarity.
the rate of bmal <dig> mrna synthesis  has the largest impact on the whole system, affecting  <dig> out of the total  <dig> reactions while vmb significantly affects  <dig> reactions, ksb affects  <dig> reactions, and v1b, v3b and ki respectively affecting  <dig> reactions .
these observations indicate that the impact of these parameters on the whole pathway are in the order of vsb >vmb >ksb >v1b >v3b >ki.
thus, we may conclude that the parameters with respect to the synthesis and degradation of bmal <dig> and its mrna exert a marked effect on producing sustained oscillations.
in other words, the clock mechanism is highly sensitive to bmal <dig>  this might be a result of the critical role of bmal <dig> promoting the circadian periodicity of the transcriptional activation, since it takes shape a positive branch of the main transcriptional autoregulatory feedback loop.
in addition, another interesting finding is that parameter c <dig> which relates to the production of mir- <dig> affects  <dig> reactions of all  <dig> reactions with |s| =  <dig>  this positively shows the importance of mir- <dig> in regulating the whole system.
figure 3b depicts this result by the red region that represents the influence of c <dig> on the model.
the small value of the sensitivity indices  reflects a fact that mirna though regulating the system in a relatively weak manner, is still crucial in controlling the dynamics of the circadian clock.
as we know, sustained oscillations only occur within an appropriate range of parameter values, and beyond this range the system tends to reach a steady state.
such an evolution is often associated with the occurrence of damped oscillations.
the mammalian circadian system is thought to be composed of a hierarchical set of oscillators with the suprachiasmatic nuclei acting as a master pacemaker  <cit>  which is independently able to both generate and sustain its own circadian oscillations.
the peripheral tissues can give rise to circadian rhythms and these peripheral rhythms also appear to be sustained  <cit> .
peripheral rhythms are damped unless they are driven by periodic signals received from the scn  <cit> .
the oscillations can readily be entrained by the periodic variation of the parameter when damped oscillations occur in the model in continuous darkness  <cit> .
all these observations can be found in the present model in figure  <dig>  which displays the time evolution curves of bn, mp, pn and mir- <dig>  these curves were obtained when c <dig> =  <dig>  nm·h- <dig> and all other parameters remained as the same values as in figure  <dig>
the effect of parameter on model dynamics
as a starting point to examine how the synthesis rate of bmal <dig> mrna  exhibits the largest effects on our system, some explorations on variation of the parameter space were carried out presently.
the value of vsb was changed as  <dig> ,  <dig>   and  <dig>  nm·h- <dig>  when values of other parameters were kept fixed.
figure  <dig> shows the obtained results where the blue, red and black curves represent the cases when vsb =  <dig> ,  <dig>  and  <dig>  nm·h- <dig>  respectively.
when vsb decreases from  <dig>  to  <dig>  nm·h- <dig>  the oscillations of bmal <dig> mrna  and myod <dig> mrna  vanish and subsequently evolve toward a steady state.
the oscillation amplitude of clock mrna  decreases accompanied with the increase of the oscillation frequency.
however, as vsb increases to  <dig>  nm·h- <dig>  an unequal amplitude of oscillation occurs for bmal <dig> mrna with a decreasing oscillation period compared to the 'basal' value of vsb =  <dig> nm·h- <dig> .
since vsb is the synthesis rate of bmal <dig> mrna, the amplitude of oscillation increases when vsb rises to  <dig>  nm·h- <dig>  the reason for the occurrence of unequal amplitude of oscillation for bmal <dig> mrna might be due to the negative feedback control on the bmal <dig> mrna production from the downstream gene like in complex .
but vsb plays an indirect role for the myod <dig> mrna production.
this also explains why myod <dig> mrna oscillates in an equal amplitude and in a constant period in regard to vsb =  <dig>  nm·h- <dig> , since there is no any negative feedback for the myod <dig> mrna .
in contrast, the oscillation of clock mrna keeps constant with only slight variations occurring in the amplitude and period with the change of vsb .
this is consistent with the clock expression dynamics to maintain normal circadian rhythmicity  <cit> .
a more interesting finding here is that the system still generates sustained oscillation even when vsb value is whether enlarged or lessened , indicating that the clock is robust to the variations of vsb.
this on one hand might explain why the biological clock period is always constant in various organisms, and on the other hand is also consistent with the fact that the clock expression is necessary to maintain the normal circadian rhythmicity  <cit> .
clearly, the robustness of the clock protein would play a crucial role in a living system to keep its normal physiological functions, since the developed biological clocks from evolution in animals, plants and others are important in regulating and coordinating the internal biological processes  <cit> .
the effect of parameter variation of mirna
previous research suggested that mir- <dig> is specifically expressed in skeletal muscle  <cit> , which is a clock-controlled gene that plays a role in regulating the peripheral circadian rhythm.
the sensitivity analysis here also shows that the system is sensitive to the variation of the mir- <dig> production rate .
therefore, an attempt has been made to investigate how the changes of the kinetic parameters associated with mir- <dig> cause changes in the dynamic behavior of the biological clock system.
it is also interesting to explore the variation of the frequency and amplitude when c <dig> was set to  <dig> ,  <dig>  and  <dig>  nm·h- <dig>  with all other rate constants kept fixed.
the obtained results are shown in figure  <dig>  in which the blue curves represent the variation of bmal <dig> mrna, myod <dig> mrna and clock mrna in the absence of mirna .
the red and black curves represent the conditions when c <dig> =  <dig>  and  <dig>  nm·h- <dig> respectively.
as seen from this figure, when c <dig> =  <dig>  nm·h- <dig>  the system still keeps relatively constant oscillations in the absence of mirna, which might indicate that the regulation of mir- <dig> in the biological clock system is relatively weak  <cit> .
but once the mir- <dig> is introduced , the oscillatory behavior of the system appears in an extremely constant manner.
when c <dig> changes from  <dig>  to  <dig>  nm·h <dig>  the amplitudes of oscillation for bmal <dig> and clock mrnas do not show any significant changes, but the amplitude of oscillation for myod <dig> mrna decreases from  <dig>  to  <dig>  nm.
the reason is that myod <dig> directly activates the expression level of mir- <dig>  thus resulting in the difference of expressions between bmal <dig> and myod <dig>  all these results reveal that the sustained oscillations occur only within a certain range of parameter values.
beyond this range, the system tends to a steady state.
so when more mirna is produced  in the system, all the above three mrnas evolve to a steady state and no oscillations occur.
clearly the over expression of mir- <dig> in the system disturbs the biological clock and makes the periodic oscillations collapsed.
this result indicates that mirnas are important in maintaining the tissue and cell normal function, and the abnormal expression of mirnas may lead to unexpected diseases.
in the next step, we carried out a detailed analysis to uncover how the system is affected by mir- <dig>  when the amplitude and the frequency of oscillation for the above three representative components are calculated.
bmal <dig> mrna, myod <dig> mrna and clock mrna are shown in black, blue and red in figure  <dig>  respectively.
the results show that the frequencies of oscillation for all three mrnas increase with the increase of c <dig>  indicating that the temporal sequence of the gene production can be interfered .
the amplitudes of oscillation for bmal <dig> and myod <dig> mrnas are almost constant and no significant changes are found though c <dig> has been changed from  <dig> to  <dig>  nm·h- <dig>  however, the clock mrna expression decreases by about  <dig> times .
this reduction in amplitude of the oscillation shows that the expression level of clock mrna can be affected when c <dig> increases.
these results show that mir- <dig> may be a stabilizing factor on the frequency and amplitude of the oscillator system, thereby regulating the dynamics of the protein production and gene expression.
in addition, the effects of the variation of parameters c <dig> , c <dig> , c <dig>  and ksc  on the whole system are also investigated, respectively.
figure  <dig> shows the obtained results.
as can be seen, when parameters c <dig> and c <dig> are increased by  <dig> folds compared to their normal values , very small variations are observed for all three representative molecules, i.e., bmal <dig> mrna, myod <dig> mrna and clock mrna.
interestingly, the changes for other two parameters  also do not significantly affect the system, and the observed evolutionary curves with different parameter values are almost totally overlapped with each other.
these results demonstrate that all the four parameters are not as significant as c <dig>  the mirna production rate, to the oscillator system, which is consistent with the findings obtained from the sensitivity analysis.
CONCLUSIONS
mirnas play an important role in various biological functions and represent a potentially novel avenue by which biological timing processes can be regulated.
in this paper, the dynamics of regulation by mir- <dig> were investigated based on the clock-myod1-mir- <dig> interacting positive and negative feedback loops controlling the timing of the circadian cycle in mammalian skeletal muscle.
to this end, a system-theoretic approach by using hill-type terms, michaelis-menten type and mass action kinetics is introduced in this work and the dynamics is implemented deterministically.
the model outlined presently not only introduces mir- <dig> into the field of circadian timing, but also accurately predicts the clock periodicity and clock entrainment.
a sustained periodic circadian oscillation occurs in the circadian clock system when the series of parameters were set with biologically reasonable values.
in addition, this model reveals the possible existence of multiple sources of biological oscillatory behavior, and provides a plausible mechanism through which tissue-specific factors such as myod <dig> and mir- <dig> can convey unique tissue requirements to the circadian clock.
in addition, the results point out the important effects of parameter variations such as the mir- <dig> synthesis on the regulation of the mammalian clock system, which not only changes the amplitude but also alters the frequency of the oscillations of the deterministic system.
in other words, mir- <dig> plays a significant role on the dynamics of regulatory systems, both by affecting the level of the gene expression and by interfering with the temporal sequence of the gene production or transmission.
in short, the essential qualitative features of these results indicate that the mirna regulation may be one of the primary means for controlling the period of oscillatory molecules and biochemical pathways within mammalian circadian clock.
we provide testable hypothesis for experimental biologists to further investigate mirna's functional roles in regulating cellular processes and development.
the challenge for future studies will be a focus on the modulation of the mirna expression as a potentially powerful diagnostic and therapeutic approach to treat cardiac and skeletal muscle disease.

 most organisms use circadian rhythms to keep the temporal order and anticipate daily variations in their environmental changes. circadian rhythms are the sustained oscillations occurring with a periodicity close to h in almost all living organisms from the cyanobacteria to plants, insects and mammals which are innately generated by an internal timing mechanism, i.e., the so-called circadian clock. in mammals, the master circadian clock is located in the suprachiasmatic nucleus of the ventral hypothalamus, where it orchestrates the diurnal changes in both the physiology and behavior. as such, the scn is capable of generating self-sustained rhythmicity in its intrinsic biological processes. the mammalian circadian clock regulates many bodily functions, such as the sleep-wake cycles, the neuroendocrine levels, mental alertness, physical strength, renal and liver activity, body temperature, blood pressure, blood viscosity and the release of appropriate hormones at different times. in addition to these widely well-known effects, circadian rhythms also play a role in the pathogenesis and guide the optimal treatment for certain diseases including the arthritis, asthma, cancer, cardiovascular disease, diabetes, duodenal ulcers, hypercholesterolemia, and seasonal affective disorders. at the molecular level, molecular and genetic studies indicate that a circadian period arises from a system of interconnected feedback loops which control the transcription of a small number of "clock" genes. it is intriguing that the mechanism of circadian rhythms relies exactly on the interaction of these negative or positive feedback loops, which has been proposed to be important for generation of the basic circadian rhythm. in other words, every oscillator has both positive and negative elements to establish the feedback loop. the positive elements of the loop activate the expression of the negative elements, in which way moving a system away from its equilibrium state and become more unstable. whereas, the negative elements feed back to block their own activation, which usually makes a system go into some equilibrium state to become more stable. importantly, the key aspects of these feedback loops can be modulated by the changes in clock. although the sustained circadian rhythms are produced by the central pacemaker located in the scn of the anterior hypothalamus, peripheral tissues such as liver, kidney or skeletal muscle can also give rise to circadian rhythms. recent work also showed that the clock timing and entrainment could be affected at a post-transcriptional level. thus, it is important to investigate the molecular basis of the post-transcriptional, i.e., mirna-mediated regulation of the circadian timing both in the scn and peripheral tissues. mirnas are small non-coding rnas, with approximately nucleotides long that are involved in post-transcriptional regulation of the gene expressions in most living organisms. they are well conserved in eukaryotic organisms and are thought to be vital and evolutionarily ancient components of the genetic regulation. mirna molecules are partially complementary to one or more mrna molecules, with a main function of down-regulating the gene expression in a variety of manners. they play a crucial role in diverse biological processes during the normal development such as the developmental timing and patterning, apoptosis, cell proliferation, organ development, as well as the pathological responses like the tumorigenesis. however, up to now for most mirnas, very little is known regarding their functions and precise regulation mechanisms. thus, great efforts have been made to discover novel mirnas and to elucidate their regulatory mechanism in recent years. lately, a large fraction of mirnas were found exhibiting strict developmental stage and tissue-dependent expression manner which is critical for their appropriate activities, and the clock-relevant mirnas are the same cases. sempere et al. proved that the canonical myomirs mir- -133a and - are highly expressed in both human and mouse heart and skeletal muscles. and mir- is, especially, specifically restricted to the skeletal muscle, but is absent or expressed at relatively low levels in other tissues, which is exactly the reason making itself a unique one among the myomir family. mir- reinforces the muscle differentiation program by reducing the levels of dna polymerase and the inhibitory hlh protein id, which functions as a negative regulator of myod. in addition, chromatin immunoprecipitation experiments also demonstrated that the expression of muscle-enriched mirnas was controlled by the myogenic regulatory factors myod and myogenin, two key components of the core circadian clock. indeed, mir- was recently found to be a direct transcriptional target of myod, a muscle-specific regulator that can stimulate the expression of mir- despite of the role as a clock-controlled gene, myod was also a constituent of the skeletal muscle circadian transcriptome. up to now, the clock gene is the only reported mir- target involved in the circadian mechanism. taken together, the unique exclusively and highly expressed muscle-specific mirna- might represent an intriguing tool to regulate the circadian rhythm process. it is known that circadian clock system involves many components and complex interactions in living organisms. clearly, this complexity renders it extremely difficult to intuitively understand the clock control mechanism or even a simple configuration of those core genes. to explore the specific effects of various genes that control the circadian clocks, it is necessary to resort to computational systems biology methods, which have emerged as powerful tools to supplement the experimental work and provide insights into the operation of the gene networks. traditionally, the modeling of chemical reactions can be achieved either by using differential equations built based on the law of mass action or by the use of their stochastic counterpart. the modeling of circadian rhythms has also begun a long time before as recently reviewed. as a result, many interesting properties of the rhythms have been uncovered by the mathematical models, like the discovery that there might exist multiple sources of periodic behavior in the genetic regulatory network controlling the circadian oscillations. recently a new deterministic model for study of the mammalian circadian clock has been proposed. this model was developed based on incorporation of the intertwined positive and negative regulatory loops involving the per, cry, bmal clock and rev-erbα genes, which account for the autonomous, sustained circadian oscillations in conditions corresponding to continuous darkness. interestingly, the importance of the mirna regulation in circadian oscillators has also been demonstrated in two dynamics models recently reported. however, up to now, little is known about the contribution effects of the translational controls to the circadian rhythms system. no further model for the mammalian circadian clock involved in the mirna-mediated regulation mechanism has been developed, particularly for the mir- a critical fine-tuner of the core circadian clock. to address this issue and in particular to explore the translational control by mirnas, in the present work we attempt to establish a detailed quantitative mathematical model for the circadian rhythms oscillators in mammalian clock, which involves a clock-myod1-mir- feedback loop controlling the timing of the circadian cycle. the process includes several aspects. firstly, the generation of the basic circadian rhythm in our model depends on two negative autoregulations of the gene expression. one is the negative autoregulatory feedback exerted by the clock-bmal on the expression of bmal gene. the other is the clock expression which is subjected to a negative autoregulation by the clock protein. secondly, besides the negative autoregulatory feedback of the gene expression, the interplay of the negative and positive feedback loops is also taken into consideration. actually, these interlocked feedback loops potentially allow for multiple inputs and outputs at different phases and help to dissect their differential roles in the system. as a result, a theoretical model was developed in this work and the dynamic mechanism it uncovered is well supported by several recent experimental studies. the model and related discovery might be helpful for deep understanding of the mechanism of mirna-mediated circadian rhythms. it points out the crucial role of the mirna-mediated control on the gene expression during the skeletal muscle development and disease, and considers the potential of mirnas as therapeutic targets.  dynamics of the pathway recent studies showed that the molecular mechanism of circadian oscillations not only relies on the feedback loops of the gene expression, but also is affected by the interfering of mirnas in the circadian rhythm choreography. therefore, in this work, a mir-206-mediated mammalian circadian clock model was developed and the properties of the oscillations involved in the model were investigated which incorporated the main components that play roles in the circadian rhythms. using a set of parameters with appropriate biological values, the modeling work was carried out presently by numerical integration of eqs. -. the reasonability of this model can be demonstrated by the simulated periods of oscillations which are close to h in continuous darkness for those key components like clock protein. these oscillations are constant and self-sustained since the conditions of the equations are constant in time. however, because some parameter values remain to be determined experimentally, the occurrence of autonomous sustained oscillations accounted by the model might suffer from a semiarbitrary choice of parameter values. despite of this, the model we developed still yields a period of oscillations in continuous darkness close to h, as all these parameters are within a reasonable physiological range. the observed dynamics of the clock system are well consistent with the general principle that the circadian rhythms in mammals can persist in continuous darkness or light. the sets and represent the cases for the risc formation rate c =, and nm-1·h- respectively. the oscillation periods of the four parameter sets are,, and h respectively. the parameter set was selected because the circadian oscillation period is consistent with the physiologic phenomenon. the black curves of figure representing those oscillations of the system in the absence of mir- disclose some other interesting information. as seen from the curves, obviously without the mir- regulation only a subtle change is observed in the oscillatory pattern of the system compared to the mir-206-mediated case, indicating that the impact of mirna regulation on the oscillatory pattern formation in circadian rhythms may not be enormous. however, the oscillatory appearance of some key components such as myod mrna and its protein changes significantly, with the amplitude increased by about 77% and 133% respectively, but the increase in the amplitude of bmal mrna, clock mrna and their proteins is less pronounced as approximately 10%. the same tendency observed in figure is also proven by figure which accurately depicts the expression increase for those key components in the pathways with or without the mir- regulations. clearly, for several crucial components like the clock mrna, bmal mrna, cytosolic and nuclear clock proteins and the complex between the nuclear clock and bmal proteins, only small variations in the amplitude of the components are observed without the mir- regulation. but for other components, like the cytosolic clock proteins, myod mrna and cytosolic myod, their expression levels are largely enhanced in the absence of mirna. this may be due to the fact that myod is a direct target of the circadian transcriptional activators in, which binds in a rhythmic manner to the core enhancer of the myod promoter. therefore, the complex in directly controls the mmy and myc expression, which is more important than the indirect mirna regulation. these results may be related to the fact that the transcription and translation of key circadian core components, on one hand are tightly connected with each other, and on the other hand may buffer each other. such a regulatory feature could also explain why a major increase in the transcription, like the one caused by the myod gene, results in only a modest increase in myod mrna abundance and probably an even more modest increase in the translated protein. from the above results, three interesting findings are presented: 1) although the mir- regulation is crucial in controlling the dynamics of the circadian rhythms system, it is not strong and fits exactly some recent data demonstrating that a drosophila mirna can function as a buffering agent against the environmental perturbations during the development. 2) sustained oscillations with a circadian period close to h occur in our model in the continuous darkness, which is precisely regulated by the interlocked positive and negative feedback loops. and these loops effectively maintain the oscillations and the stability of the system, as indicated by the small variation in the amplitude of several key components. but the expression levels of cytosolic clock proteins, myod mrna and cytosolic myod are largely enhanced in the absence of mirna, reflecting the fact that the complex in's direct effects on the mmy and myc expression is more important than the indirect mirna regulation. these observations may provide a plausible mechanism through which tissue-specific factors such as myod and mir- can convey unique tissue requirements to the circadian clock. 3) once again, it is confirmed that mir- is necessary for accurate circadian timekeeping, since in the absence of mirna, the oscillation period of the system as shown by the three representative components, bmal myod and clock mrnas and their corresponding proteins is not very constant and changes with time. at h, the period is h. but when the simulation time increases to h, the period has changed up to ~ h, and even up to ~ h at h or much longer time. however the system is more stable in the presence of mirna when the period of the oscillatory system is always h, which further indicates the important role of mir- plays in the accurate circadian timekeeping. dynamic sensitivity analysis having many successfully applications in the study of biochemical systems, sensitivity analysis provides a systematic framework to investigate how changes of a parameter cause changes of the dynamic behavior of a pathway, and which parameters are the most crucial ones impacting the whole system. in the present work, a sensitivity analysis for all parameters of the mammalian circadian pathway mediated by mir- was conducted, when a total of local sensitivities were calculated and normalized with scaled sensitivity absolute values. as a result, rate constants are found larger than, where negative s indicates that the reaction output decreases with the increasing rate constant. in figure 3a, only those reactions or parameters with significant effects as revealed by the sensitivity analysis on the system are shown, and others with weak or no influences on the pathway were omitted for clarity. the rate of bmal mrna synthesis has the largest impact on the whole system, affecting out of the total reactions while vmb significantly affects reactions, ksb affects reactions, and v1b, v3b and ki respectively affecting reactions. these observations indicate that the impact of these parameters on the whole pathway are in the order of vsb >vmb >ksb >v1b >v3b >ki. thus, we may conclude that the parameters with respect to the synthesis and degradation of bmal and its mrna exert a marked effect on producing sustained oscillations. in other words, the clock mechanism is highly sensitive to bmal this might be a result of the critical role of bmal promoting the circadian periodicity of the transcriptional activation, since it takes shape a positive branch of the main transcriptional autoregulatory feedback loop. in addition, another interesting finding is that parameter c which relates to the production of mir- affects reactions of all reactions with |s| = this positively shows the importance of mir- in regulating the whole system. figure 3b depicts this result by the red region that represents the influence of c on the model. the small value of the sensitivity indices reflects a fact that mirna though regulating the system in a relatively weak manner, is still crucial in controlling the dynamics of the circadian clock. as we know, sustained oscillations only occur within an appropriate range of parameter values, and beyond this range the system tends to reach a steady state. such an evolution is often associated with the occurrence of damped oscillations. the mammalian circadian system is thought to be composed of a hierarchical set of oscillators with the suprachiasmatic nuclei acting as a master pacemaker which is independently able to both generate and sustain its own circadian oscillations. the peripheral tissues can give rise to circadian rhythms and these peripheral rhythms also appear to be sustained. peripheral rhythms are damped unless they are driven by periodic signals received from the scn. the oscillations can readily be entrained by the periodic variation of the parameter when damped oscillations occur in the model in continuous darkness. all these observations can be found in the present model in figure which displays the time evolution curves of bn, mp, pn and mir- these curves were obtained when c = nm·h- and all other parameters remained as the same values as in figure the effect of parameter on model dynamics as a starting point to examine how the synthesis rate of bmal mrna exhibits the largest effects on our system, some explorations on variation of the parameter space were carried out presently. the value of vsb was changed as, and nm·h- when values of other parameters were kept fixed. figure shows the obtained results where the blue, red and black curves represent the cases when vsb =, and nm·h- respectively. when vsb decreases from to nm·h- the oscillations of bmal mrna and myod mrna vanish and subsequently evolve toward a steady state. the oscillation amplitude of clock mrna decreases accompanied with the increase of the oscillation frequency. however, as vsb increases to nm·h- an unequal amplitude of oscillation occurs for bmal mrna with a decreasing oscillation period compared to the'basal' value of vsb = nm·h-. since vsb is the synthesis rate of bmal mrna, the amplitude of oscillation increases when vsb rises to nm·h- the reason for the occurrence of unequal amplitude of oscillation for bmal mrna might be due to the negative feedback control on the bmal mrna production from the downstream gene like in complex. but vsb plays an indirect role for the myod mrna production. this also explains why myod mrna oscillates in an equal amplitude and in a constant period in regard to vsb = nm·h-, since there is no any negative feedback for the myod mrna. in contrast, the oscillation of clock mrna keeps constant with only slight variations occurring in the amplitude and period with the change of vsb. this is consistent with the clock expression dynamics to maintain normal circadian rhythmicity. a more interesting finding here is that the system still generates sustained oscillation even when vsb value is whether enlarged or lessened, indicating that the clock is robust to the variations of vsb. this on one hand might explain why the biological clock period is always constant in various organisms, and on the other hand is also consistent with the fact that the clock expression is necessary to maintain the normal circadian rhythmicity. clearly, the robustness of the clock protein would play a crucial role in a living system to keep its normal physiological functions, since the developed biological clocks from evolution in animals, plants and others are important in regulating and coordinating the internal biological processes. the effect of parameter variation of mirna previous research suggested that mir- is specifically expressed in skeletal muscle, which is a clock-controlled gene that plays a role in regulating the peripheral circadian rhythm. the sensitivity analysis here also shows that the system is sensitive to the variation of the mir- production rate. therefore, an attempt has been made to investigate how the changes of the kinetic parameters associated with mir- cause changes in the dynamic behavior of the biological clock system. it is also interesting to explore the variation of the frequency and amplitude when c was set to, and nm·h- with all other rate constants kept fixed. the obtained results are shown in figure in which the blue curves represent the variation of bmal mrna, myod mrna and clock mrna in the absence of mirna. the red and black curves represent the conditions when c = and nm·h- respectively. as seen from this figure, when c = nm·h- the system still keeps relatively constant oscillations in the absence of mirna, which might indicate that the regulation of mir- in the biological clock system is relatively weak. but once the mir- is introduced, the oscillatory behavior of the system appears in an extremely constant manner. when c changes from to nm·h the amplitudes of oscillation for bmal and clock mrnas do not show any significant changes, but the amplitude of oscillation for myod mrna decreases from to nm. the reason is that myod directly activates the expression level of mir- thus resulting in the difference of expressions between bmal and myod all these results reveal that the sustained oscillations occur only within a certain range of parameter values. beyond this range, the system tends to a steady state. so when more mirna is produced in the system, all the above three mrnas evolve to a steady state and no oscillations occur. clearly the over expression of mir- in the system disturbs the biological clock and makes the periodic oscillations collapsed. this result indicates that mirnas are important in maintaining the tissue and cell normal function, and the abnormal expression of mirnas may lead to unexpected diseases. in the next step, we carried out a detailed analysis to uncover how the system is affected by mir- when the amplitude and the frequency of oscillation for the above three representative components are calculated. bmal mrna, myod mrna and clock mrna are shown in black, blue and red in figure respectively. the results show that the frequencies of oscillation for all three mrnas increase with the increase of c indicating that the temporal sequence of the gene production can be interfered. the amplitudes of oscillation for bmal and myod mrnas are almost constant and no significant changes are found though c has been changed from to nm·h- however, the clock mrna expression decreases by about times. this reduction in amplitude of the oscillation shows that the expression level of clock mrna can be affected when c increases. these results show that mir- may be a stabilizing factor on the frequency and amplitude of the oscillator system, thereby regulating the dynamics of the protein production and gene expression. in addition, the effects of the variation of parameters c, c, c and ksc on the whole system are also investigated, respectively. figure shows the obtained results. as can be seen, when parameters c and c are increased by folds compared to their normal values, very small variations are observed for all three representative molecules, i.e., bmal mrna, myod mrna and clock mrna. interestingly, the changes for other two parameters also do not significantly affect the system, and the observed evolutionary curves with different parameter values are almost totally overlapped with each other. these results demonstrate that all the four parameters are not as significant as c the mirna production rate, to the oscillator system, which is consistent with the findings obtained from the sensitivity analysis.  mirnas play an important role in various biological functions and represent a potentially novel avenue by which biological timing processes can be regulated. in this paper, the dynamics of regulation by mir- were investigated based on the clock-myod1-mir- interacting positive and negative feedback loops controlling the timing of the circadian cycle in mammalian skeletal muscle. to this end, a system-theoretic approach by using hill-type terms, michaelis-menten type and mass action kinetics is introduced in this work and the dynamics is implemented deterministically. the model outlined presently not only introduces mir- into the field of circadian timing, but also accurately predicts the clock periodicity and clock entrainment. a sustained periodic circadian oscillation occurs in the circadian clock system when the series of parameters were set with biologically reasonable values. in addition, this model reveals the possible existence of multiple sources of biological oscillatory behavior, and provides a plausible mechanism through which tissue-specific factors such as myod and mir- can convey unique tissue requirements to the circadian clock. in addition, the results point out the important effects of parameter variations such as the mir- synthesis on the regulation of the mammalian clock system, which not only changes the amplitude but also alters the frequency of the oscillations of the deterministic system. in other words, mir- plays a significant role on the dynamics of regulatory systems, both by affecting the level of the gene expression and by interfering with the temporal sequence of the gene production or transmission. in short, the essential qualitative features of these results indicate that the mirna regulation may be one of the primary means for controlling the period of oscillatory molecules and biochemical pathways within mammalian circadian clock. we provide testable hypothesis for experimental biologists to further investigate mirna's functional roles in regulating cellular processes and development. the challenge for future studies will be a focus on the modulation of the mirna expression as a potentially powerful diagnostic and therapeutic approach to treat cardiac and skeletal muscle disease. 
a detailed dynamic model of mir-206-mediated mammalian circadian clock system was developed presently by using hill-type terms, michaelis-menten type and mass action kinetics. although mirnas are known to regulate various biological processes ranging from development to cancer, much less is known about their role in the dynamic regulation of the mammalian circadian clock. mir- has a profound effect on the dynamic mechanism of the mammalian circadian clock, both by control of the amplitude and control or alteration of the frequency to affect the level of the gene expression and to interfere with the temporal sequence of the gene production or delivery. based on a system-theoretic approach, the model accurately predicts both the periodicity and the entrainment of the circadian clock. our results show that mir- is an important regulator of the circadian clock in skeletal muscle, and thus by study of mir- the main features of its mediation on the clock may be captured. this undoubtedly uncovers a new mechanism for regulation of the circadian clock at a post-transcriptional level and provides important insights into the normal development as well as the pathological conditions of skeletal muscles, such as the aging, chronic disease and cancer. it also explores the dynamics properties of the oscillations mediated by mir- by means of sensitivity analysis and alterations of parameters. recent work has revealed that mirnas are essential for the development and functioning of the skeletal muscles where they are. in particular, mir- has not only been identified as the only mirna expressed in skeletal muscles, but also exhibited crucial roles in regulation of the muscle development. as a group of highly conserved small non-coding rnas with a length of 21~ nucleotides, micrornas regulate the gene expression post-transcriptionally by base pairing with the partial or full complementary sequences in target mrnas, thus resulting in the repression of mrna translation and the acceleration of mrna degradation.
 as a group of highly conserved small non-coding rnas with a length of 21~ nucleotides, micrornas regulate the gene expression post-transcriptionally by base pairing with the partial or full complementary sequences in target mrnas, thus resulting in the repression of mrna translation and the acceleration of mrna degradation. recent work has revealed that mirnas are essential for the development and functioning of the skeletal muscles where they are. in particular, mir- has not only been identified as the only mirna expressed in skeletal muscles, but also exhibited crucial roles in regulation of the muscle development. although mirnas are known to regulate various biological processes ranging from development to cancer, much less is known about their role in the dynamic regulation of the mammalian circadian clock.  a detailed dynamic model of mir-206-mediated mammalian circadian clock system was developed presently by using hill-type terms, michaelis-menten type and mass action kinetics. based on a system-theoretic approach, the model accurately predicts both the periodicity and the entrainment of the circadian clock. it also explores the dynamics properties of the oscillations mediated by mir- by means of sensitivity analysis and alterations of parameters. our results show that mir- is an important regulator of the circadian clock in skeletal muscle, and thus by study of mir- the main features of its mediation on the clock may be captured. simulations of these processes display that the amplitude and frequency of the oscillation can be significantly altered through the mir-206-mediated control.  mir- has a profound effect on the dynamic mechanism of the mammalian circadian clock, both by control of the amplitude and control or alteration of the frequency to affect the level of the gene expression and to interfere with the temporal sequence of the gene production or delivery. this undoubtedly uncovers a new mechanism for regulation of the circadian clock at a post-transcriptional level and provides important insights into the normal development as well as the pathological conditions of skeletal muscles, such as the aging, chronic disease and cancer. 