Mathematical Modeling in Chronobiology

Abstract

Circadian clocks are autonomous oscillators entrained by external Zeitgebers such as light–dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic nucleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three different levels (1) rhythm generation in single cells by delayed negative feedbacks, (2) synchronization of cells via external stimuli or cell–cell coupling, and (3) optimization of chronotherapy.

Appendices

Appendix A: Oscillations Due to Delayed Negative Feedback

1.1 A.1 The Model

One of the simplest models for a self-suppressing gene reads

$$ \frac{{\mathrm{ d}x(t)}}{{\mathrm{ d}t}}=\frac{a}{{1+b{x^n}(t-\tau )}}-dx(t). $$

(3)

Here, the time-dependent state variable x(t) corresponds to the mRNA level of a clock gene, for instance, Per2 at time t. The positive parameter d is the mRNA degradation rate, large values correspond to a rapid degradation, whereas small values model more stable mRNAs. Parameter a determines the basal transcription rate in the absence of the inhibitor.

The self-inhibition is modeled in the following way: For simplicity, we consciously refrain from modeling all intermediate steps, which lead from the mRNA to its protein product translocated back into the nucleus. We merely postulate that the nuclear protein abundance is proportional with the factor b to the amount of mRNA τ hours earlier. The power n is the cooperativity index, which in the case of dimerization is given by n = 2. The self-inhibition is reflected by the delayed state variable x(t − τ) appearing in the denominator. Its high values decrease the net production rate of the mRNA ${{{\mathrm{ dx(t)}}}}/{{ {\mathrm{ dt}}}}$. Asymptotically, for very large values of x(t − τ), the production rate of mRNA tends to zero.

A typical choice of parameters is given by the following values: The basal transcription rate can be set to a = 1, because we have arbitrary units, the degradation rate to d = 0.2 h⁻¹, which corresponds to a typical mRNA half-life (Sharova et al. 2009), and the time delay to τ = 8 h, which is a characteristic delay between Per2 and phosphorylated nuclear PER2.

We stress that the model given by Eq. (3) is qualitative and we do not expect an exact quantitative correspondence of its predictions with the numerical values from experiments. However, many features of the oscillations can be predicted by the model equation. For example, parameter ranges of the delay τ and the degradation rate can be determined that allow the generation of oscillations.

1.2 A.2 Steady State and Its Stability

We generalize our model given by Eq. (1) to a one-dimensional DDE as follows:

$$ \frac{{\mathrm{ d}x(t)}}{{\mathrm{ d}t}}=g(x(t-\tau ))-d\cdot x(t), $$

(4)

where τ is a time delay, g(·) is a nonlinear function, and d > 0 is a degradation constant. An example is the nonlinear feedback in the form of

$$ g(x)=\frac{a}{{1+b{x^n}}}, $$

(5)

with the parameters a, b, n as discussed above.

A steady state of Eq. (4) satisfy $\frac{\text{$\scriptstyle {\mathrm{ d}x(t)}$}}{\text{$\scriptstyle {\mathrm{ d}t}$}}=0$ and is given by the nonlinear equation

$$ g(x)-dx=0, $$

which is in the case of Eq. (5) equivalent to

$$ a-d(1+b{x^n})x=0. $$

(6)

This is a nonlinear equation, which can be analytically solved only for small values of the exponent n. Generally, for arbitrary n, the steady state can be determined numerically.

Suppose that we have solved the steady state equation and the equilibrium is given by x = x ₀. We are now interested in the question of the stability of x ₀: that is, whether the system in the course of time will return back to equilibrium x ₀ or depart from it. For this purpose, we introduce the ansatz

$$ x(t)={x_0}+y(t), $$

(7)

with a small function of time y(t), which is the deviation of x(t) from its steady state value x ₀. In order to determine the stability of x ₀, we need to see, whether the derivation, y(t), would grow or decay in time. We emphasize that we are interested in what happens in the intermediate neighborhood of x ₀, which implies that y(t) is small.

Let us introduce our ansatz into the equations. We have for the left-hand side of Eq. (4)

$$ \frac{{\mathrm{ d}x(t)}}{{\mathrm{ d}t}}=\frac{{\mathrm{ d}y(t)}}{{\mathrm{ d}t}} $$

and correspondingly for its right-hand side by using a Taylor expansion up to the first order:

$$ \begin{array}{lll} g(x(t-\tau ))-dx(t) =g({x_0}+y(t-\tau ))-d{x_0}-dy(t)\approx \\ \approx g({x_0})+Jy(t-\tau )-d{x_0}-dy(t) =\\ =Jy(t-\tau )-dy(t).\end{array}$$

Here, J is the slope of the nonlinear function g in the steady state x ₀ given by

$$ J=\frac{\mathrm{ d}}{{\mathrm{ d}x}}g({x_0}). $$

Putting both sides together results in

$$ \frac{{\mathrm{ d}y(t)}}{{\mathrm{ d}t}}=Jy(t-\tau )-dy(t). $$

(8)

This is a DDE for the unknown function y(t). This equation is linear, and we can solve it by an exponential ansatz

$$ y(t)={y_0}{{\mathrm{ e}}^{{\lambda t}}}, $$

with the unknown complex number λ. The last ansatz, when substituted in Eq. (8), leads to

$$ {y_0}\lambda {{\mathrm{ e}}^{{\lambda t}}}=J{y_0}{{\mathrm{ e}}^{{\lambda (t-\tau )}}}-d{y_0}{{\mathrm{ e}}^{{\lambda t}}}, $$

and after dividing by y ₀e^λt it results in the transcendental characteristic equation for λ:

$$ \lambda =J{{\mathrm{ e}}^{{-\lambda \tau }}}-d. $$

(9)

If we find a value λ which solves Eq. (9), the function y(t) = y ₀e^λt would be a solution to Eq. (8). The growth or decay of y(t) = y ₀e^λt is determined by the sign of the real part of λ. If Re λ < 0, the function y(t) will decay in the course of time, which would correspond to a stable steady state x ₀. If Re λ > 0, the function y(t) grows, which means that the system departs from steady state x ₀ and the latter is unstable.

To sum up, given steady state x ₀, we have to solve Eq. (9) for the unknown λ, whose real part determines the stability of x ₀. Note that Eq. (9) depends on the steady state x ₀ through J, on the value of the time delay τ, and on the degradation rate d. Thus we expect that the stability of the steady state can be changed by tuning any of those parameters.

1.3 A.3 Oscillation Onset (Hopf Bifurcation)

Here, we are interested in a special situation, where the complex exponent λ has a zero real part. This corresponds to a parameter set, for which the stability of the steady state changes: if we change one of the parameters slightly, the real part will become nonzero and the steady state would either loose or gain stability, depending on the direction of the parameter change.

We introduce λ = μ + iω, which, when substituted in Eq. (9), results in

$$ \begin{array}{llll} \mu =J{{\mathrm{ e}}^{{-\mu \tau }}}\cos (\omega \tau )-d, \hfill \\ \omega =-J{{\mathrm{ e}}^{{-\mu \tau }}}\sin (\omega \tau ). \end{array} $$

We are interested in the situation when μ = 0, since it is associated with a change of stability of the steady state. At the same time when the steady state looses its stability, a small limit cycle emerges from the steady state. The period T of this limit cycle is close to 2π/ω. This scenario is known as a Hopf bifurcation. Going on with our calculations, the condition μ = 0 simplifies the above two equations to

$$ \begin{array}{lllll}\cos (\omega \tau )-d=0, \hfill \\-J\sin (\omega \tau )=\omega. \end{array}$$

Using cos² (ωτ) + sin² (ωτ) = 1, we have

$$ {J^2}={d^2}+{\omega^2}, $$

and $d=\sqrt{{{J^2}-{\omega^2}}}$. This in turn leads to the expression for the critical value of delay:

$$ \cos (\omega \tau )=\frac{{\sqrt{{{J^2}-{\omega^2}}}}}{J}=\sqrt{{1-\frac{{{\omega^2}}}{{{J^2}}}}}. $$

Moreover, we can express $\omega =\sqrt{{{J^2}-{d^2}}}$ and have cos (ωτ) = d/J, which gives the value for the critical delay

$$ \tau =\frac{{\arccos (d/J)}}{\omega }=\frac{{\arccos (d/J)}}{{\sqrt{{{J^2}-{d^2}}}}}. $$

We can analyze this equation a bit further: Owing to d ≥ 0 and J < 0, d/J is negative or zero. Hence, arccos (d/J) assumes values in between π/2 (corresponding to d/J = 0) and π (corresponding to d/J = −1). Thus, the value of τ is in between ${{ 2{ \pi }}}{{ /\omega }}$ and ${{ {\pi }}}{\rm{ /\omega }}$, which is exactly one-fourth to one-half of $T={{ {2\pi }}}{\rm{ /\omega }}$. Here, T approximates the period of the limit cycle, which emerges from the steady state in a Hopf bifurcation with λ = 0 + iω.

Our analytical calculations allowed us to specify the parameters where a Hopf bifurcation occurs: From $J=\sqrt{{{d^2}+{\omega^2}}}$ we see that a certain slope is needed. Furthermore, the delay must exceed a quarter of a period (6 h for circadian rhythms). Finally, the period is approximately proportional to the delay.

Appendix B: Explicit Delays Versus Reaction Chains

In the main text we studied the DDE

$$ \frac{{\mathrm{ d}x(t)}}{{\mathrm{ d}t}}=g(x(t-\tau ))-d\cdot x(t). $$

(10)

If x(t) represents the mRNA of a clock gene, the transcriptional inhibition is executed by its time-delayed value x(t − τ). In reality, the mRNA is spliced, exported, and translated to a protein. The protein forms complexes, can be posttranslationally modified, and will be translocated to the nucleus, where it regulates transcription. This series of events can be modeled in principle by studying all the corresponding intermediate concentrations and the resulting inhibitory complex. Since many quantitative details are not known, we introduced here the shortcut with an explicit delay.

It turns out that variables with explicit delays can be approximated by a chain of k intermediate auxiliary variables y _i (t):

$$ \begin{array}{lllll} {\frac{{\mathrm{ d}x(t)}}{{\mathrm{ d}t}} =g({y_k}(t))-dx(t),} \\ {\frac{{\mathrm{ d}{y_1}(t)}}{{\mathrm{ d}t}} =h(x(t)-{y_1}(t)),} \\ {\frac{{\mathrm{ d}{y_2}(t)}}{{\mathrm{ d}t}} =h({y_1}(t)-{y_2}(t)),} \\ \ldots \\ {\frac{{\mathrm{ d}{y_k}(t)}}{{\mathrm{ d}t}} =h({y_{k-1 }}(t)-{y_k}(t)).} \\ \end{array} $$

(11)

If we choose $h=k/\tau$, the chain of ODEs approximates the DDE (10) [this transformation is called the linear chain trick (MacDonald et al. 2008; Smith 2010)]. Here we sketch a short explanation for that claim.

We begin by ad hoc introducing a family of gamma functions G _h,q by

$$ {G_{h,q }}(t)=\frac{{{h^q}{t^{q-1 }}{{\mathrm{ e}}^{-ht }}}}{(q-1)! }. $$

A first useful observation is that the time derivative of the gamma functions satisfies the following relation

$$ \frac{\mathrm{ d}}{{\mathrm{ d}t}}{G_{h,q }}(t)=h({G_{h,q-1 }}(t)-{G_{h,q }}(t)),\quad q=2,3,\ldots,k, $$

which formally reminds the last k equations in (11). Using this result, a straight-forward differentiation shows that functions, y _q (t), given by the convolution integrals,

$$ {y_q}(t)=\int\limits_{{-\infty}}^t {x(s){G_{h,q }}(t-s)\mathrm{ d}s,\quad q=1,2,\ldots,k,} $$

(12)

indeed solve the last k equations in (11).

We now turn to the properties of y _k (t). It is formed by convolution integrals of x(t) with the gamma function G _h,k (t). These functions have mean value at $t=k/h$ and variance proportional to k. Thus, for large k, the functions G _h,k (t) become narrower, approximating a peak centered at $k/h$.

In the above integral (12) for q = k, we aim at localizing x(s) at the time moment t − τ by properly choosing G _h,k (t − s). The condition s = t − τ results in G _h,k (t − s) = G _h,k (τ). By an appropriate choice of parameter h, we tune the gamma function in such a way that its mean value is at the delayed time point t − τ. Owing that the mean value of G _h,k (τ) is at $\tau =k/h$, this leads to the sought condition for h: $h=k/\tau$. We conclude that the solution of the last equation of system (11), given by

$$ {y_k}(t)=\int\limits_{{-\infty}}^t {x(s){G_{h,k }}(t-s)\;\mathrm{ d}s} $$

with $h=k/\tau$ indeed approximates the delayed value of x: y _k (t) ≈ x(t − τ). The approximation becomes better for larger chain lengths due to narrower G _h,k for large k.

These calculations illustrate that chains of ODEs as studied in most clock models are closely related to DDEs analyzed above. The fact that long chains (i.e., large number k of the ODE equations) lead to sharper delays could be related to the observation that many posttranslational modifications (Vanselow et al. 2006), complex formations (Zhang et al. 2009; Robles et al. 2010), and epigenetic modification (Bellet and Sassone-Corsi 2010) are involved in generation of 24 h rhythms. We also refer to Forger (2011) for a somewhat similar study of the Goodwin model as a chain of three interconnected steps.

Appendix C: Modeling Details of Single Cell Oscillators

The dynamics of single cells was described either by a noisy limit cycle model or a noise-driven weakly damped oscillator model. For N cells, the governing deterministic differential equations read:

$$ \begin{array}{llll} \frac{{\mathrm{ d}{r_i}}}{{\mathrm{ d}t}} =-{\lambda_i}({r_i}-{A_i}), \hfill \cr \frac{{\mathrm{ d}{\varphi_i}}}{{\mathrm{ d}t}}=\frac{{2\pi }}{{{\tau_i}}},\quad i=1,2,\ldots,N. \end{array}$$

(13)

Here, λ _i is the radial relaxation rate, τ _i is the cell’s period, and A _i is the cell’s signal amplitude. All three parameters were estimated from experimental data as explained in (Westermark et al. 2009). Limit cycle oscillators have a nonzero amplitude A _i , whereas for damped oscillators we set A _i  = 0. The cell stochasticity was modeled by Gaussian noise sources added to the right-hand sides of Eq. (13). The variances of the noise terms were also estimated from experimental data as in (Westermark et al. 2009). For time integration of the resulting stochastic differential equation, we used the Euler–Murayama method.

For both limit cycle oscillators and weakly damped ones, three simulation protocols were realized.

• Synchronization by a pulse: At a certain time moment, we simultaneously shifted each oscillator in a specific direction by 120 dimensionless units (see Fig. 4).

• External periodic forcing: For the results presented in Fig. 5, we subjected oscillators to an external periodic force with a 24 h period and an amplitude of 0.5 dimensionless units. This driving force is much smaller than the typical oscillator amplitude of 10–20 (dimensionless units).

• Synchronization via mean field: In the third protocol (see Fig. 6), the oscillators were subjected to the mean field Z, which resulted from averaging across the ensemble:

$$ Z=\frac{1}{N}\sum\limits_N {{r_i}{{\mathrm{ e}}^{{\mathrm{ i}{\varphi_i}}}}}. $$

For linear damped oscillators, a saturation of the mean field at 20 dimensionless units was introduced in order to avoid amplitude explosion due to the linearity of the model.