or
∆φx= arctan ω
γ
.
(A.8)
The last expressions show that in the case of constant degradation, the mean concentration equals the non-oscillatory steady-state (i.e., Aprod=Adeg = 0 anddx/dt = 0 in Equation A.5).
In Equations A.8, the range of ∆φx is limited to the range of the arctan function for positive argument, which is between 0 and π/2, so that ∆φx can only vary between 0 and 6 hours on the circadian time scale. Another direct conclusion from Equations A.8 is that M → ∞ and Ax → 0 for γ → 0. This means that for long half-lives, magnitudes increase but amplitudes vanish, so that circadian rhythms are lost.
A.1.2 Rhythmic Degradation:
Approximation with Fourier Expansion and Harmonic Balancing
Floquet theory guarantees that the solution to Equation A.1 is an ω-periodic function [255].
Thus, after an initial transient, the abundance x can be approximated by Fourier expansion to the order of n:
x(t)≈A0+A1cos(ωt) +B1sin(ωt) +...+Ancos(nωt) +Bnsin(nωt). (A.9) The exact Fourier coefficients in such a truncated Fourier expansion cannot be calculated in closed form. Therefore, we use the following idea: We plug the truncated Fourier expansion Equation A.9 into the model, Equation A.1, and compare the coefficients. This leads to a system of linear equations which can be solved to obtain the approximate coefficients A0, A1, B1, ...,An,Bn. These coefficients are not the exact Fourier coefficients, since we use a truncated Fourier series to obtain the describing linear equations. Below, we show that the error of this approximation vanishes as the expansion becomes longer, and that the coefficients then converge to the exact Fourier coefficients.
Numerical solutions generally exhibit shapes close to cosine-functions, this means that al-ready a Fourier expansion to the first order is generally sufficient, see also numerical validation,
87
Section A.1.4, to describe the solution of the full model Equation A.5. Hence we obtain A0 =k
γ
2(γ2+ω2) +AprodAdegγ(ωsin(∆φdeg)−γcos(∆φdeg)) 2(γ2+ω2)−A2degγ2
A1 =2kγ(Aprod−Adegcos(∆φdeg))−2kAdegωsin(∆φdeg) 2(γ2+ω2)−A2degγ2 − AprodA2degkγsin2(∆φdeg)
2(γ2+ω2)−A2degγ2
B1 =2kω(Aprod−Adegcos(∆φdeg)) + 2kAdegγsin(∆φdeg) 2(γ2+ω2)−A2degγ2 − AprodA2degkγsin(∆φdeg) cos(∆φdeg)
2(γ2+ω2)−A2degγ2 .
(A.10)
The relative amplitudes Aprod and Adeg take values smaller than or equal to 1, and the same is true for the absolute values sin(∆φdeg) and cos(∆φdeg). Mixed products of these terms as AprodA2degsin(∆φdeg) cos(∆φdeg) and AprodA2degsin2(∆φdeg) are small, and are therefore ne-glected in the following, to obtain a more convenient approximation:
A0 = k γ
2(γ2+ω2) +AprodAdegγ(ωsin(∆φdeg)−γcos(∆φdeg)) 2(γ2+ω2)−A2degγ2
A1 ≈ 2kγ(Aprod−Adegcos(∆φdeg))−2kAdegωsin(∆φdeg) 2(γ2+ω2)−A2degγ2
B1 ≈ 2kω(Aprod−Adegcos(∆φdeg)) + 2kAdegγsin(∆φdeg) 2(γ2+ω2)−A2degγ2 .
(A.11)
A Fourier Series to the first order describes a harmonic function as linear combination of a cosine function and a sine function. We reformulate this linear combination to gain a cosine function with a relative amplitude Ax and a phase ∆φx [256] (cp. also Equation A.7):
x(t) =A0+A1cos(ωt) +B1sin(ωt) =Mx(1 +Axcos(ωt−∆φx)), (A.12) where
Mx =A0
Ax = 1 A0
q
A21+B12
∆φx= arctan 2 (B1, A1).
(A.13)
The function arctan 2(y, x) is the arctangent function with two arguments, which computes the principal value of the argument function applied to the complex number x+iy. The definition of arctan 2(y, x) is given in Figure A.1. With this ansatz we neglect higher order terms in the Fourier expansion, which is justified below.
A.1 ODE Model
Figure A.1: Definition of the function arctan 2(y, x). The function arctan(y/x) covers a range from−π/2 to +π/2, which would not be suitable to describe an arbitrary phase shift. To consider all possible phases, we use the function arctan 2(y, x), which covers the full range from−πto +π and is defined as follows: ∆φx= arctan 2(y, x) is the angle between (x, y) and (y= 0, x >0) in the x-y plane. If y >0, the angle is taken counterclockwise from 0 toπ, and ify <0, then the angle is taken clockwise from 0 to−π.
With Equations A.11 we obtain the following expressions:
Mx =k γ
2(γ2+ω2−Aprod2Adegγ(ωsin(∆φdeg) +γcos(∆φdeg))) 2(γ2+ω2)−γ2A2deg
Ax =
γqA2prod+A2deg−2AprodAdegcos(∆φdeg) pγ2+ω2−Aprod2Adegγ(ωsin(∆φdeg) +γcos(∆φdeg))
∆φx= arctan 2 ω(Aprod−Adegcos(∆φdeg))−Adegγsin(∆φdeg), γ(Aprod−Adegcos(∆φdeg)) +Adegωsin(∆φdeg)
! .
(A.14)
For the special case of constant degradation (Adeg = 0), these equations reduce to the exact expressions, Equations A.8. As defined previously, we used the relative phase ∆φdeg = φdeg−φprod.
Vector Representation of Phase and Amplitude
The Fourier expansion resulted in a good approximation for biologically relevant parameter val-ues, see also numerical validation, Section A.1.4, and if needed it is possible to use the same ap-proach and expand to a higher order to achieve higher accuracy. However, Equations A.14 do not provide intuitive insight into the properties of phase and amplitude of a rhythmically degraded biomolecule. In the main text (see also Section 2.5), we introduce phase and relative amplitude as a result of a vector addition in the complex plane (Equations 2.10, 2.11. In the following we derive this description from the Fourier expansion derived above.
The phase ∆φxis formulated in Equations A.14 with an arctan 2(y, x) function. This equation can be reinterpreted as a calculation in a two-dimensional vector space. Then, the phase is the
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angle of a vector with the entriesx and y:
∆φx= arctan 2(y, x)def= angle x y
! .
Now the phase is described as the result of a vector calculation. The expression in Equation A.14 can be rewritten as an angle:
∆φx= angle
Here,R(∆φx) is a rotation matrix that describes a counterclockwise rotation by the angle ∆φx: R= cos (∆φx) −sin (∆φx)
sin (∆φx) cos (∆φx)
! .
From the two-dimensional vector space to the complex plane, a rotation matrix by the angle φ translates to eiφ and a vector with positive entries x and y to px2+y2eiarctan(y/x). Thus,
To derive the Fourier approximation we reduced the number of parameters by describing all phases in relation to the phase of the production. We now return to a description in the absolute time frame and rewrite ∆φdeg=φdeg−φprodand ∆φx=φx−φprod. This gives for the calculation A.16 in the complex space:
φx= argAprodeiφprod −Adegeiφdeg+ arctan ω
γ
. (A.17)
This is the vector representation of the phase in the main text (Equation 2.11).
We introduced the termAprodeiφprod −Adegeiφdeg in the main text as the so-called “production-degradation vector”. It describes the influence of only the oscillatory part of the production and degradation rate independent from their mean rates. The absolute value of the effective production is:
Aprodeiφprod−Adegeiφdeg=qA2prod+A2deg−2AprodAdegcos(φdeg−φprod). (A.18) Replacing ∆φdeg = φdeg −φprod, the square root in Equation A.18 can be identified in the previously derived amplitude of x (see Equation A.14), which yields:
Ax=
γAprodeiφprod−Adegeiφdeg
pγ2+ω2−Aprod2Adegγ(ωsin(φdeg−φprod) +γcos(φdeg−φprod))
. (A.19)
A.1 ODE Model
We refer to the factor Aprod2Adegγ(ωsin(φdeg−φprod) +γcos(φdeg−φprod)) as a correction fac-tor and replace it by C in the main text. With this step we have derived the vector description of the amplitude (Equation 2.10 in the main text).