A.2 Additional Results from the ODE model
A.2.1 Error Propagation of Half-Lives for the Production-Degradation Vector . 100
Here we derive the point estimate and the covariance matrix for the production-degradation vector. These estimates are needed for the chi-squared test described in Section 2.4.1 and implemented in the R package “patest”.
We estimate amplitudes and phases using harmonic regression: Ordinary least squares fitting to a linear combination of cosine- and sine-functions:
x(t) =m+acos(ωt) +bsin(ωt) +, (A.32) where x(t) stands for the abundance of a molecule at time t, and is an error term. The parameters aand bdefine (span) the circadian phase plane and constitute the vector x
x= a b
! ,
and the fitting procedure results in estimates of aand bdenoted µxwith symmetric covariance matrix covx:
µx= µa
µb
!
, (A.33)
covx= σa2 σa,b
σa,b σ2b
!
. (A.34)
If the measured time series is evenly sampled and consists only of fully measured periods the covariance matrix reduces to a diagonal matrix:
covx= σab2 0 0 σab2
!
. (A.35)
Estimates µx and covx are obtained from the residuals of the fit [215, 258]. In the same way, a vector p representing production phase and amplitude is obtained from measurements of transcriptional activity for the case of mRNAs (in practice, these are often also abundances, for example of nascent mRNA), together with an estimated covariance matrix covp.
In our model for rhythmic degradation we define the so called production-degradation vector pd, which is determined by a shift in the x- and y-coordinate of x depending on the half-life τ. Our chi-squared test is based on the difference between the production-degradation vector and the measured production vector. The null distribution for this difference can be estimated if an estimation for the covariance matrix of the production-degradation vector is available.
Such an estimate can be obtained from the abundance vector (harmonic regression estimates from mature mRNA-seq data in the present study), the production vector (harmonic regression estimates from nascent mRNA-seq data), and the estimated covariance matrices for these. Also needed is the estimated half-life of the mRNA molecule, and the variance of this estimate. The
A.2 Additional Results from the ODE model
half-lifeτ of the molecule is measured with meanµτ and standard deviation στ.
Thex-component of the production-degradation vectorpd follows:
xpd=
Here, γ is the mean degradation rate which is inversely related to the half-life: γ = log(2)/τ. The y-component of the vectorpdfollows:
ypd=b− aω
γ . (A.37)
We search the estimate µpd and covariance matrix covpd for the vectorpd:
µpd= µxpd
The components of the vector pd are functions of three variables, a, b, andτ. The mean µpd is then obtained from the definitions of pd, Eqns. (A.36 and A.37)
µpd= µa+ log(2)µbµτ µb− µlog(2)aωµτ
!
. (A.40)
To derive the covariance matrix we apply multivariate error propagation. In the general case of m functions yk innvariables x1, ..., xn=x, each function yk is linearized around the mean µx
To derive the covariance matrix cov of the functionsyk, the definition of variance is used:
where cov(y) is the covariance matrix of y,cov(x) is the covariance matrix of the variables x, and J|µ
x is them×n JacobianJij = ∂y∂xi
i evaluated at the mean of x. Using this method, the covariance matrix ofpd can be derived:
covpd=
If the time series is evenly sampled and over an integer number of periods, the covariance matrix has a simpler form:
A.2 Additional Results from the ODE model
A.2.2 Sensitivity
In Section 2.5 we describe how a small change of the relative amplitudes Aprod and Adeg can have a large effect on the phase φx. This can be quantified as the sensitivity of the phase φx
with respect to changes in the input parametersAprod,Adeg,φprod andφdeg. In the following we derive the sensitivity coefficient for the phase φx as a measure of the sensitivity.
First, we introduce the ratio between the amplitudes of the production term and the degradation rate: c=Aprod/Adeg. Then, the phase φx is calculated from Equation A.17:
φx= arctan 2
ccosφprod+ arctanωγ−cosφdeg+ arctanωγ, csinφprod+ arctanωγ−sinφdeg+ arctanωγ
. (A.48)
We now define the sensitivity coefficient Scφx as the derivative of the phase φx with respect to the relation c:
Scφx def= ∂φx
∂c = sin(φdeg−φprod)
1 +c2−2ccos(φdeg−φprod)) . (A.49) In Figure 2.2C we plotted the sensitivity coefficient for different values of c. Here, a large value means that a small change in the input parameters results in a large change in the phase φx. The sensitivity is large when production and degradation peak roughly at the same time, i.e.
φprod−φdeg ≈0. This effect becomes very pronounced when the amplitudesAprodand Adeg are similar, i.e. c≈1. The large sensitivity coefficient implies a phase inversion in this range of c.
This means that a small change in the relation of Aprod and Adeg can cause a phase shift of up to 12 h.
103
Appendix B: PDE Model - Aging Molecules
In this section I derive the PDE model used in Section 3.3, discuss different rate functions which describe different scenarios and present an analytical solution of the PDE for a specific case.
Furthermore, I formulate the inverse problem which can be used to recalculate unmeasured deadenylation rates TAIL-seq data.
B.1 Derivation of the PDE Model
Let x(t, d0) be the number of molecules of damage (d, d+ dd) at time t. Therefore x(t, d) is a density function in the damage d. If we integrate x(t, d) over all possible damage values we receive the total molecules concentration at time t:
X(t) = Z ∞
d=0
x(t, d) dd. (B.1)
The rate of change in molecules of a given damage interval ∆dis defined by
∂x(t, d)
∂t ∆d=
+ rate of entry atd
−rate of departure atd+ ∆d
−degradation
(B.2)
or
∂x(t, d)
∂t ∆d=J(t, d)−J(t, d+ ∆d)−v(t, d)x(t, d)∆d, (B.3) where v(t, d) is the degradation rate of molecules with damage dat time t. J(t, d) is a flux of molecules with damagedat time tin more detail examined below. Division by ∆dyields
∂x(t, d)
∂t =−J(t, d)−J(t, d+ ∆d)
∆d −v(t, d)x(t, d). (B.4)
Taking the limit ∆d→0 we receive a ’transport equation with advective flux’ for the molecules
∂x(t, d)
∂t =−∂J(t, d)
∂d −v(t, d)x(t, d). (B.5)
The flux J is not a flux in space as in the original transport equation but a flux in the damage domain and describes the movement of molecules towards higher damage. As a flux in space the damage-dependent flux depends on the molecule density x(t, d) and the speed of damage accumulationq(t). The speedq(t) is defined by the change of damage per time unitq(t, d) = dddt
105
and the flux J therefore is given by:
J(t, d) =x(t, d)q(t, d). (B.6)
The model equation Equation B.5 calculates thus to
∂x(t, d)
∂t =−q(t, d)∂x(t, d)
∂d −x(t, d)∂q(t, d)
∂d −v(t, d)x(t, d). (B.7) This equation is defined for positived, we assume there is no negative damage, and the damage is steadily increasing, hence we need a boundary conditionx(t,0) where molecules with no damage enter the system.
x(t, d) describes the molecule density in damage. For any acceptable solution x(t, d) the integral Equation B.1 is well defined and hence converges which requires that
d→∞lim x(t, d) = 0. (B.8)
The integral of the model equation EquationB.7 over damage dcalculates then Z ∞ The first integral on the right hand-side can be reduced by using partial integration
Z
Taking into account borders and Equation B.8 we receive Z ∞
The time development of molecule concentration reads therefore (EquationsB.9,B.13) dX
dt =x(t,0)q(t,0)− Z ∞
0
v(t, d)x(t, d)dd . (B.14) The equation for a general time development of a molecule concentration X where molecules