### Book Chapter

### Reference

### A minimal model for the rhythmic protein per expression in Drosophila

### DEGLI AGOSTI, Robert

### DEGLI AGOSTI, Robert. A minimal model for the rhythmic protein per expression in Drosophila.

### In: Greppin H., Penel C., Broughton W, Strasser R.J. *Integrated plant systems* . Genève : University of Geneva, 2000. p. 229-236

Available at:

http://archive-ouverte.unige.ch/unige:42875

Disclaimer: layout of this document may differ from the published version.

### A minimal model for the rhythmic protein *per * expression in *Drosophila *

R. DEGLI AGOSTI

*Laboratory of Plant Biochemistry and Physiology, Plant Physiomatics, University of Geneva, *
*3 Place de l'Université, CH-1211 Genève 4, Switzerland E-mail: *

*Robert.DegliAgosti@bota. uni ge. ch *

### Introduction

Metabolic models with negative feedback regulation have been since a long time
proposed to explain sorne rhythmic phenomena's [ 4, 12]. They were temporarily
discarded from consideration in circadian rhythms on the basis of theoretical grounds
[see 2, for a discussion of this aspect]. However, it clearly appears now from
experimental results obtained in the insect *Drosophila, *the fungus *Neurospora *and
man that such kind of feedback's are involved in circadian rhythms or even
constituting the core of biological clocks [ 1,2,5,8]. Among the current most studied
systems is the *Drosophi/a *one. An important gene has been identified early and found
to be closely involved in the timing mechanisms: the *per *gene. Its mRNA and the
corresponding *per *protein oscillate in a circadian ( -24 h) manner. Since then, other
genes have also been identified [7,9,11]. We use here the concept of minimal modeling
approach to model the *per *rhythmic circadian expression with the minimum number of
variables (elements or state variables). W e will use the method of metabolic network
modeling with a negative feedback regulation [2].

The purpose of this paper is to illustrate how we can construct such models, starting from experimental data, to fmally obtain coherent simulations on computers. This is not so trivial since from the experimental model (Fig.1) it is not intuitively clear how this can effectively lead to a correct simulation of the observed (measured) variables.

In other words, how many people when observing Figure 1 could positively answer to
the question: is this model oscillating with a circadian (-24 h) rhythmicity? To answer
this question one *must *construct a mathematical model, obtain the evaluation of
parameters and then simulate it. To be precise, we should mention here that methods
do exist that extract values of parameters from raw data with just the mathematical
model only ( e.g. by non linear multiple least squares regression). This method will
allow to estimate parameters where often not enough experimental data is available
and is therefore very useful. However, the drawback of this is that the simulation will
be a little biased to necessarily work with artificially estimated parameters. ln this

R. Degli Agosti

paper we will follow the frrst method, namely obtain successively each parameters estimation from experimental data available.

### 1. The model

From an early "experimental" biological model presented in the literature for the
*per *gene regulation in *Drosophi/a *the following network is presented in Figure 1.

*Figure 1. Early model of the per protein regulation involved in circadian rhythms in *
*Drosophila. Very simplified metabolic network of the of protein per regulation *
according to [5]. RNA and per represent the relative concentrations of mRNA for the
*per protein and the protein per itself, respectively. RNA is synthesized with a basic *
flux (v o> controlled by the inhibition with the per protein (negative feedback f(pery).

The RNA is degraded with a first order kinetic law (k1). The synthesis of the protein
*per depends on the. RNA concentration according to a first order kinetics reaction *
law (k2), a delay has been experimentally observed in this reaction (-8 h). Finally
the per protein is destroyed with a first order kinetics reaction law (k3).

The corresponding differentiai equations that allow the modeling and the simulation of the dynamics of the concentrations of the 2 substances may be written as follows:

*d.RNA(t) * *v*_{0}*f(per(t))- k*_{1}*RNA(t) *
*dt *

*dper(t) *= *k*

*2**RNA(t- delay)- k*

*3**per(t) *
*dt * .

### 2. Evaluation of the model parameters

(1)

(2)

The data which will allow us to evaluate the parameters of the models are available
in the literature. W e will describe this important aspect in details here for illustrating
purposes. Data for RNA is found in [5] and for *per *protein in [13]. We have
normalized the original values to "normalized units of concentrations (n.u.c)", by
eliminating baselines and dividing each protein or mRNA *per *value by their respective
maximal values. They are presented in Figure 2.

Subj. Light

### -

~=! 0.8

### .s

*tA *
1: 0.6

### ii

0_{Loo }

## -

^{1: }

^{cu }

_{u }0.4

1: _{0 } 0.2

0

0.0

0 4

### a

12 16 20 24Time (hours)

Figure 2. Experimental data expressed in normalized units of concentrations
(n.u.c) of the *per *protein and *per *mRNA according to [13] and [6]. One might
observe that the synthesis and degradation of *per *rn RNA are triggered around
a threshold of 0.2 n.u.c. of perprotein.

2.1 The type of regula tory fonction

The regulatory function (negative feedback) of *per mRNA synthesis may be *
identified directly by a closer look to these experimental results. Indeed, when a
concentration higher of 0.2 n.u.c. of per is present the synth~sis is switched off.

Whereas, when a concentration lower than 0.2 n.u.c is present the synthesis is switched on (degradation is always active). This corresponds typically to one· of the most simple threshold regulatory function as follows:

*f(per(t)) *= {1 if *per(t) *~ 0.2
0 if *per(t) > 0.2 *
2.2 Estimating k1

(3)

When the synthesis ofmRNA is inhibited (whenper> 0.2 n.u.c), then using (3), the frrst term in equation (1) becomes 0, and we have:

*dARN(t) -k ARN(t) *

*dt * ^{1 } (4)

Which after integration gives the weil known solution:

(5)
This is a frrst order decreasing kinetics. From the original data one plots the natural
logarithm as function of the time and estimates k1 from the slope (Fig. 3) as 0.402 ±
0.049 h-^{1 }(correlation= 0.95).

R. Degli Agosti

0.5 . . . . - - - -- - - - -- - - , -0.5

~ -1.5

~

### .5

^{-2.5 }

-3.5

corr. = 0.95 0

0

-4.5 +---.----r---r----r---~---1

0 5 10 15 20 25 30

Time (hours)

2.3 Estimating vo

Figure 3. Evaluation of
*the per mRNA degra-*
dation kinetics constant k1
by the slope of the RNA
degradation. Data is the
same as in Figure 2, but
plotted on a natural
logarithmic scale. The
correlation coef-ficient is
0.95.

*When the synthesis ofper mRNA is switched on (i.e.,f(per(t)= 1), the equation (1) *
becomes:

*dRNA(t) *= *U*

*0 *- *0.402RNA (t) *
*dt *

Which after integration gives:

(1-*eo.402t) *
*RNA(t) *= *V** _{0 }*~----'-

0.402

This could be linearized in:

*RNA(t) *

### =

*u*

_{0}*B(t)*

(6)

(7)

(8)
Thus, the plot of experimentally measured RNA as a fonction of calculated B(t)
should be aline and its slope will be v0• It equals to 0.358 ± 0.015 n.u.c. h-^{1 }(Fig. 4).

1 ^{~ } 0~

Figure 4. Plot of the

### ,

Cl)### ...

^{::1 }

_{en }

_{0.8 }~ corr.

### =

0.93 from Figure 2) as a function*measured per mRNA (data*

0.6 0 of theoretical B(t) (see text).

### "'

Cl)

E 0.4 The slope of the line is v_{0. }

<( The correlation coefficient is

z 0:::

0.2- 0.93.

0 0

0 0.5 1.5 2 2.5 3

B(t) function

2.4 Estimating the *delay *

The *delay *value is an experimentally observed feature. It represents a time latency
between the observed maxima of RNA and *per *protein (see [5]), *delay= *-8 h.

2.5 Evaluation of k2 and k3•

These evaluations are more difficult to obtain, since datais far less precise (see Fig.

2). It is possible to obtain an estimation of k_{3 }*(per *protein degradation) from Figure 2
by estimating that the half-life (t112) of the degradation kinetics is -2 h. This represents
a k3 of 0.35 h-^{1 }(k3 = ln(2)/t112). A rough estimation of k2 is obtained if we consider that
a steady state is observed at 1.0 n.u.c. Indeed, at *per *steady state *(per(teq)), *equation (2)
becomes:

(9)
*per(teq) *= 1.0 n.u.c, k_{3 }=0.35 h-^{1}, the value of *RNA(teq-8) *can be estimated directly from
Figure 2, since *per(teq) *is observed at- 22 to 24 h, then *RNA(teq-8) *at 14 to 16 his 1.0
n.u.c. Substituting these numbers in (9), it cornes that k_{2 }= 0.35 h-^{1}.

### 3. Model simulation

Once each parameter has been estimated, the dynamics of the system is described by the following equations:

### ~:(t)

*0.358/(per(t))- 0.402RNA(t)*

*dper(t) *= *0.35RNA(t-8)- 0.35 per(t) *
*dt *

(10)

(11)

The regulation function *f(per(t)) * is defmed by (3). It is clear that, from these
equations, it is rather difficult to intuitively see if they oscillate and, more difficult
even, is at which period. One way is to simulate their resolution by numerical methods
(numerical integration by a runge-kutta 4th order method, e.g.) in order to obtain the
*RNA *and *per *dynamics.

The simulation is presented in Figure 5: the system oscillates with a periodicity of
-26 h, which is very near the one experimentally observed (-24 h). A rough sensitivity
study has been realized by varying sorne parameters to check the stability of the
model. With a *delay *of 6 h instead of 8 h, the period is still circadian (21.6 h).

Changing k_{2 }= k_{3 }(actually at 0.35 h-^{1}) to 0.175 h-^{1 }orto 0.70 h-^{1 }gives a periodicity of
29.8 h and 23.2 h respectively. The model is thus robust with respect to changes in
cri ti cal parameters (particularly the ones that were the most difficult to evalua te).

R. Degli Agosti

1.0 0.8 0.6 0.4 0.2

0.0

0 10 20 30 40 50

Time (hours)

*Figure 5. Simulation of the per protein and its mRNA dynamics in the Drosophila *
model. The period of oscillation is circadian (26 h). The conœptual and mathematical
(before and after evaluation of the parameters from the experimental data) are
presented in Figure 1, equations (1 & 2) and equations (3,10 & 11) respectively.

### 4. Comparison between real and simulated data

Figure 6 shows the comparison between the original and simulated data for *per *
mRNA (see also Figure 2). From visual inspection the fitting seems to be rather
correct.

• 0 1.0

~

### "2

^{0.8 }

### 5-ro

~ :J 0.6

### cv

E E lii 0.4<(<(

### z z

0.2~~

0.0

0 10

Time (hours)

*Figure 6. Comparison between the original experimental data of per mRNA *
*and the results of the simulated per mRNA according to equations *
(3, 10, 11). Units of mRNA are in n.u.c.

### Conclusion

More recent works have shown that the oscillation of the *per *protein is also
dependent of another essential element: the protein *tim *[7, 9, 11 ], that possess a specifie
gene distinct from *per. *As it is possible to observe in this work, the model we have
elaborated doesn't contain these informations. However, it seems to rather correctly

describe the endogenous oscillation. The model is systemic, it possess a great power of simplification (aggregation) of a complex phenomenon. This choice is deliberate.

More precisely, the existence of a delay factor is actually corresponding to a lot of molecular 1 physiological 1 biochemical reactions that were introduced in this kind of model only by their fmal simplified result, but with a correct dynamical equivalent.

IndeecL as sorne authors have suggested, this delay might be explained by the necessity for per to be stabilized by tim for the transport of this complex from the cytoplasm to the nucleus [9].

In other words, in order to elaborate a functional (imperfect, but useful) systemic model, it is not necessary to know ali the detailed molecular mechanisms of the phenomenon, but only sorne essential dynamical characteristics, only sorne aggregating parameters are needed. It is clear that the identification of such key parameters is of the utmost importance for modeling complex phenomena's with minimal systemic modeling. Other important domains were complexity is the rule like ecological modeling are using this kind of approach with success since a long time now [10].

There are little doubts that once any system has been somehow reduced in sorne of its components, one needs again someway to integrate them in order to test the modeled "behavior" with the experimental observations. One method, among others, to achieve integrated approaches is by modeling and simulating techniques.

### References

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R. Degli Agosti

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