a. Principle of this model

The signal ﬂow model is a low-level model. This kind of model is used to predict its behavior with accuracy and is based on equations to compute different concentrations. All biological mechanisms are based on chemical equations linking the concentration of chemical species involved to each other. These chemical equations can be transformed into ordinary differential equations (ODEs), which are integrated to the model.

First we will see a presentation of the system and the different interactions between states. Then we will explain the biological equations used to make the model and we will show simulation results for each part of the system.

b. System's presentation

Our system is made up of a block named PCC (including the Phytochrome, ClpX and ClpP) and an other block, the DAS_GPF_ PIF chain where GFP is the TAG protein which is used to illustrate the mechanism. This system can be boiled down to the following four states scheme:

The first state is the state where PCC and DAS_GFP_PIF are free. Then PCC can form a complex with DAS, called PCC_DAS, with k

_{1,0} the coefficient of this complexation, or with PIF, called PCC_PIF, with k

_{0,1} the coefficient of this complexation. The last state is the PCC_DAS_PIF complex which is reached from PCC_DAS state by the complexation coefficient k

_{0,1}*c

_{1,1} or from PCC_PIF by k

_{1,0}*c

_{1,1}, where c

_{1,1} is the the coupling factor between the two sites of complexation.

The TAG protein can only be degraded in PCC_DAS or PCC_DAS_PIF conformation, because DAS must be linked to launch the protein degradation process. This is why there are single arrows between states PCC and PCC_DAS and between PCC_PIF and PCC_DAS_PIF.

c. Biological mechanisms

• Light-sensitive coefficient

k

_{1,0} is very small so the formation of PCC_DAS is scarce. k

_{0,1} is a coefficient which depends on the light's wavelength and this variance corresponds to the different physical structures of the PIF receptor (active or passive). With a 660 nm red light, PIF receptor is active and this coefficient is high but with a 730 nm infra-red light, PIF receptor is inactive and k

_{0,1} becomes very small. So to model this coefficient we use a Gaussian function centered on 660:

where μ = 660, σ = 5 and "light" the wavelength of the light. k

_{730} is the constant of complexation at 730 nm and is very small.

• Complexation reactions

To find the ODEs for the different reactions of complexation brought into play, we start with a simple complexation between two species, A and B:

where k

_{on} is the coefficient of the complex’s formation and k

_{off} the coefficient of the complex’s dissociation. The ODE of this reaction which gives the concentration of the complex AB is the following one:

k

_{degr} is only the constant of natural degradation of the complex. Using the same principle, we deduced the following equations for our system:

We insert these equations into our model and we obtained these simulation results:

• Computed concentrations of present species

To start, we will see the protein synthesis mechanism. We consider the synthesis of a P protein regulated by a set of protein X

_{p}. We obtain a set of two equations. The transcription of DNA into mRNA is based on Hill’s equation:

where k

_{tr} is the kinetic constant of transcription, K

_{p} the Hill’s constant representing the strength of the activator or repressor, n

_{p} the Hill’s coefﬁcient (positive for an activator and negative for a repressor) and d

_{mRNA} the degradation’s coefﬁcient of mRNA.

The same way, the synthesis of P protein from mRNA is deﬁned by:

where k

_{tl} is the kinetic constant of translation and d

_{P} the degradation coefﬁcient of protein P.

For each species, PCC, PIF and DAS, we use this mechanism to compute the species concentration produced, respectively PCC_prod, PIF_prod and DAS_prod. We must now compute the effective concentration of these species with these equations:

where PIF_l and DAS_l are the concentrations of PIF and DAS linked to the DAS_GFP_PIF chain already complexed at the other boundary (respectively DAS and PIF). We obtain the same equation for the different species. This is because we compute separately DAS, PIF and PCC’s concentrations for the complexation mechanism but we do have the same DAS_GFP_PIF free chain concentration as the free phytochrome concentration. So for the simulation results we just show the PCC and the DAS_GFP_PIF concentration because DAS and PIF concentrations are the same:

• TAG protein degradation

The last but the most important concentration to show is the TAG protein's (GFP) concentration. First we compute the concentration of GFP produced, GFP_prod, with the same mechanism introduced in the previous part. Then, the concentration of PCC fixed GFP (ready to be degraded), GFP_l, is given by:

Finally we obtain the effective concentration of GFP with the following equation, including the degradation (with d

_{GFP} degradation coefficient) of the fixed GFP:

The simulation of these equations gives the following result:

We have seen that this model is accurate but it has some drawbacks too. It is based on ODEs so it requires the use of a numerical solver, leading to slow simulations. It depends on parameters that should be estimated with experiments and the more the parameters are accurate, the more the simulations are close to real cell behavior.