Marc Sturrock
email: msturrock@maths.dundee.ac.uk

Alan J Terry

Dimitris P Xirodimas

Alastair M Thompson

Mark A J Chaplain

Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways

Keywords: Intracellular signalling, Negative feedback, Oscillatory dynamics, Spatial model, Hes1, p53, Cancer

come

Introduction

Negative feedback loops controlling the concentrations of key intracellular proteins are prevalent in a diverse range of important cellular processes. Examples include inflammation, meiosis, apoptosis and the heat shock response [START_REF] Alberts | Molecular Biology of the Cell[END_REF][START_REF] Lahav | Dynamics of the p53-Mdm2 feedback loop in individual cells[END_REF][START_REF] Fall | Computational Cell Biology[END_REF]. Experimental data reveal that pathways containing negative feedback loops exhibit sustained oscillations [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF][START_REF] Geva-Zatorsky | Oscillations and variability in the p53 system[END_REF][START_REF] Nelson | Oscillations in NF-κB signaling control the dynamics of gene expression[END_REF][START_REF] Shankaran | Rapid and sustained nuclear-cytoplasmic ERK oscillations induced by epidermal growth factor[END_REF]). This is not unexpected given the interactions involved in a negative feedback loop.

[Fig. 1

about here.]

A generic example of a negative feedback loop with variables x and y is shown in Fig. 1. An increase in x causes y to increase, which in turn results in the inhibition of x. After x begins to decrease, this will also cause y levels to diminish, eventually allowing x to increase again. This process repeats and thus produces oscillations in x and y. We will consider intracellular negative feedback loops in this paper, specifically those involving transcription factors, critical contributors to cellular homeostasis and, when dysfunctional, to disease processes. Transcription factors are regulatory proteins required to initiate or regulate transcription in eukaryotic cells. They act by binding to specific DNA sequences in the nucleus, either promoting or inhibiting the binding of RNA polymerase to DNA and are of fundamental importance for normal cellular function. mRNA is transcribed in the nucleus and subsequently protein is translated in the cytoplasm.

Most previous mathematical models examining intracellular negative feedback systems have taken a simplified approach using ordinary differential equations (ODEs) and have not considered the different spatial structures within a cell.

Such ODE models have used delays to account for the processes of transcription, translation and transport within the cell. However, in this paper we consider modelling the spatial interactions explicitly, using partial differential equation (PDE) models, with the knowledge that the localisation of certain proteins is critical for normal cellular functioning.

The format of this paper is as follows. In the next section, we begin by considering a simple example of a feedback inhibition system, namely that of hes1 mRNA and Hes1 protein. We present a mathematical model of this system, showing how our model builds on previous work to reflect the biology in greater depth and present our computational simulation results. In the subsequent section we consider the p53-Mdm2 system, where localisation of proteins is of particular importance since it has implications for cancer. Again we develop a mathematical model, justifying the formulation and present our simulation results. In the final section, we conclude with a discussion of our results and an indication of future work in this area.

The Hes1 System

One of the most thoroughly investigated feedback inhibition systems, and therefore a good place to begin modelling, is the system involving the transcription factor Hes1 [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF][START_REF] Momiji | Dissecting the dynamics of the Hes1 genetic oscillator[END_REF][START_REF] Agrawal | Computational models of the notch network elucidate mechanisms of context-dependent signaling[END_REF]). Hes1 plays a role in somitogenesis, the developmental process whereby the vertebrate embryo becomes segmented. Somitogenesis depends on a segmentation clock. The existence of a direct feedback inhibition loop, coupled with available experimental data [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF], suggests that the oscillatory expression of Hes factors play a central role in maintaining the segmentation clock. Hes1 represses the transcription of its own gene through direct binding to regulatory sequences in the hes1 promoter. The basic interactions of this system (see Fig. 2) are the same as those described previously for the generic variables x and y. Hes1 protein is produced by hes1 mRNA and then goes on to inhibit its own mRNA and so forth, with the result that the system oscillates with a period of around 120 minutes.

[Fig. 2 about here.]

Hes1 Mathematical Model

Mathematical modelling of intracellular regulatory systems began with the work of Goodwin [START_REF] Goodwin | Oscillatory behavior in enzymatic control processes[END_REF]). Mahaffy and co-workers developed this work introducing spatial structure (although only in 1-dimension) and also delays accounting for transcription and translation [START_REF] Mahaffy | Models of genetic control by repression with time delays and spatial effects[END_REF][START_REF] Busenberg | Interaction of spatial diffusion and delays in models of genetic control by repression[END_REF][START_REF] Mahaffy | Genetic control models with diffusion and delays[END_REF]). However, Monk was the first to develop a mathematical model of the Hes1 system and validate it with biological data [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF]. Denoting by [m] and [p] the concentrations of hes1 mRNA and Hes1 protein respectively, the basic reaction kinetics for this system can be modelled using ordinary differential equations (ODEs) as follows:

d[m] dt = α m 1 + ([p]/ p) h -μ m [m],
(1)

d[p] dt = α p [m] -μ p [p]. (2)
The first term on the right hand side of Eq. ( 1) is a Hill function, which decreases as the protein concentration increases, modelling repression by the Hes1 protein.

The parameter α m is the basal rate of transcript initiation in the absence of Hes1 protein and p is the concentration of Hes1 that reduces the rate of initiation of hes1 transcripts to half of its basal value (the repression threshold) and h is a Hill coefficient. The second term represents the natural degradation of the hes1 mRNA, with parameter μ m . The first term on the right hand side of Eq. ( 2) is the Hes1 protein production term from translation of hes1 mRNA with parameter α p and the second term represents Hes1 protein degradation with parameter μ p . It can be proved mathematically that it is impossible for Eqs. ( 1), (2) to generate sustained oscillatory dynamics [START_REF] Lewis | Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator[END_REF]. In order to model the intracellular processes more accurately, Monk [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF] introduced delays to Eqs. ( 1), (2) to account for the processes of transcription and translation. This leads to a system of delay differential equations (DDEs):

d[m] dt = α m 1 + ([p(t -τ m )]/ p) h -μ m [m],
(3)

d[p] dt = α p [m(t -τ p )] -μ p [p], (4)
where τ m and τ p are transcriptional and translational delays, respectively. With the addition of the delay terms, Monk showed that it was possible to obtain sustained oscillations [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF]. These results accurately reflect experimental data well but shed no light on where the mRNA and protein are located in the cell. We now extend the above models and consider spatial interactions within the cell explicitly as shown schematically in Fig. 3, i.e., we consider the nucleus and cytoplasm as two distinct spatial compartments separated by the nuclear membrane, and the cytoplasm enclosed within the outer cell membrane. Transcription occurs exclusively in the nucleus and protein synthesis occurs exclusively in the cytoplasm. We assume that the main mechanism governing the spatial movement of mRNA and protein between the nucleus and cytoplasm is diffusion. Denoting by [m n ], [m c ] and [p n ], [p c ] the concentrations of nuclear and cytoplasmic hes1 mRNA and nuclear and cytoplasmic Hes1 protein respectively, the system of equations describing the spatio-temporal evolution of hes1 mRNA and Hes1 protein concentrations is now

∂ [m n ] ∂t = D m n ∇ 2 [m n ] + α m 1 + ([p n ]/ p) h transcription -μ m [m n ], (5)
∂ [m c ] ∂t = D m c ∇ 2 [m c ] -μ m [m c ], (6)
∂ [p c ] ∂t = D p c ∇ 2 [p c ] + α p [m c ] synthesis -μ p [p c ], (7)
∂ [p n ] ∂t = D p n ∇ 2 [p n ] -μ p [p n ], (8)
where parameters D i denote the diffusion coefficients for each species. We apply continuity of flux boundary conditions across the (internal) nuclear membrane and zero-flux boundary conditions at the outer cell membrane:

D m n ∂ [m n ] ∂ n = D m c ∂ [m c ] ∂ n and [m n ] = [m c
] at the nuclear membrane, ( 9)

D p n ∂ [p n ] ∂ n = D p c ∂ [p c ] ∂ n and [p n ] = [p c
] at the nuclear membrane, ( 10)

∂ [m c ] ∂ n = 0 at the cell membrane, (11)
∂ [p c ] ∂ n = 0 at the cell membrane, ( 12
)
where n is a unit normal. In biological terms the boundary conditions allow import and export of mRNA and protein across the nuclear membrane, while ensuring that all molecules remain within the cell and are not exported across the cell membrane. The system is closed by imposing appropriate initial conditions for the mRNA and protein concentrations.

Eqs. ( 5) -(8) represent a system of reaction-diffusion equations modelling the spatio-temporal evolution of the Hes1 system. The same reaction kinetics from the ODE model ( 1), (2), are retained but are now also coupled with diffusion to model explicitly protein and mRNA transport within a cell, i.e., molecules move from the nucleus to the cytoplasm and cytoplasm to nucleus across the nuclear membrane. The PDE system reflects the reality mRNA is transcribed from DNA exclusively in the nucleus and that protein is translated from mRNA exclusively in the cytoplasm, i.e., there are production terms only for [m n ] (in equation ( 5)) and [p c ] (in equation ( 7)). Finally, we make the assumption that the translation of proteins from mRNA in the cytoplasm occurs some distance away from the nucleus and outside the endoplasmic reticulum (ER), since proteins produced in the ER are mainly either exported to the exterior of the cell or transported to other membrane structures such as the Golgi apparatus, lysosomes and endosomes (see Appendix for details) [START_REF] Alberts | Molecular Biology of the Cell[END_REF][START_REF] Mahaffy | Models of genetic control by repression with time delays and spatial effects[END_REF][START_REF] Busenberg | Interaction of spatial diffusion and delays in models of genetic control by repression[END_REF]. In order to model this, we modify Eq. ( 7) as follows:

∂ [p c ] ∂t = D p c ∇ 2 [p c ] + H 1 (x, y)α p [m c ] -μ p [p c ], (13)
where H 1 (x, y) is a function localising the protein production whose specific form will be given after the nondimensionalisation of the system.

[Fig. 3 about here.]

We nondimensionalise ( 5), ( 6), ( 13) and ( 8) with appropriate reference values (scaling variables) as follows:

[m n ] = [m n ] [m 0 ] , [m c ] = [m c ] [m 0 ] , [p n ] = [p n ] [p 0 ] , [p c ] = [p c ] [p 0 ] , t = t τ , x = x L , y = y L , (14)
where [m 0 ], [p 0 ] are reference concentrations, τ is a reference time, and L is a reference length. Using this scaling, ( 5), ( 6), ( 13) and ( 8) become:

∂ [m n ] ∂t = D * m n ∇ 2 [m n ] + α * m 1 + (p * [p n ]) h -μ * m [m n ], (15)
∂ [m c ] ∂t = D * m c ∇ 2 [m c ] -μ * m [m c ], (16)
∂ [p c ] ∂t = D * p c ∇ 2 [p c ] + H 1 (x, y)α * p [m c ] -μ * p [p c ], (17)
∂ [p n ] ∂t = D * p n ∇ 2 [p n ] -μ * p [p n ], (18)
where

D * m n = τD m n L 2 , α * m = τα m [m 0 ] , p * = [p 0 ] p , μ * m = τμ m , D * m c = τD m c L 2 , ( 19
)
D * p c = τD p c L 2 , α * p = τ[m 0 ]α p [p 0 ] , μ * p = τμ p , D * p n = τD p n L 2 ,
and

H 1 (x, y) = ⎧ ⎪ ⎨ ⎪ ⎩ 0, if x 2 2 + y 2 ≤ 0.25, 1, if x 2 2 + y 2 > 0.25.
H 1 (x, y) is illustrated graphically in the Appendix. We apply zero initial conditions, zero-flux boundary conditions at the cell membrane and flux continuity boundary conditions across the nuclear membrane:

[m n ] = [m c ] = [p n ] = [p c ] = 0, at t = 0, ( 20
)
D * m n ∂ [m n ] ∂ n = D * m c ∂ [m c ] ∂ n and [m n ] = [m c ] at nuclear membrane, ( 21
)
D * p n ∂ [p n ] ∂ n = D * p c ∂ [p c ] ∂ n and [p n ] = [p c ] at nuclear membrane, (22) ∂ [m c ] ∂ n = 0 at cell membrane, (23)
∂ [p c ] ∂ n = 0 at cell membrane. ( 24
)
In the absence of available experimental data (cf. [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF]), we take reference concentrations to be [m 0 ] = 0.05μM and [p 0 ] = 1μM (in line with the relative concentrations of mRNA and protein, and also in line with estimates for equivalent p53 data which will be discussed in Section 3.1, [START_REF] Ma | A plausible model for the digital response of p53 to DNA damage[END_REF]). In preliminary simulations of ( 15) -( 18) it was observed that a period of oscillation was approximately 200 time units. Hence, knowing that the period of oscillation of Hes1 is approximately 2 hours [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF], we obtain an appropriate reference time τ as follows: 200τ = 2hrs which implies τ = 36s.

For our simulations we use a 2-dimensional cell domain composed of two ellipses to represent the nucleus and cytoplasm. The nucleus has a major axis of length 0.8 units and minor axis of length 0.5 units. The cytoplasm has a major axis of length 3 units and a minor axis of length 2 units. We assume a cell to be of length 30μm, and hence take the reference length L = 10μm, i.e., in Figs. 6 and7 the nondimensional cell width is equal to 3 units or 3L = 30μm.

Parameter Estimation

The following parameter values were used in our simulations of the nondimensional Hes1 system: 19) and ( 25) the dimensional value of the diffusion coefficient, D m n , can be calculated:

D * m n = D * m c = D * p c = D * p n = 7.5 × 10 -4 , α * m = 1, (25) h = 5, p * = 1, μ * m = μ * p = 0.03, α * p = 2. From (
D m n = L 2 D * m n τ = 2.08 × 10 -11 cm 2 s -1 .
Similarly, we are able to calculate the remaining dimensional parameter values:

D m n = D m c = D p c = D p n = 2.08 × 10 -11 cm 2 s -1 , α m = 0.02Ms -1 , h = 5, p = 1 × 10 -6 M, μ m = μ p = 8.33 × 10 -4 s -1 , α p = 1.11s -1 .
From a number of simulations carried out on system (15) -( 18), we have calculated a range of diffusion coefficients (if all other parameters remain unchanged) for mRNA and protein for which the system exhibits oscillatory dynamics: 1.67 × 10 -11 to 9.72 × 10 -11 cm 2 s -1 . We have also calculated a range of mRNA degradation rates (1.67 × 10 -4 to 1.17 × 10 -3 s -1 ), protein degradation rates (1.94 × 10 -4 to 1.06 × 10 -3 s -1 ) and Hill coefficients (h ≥ 4), for which the system exhibits oscillations.

We note that the kinetic parameters we have used are in line with previous estimates [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF]. Regarding the values of our diffusion coefficients, we note that the results of [START_REF] Klonis | Fluorescence photobleaching analysis for the study of cellular dynamics[END_REF] show that diffusion rates of macromolecules in the cytoplasm and nucleus are up to 100-times slower than in aqueous buffers, which would then bring our estimated values of around 10 -11 -10 -10 cm 2 s -1 in line with the estimates of [START_REF] Seksek | Translational diffusion of macromolecule-sized solutes in cytoplasm and nucleus[END_REF] (10 -8 cm 2 s -1 ) and [START_REF] Matsuda | Direct measurement of protein dynamics inside cells using a rationally designed photoconvertible protein[END_REF] (10 -9 cm 2 s -1 ).

Hes1 Numerical Simulations

We solved PDE system (15) -(18) numerically using the COMSOL/FEMLAB package, which uses the finite element technique. Triangular basis elements and Lagrange quadratic basis functions along with a backward Euler time-stepping method for integrating the equations were used in all simulations.

Fig. 4 shows the total concentrations of hes1 mRNA and Hes1 protein over time in the nuclear compartment, while Fig. 5 shows the total concentrations in the cytoplasm. Both sets of results show oscillatory dynamics of the Hes1 system. The plots presented in Figs. 6 and7 show how the hes1 mRNA and protein concentrations vary spatially as well as temporally within the cell. The mRNA is produced inside the nucleus and by t = 60 minutes has started to cross the nuclear membrane to enter the cytoplasm (Fig. 6). In the cytoplasm the mRNA is translated into protein, which then diffuses back into the nucleus and represses the production of its own mRNA (t = 120 minutes). The mRNA concentration has clearly depleted by t = 120 minutes, reflecting the period of the temporal oscillation seen in Figs. 4, 5. As can be seen from Fig. 7, there is a delay in the rise of protein concentration after t = 0 as it takes time for the mRNA to be produced and exported to the cytoplasm. By t = 60 minutes the protein levels have clearly risen in the cytoplasm and have reached the nucleus. At t = 120 minutes the protein concentration has decreased significantly, due to the inhibition of mRNA transcription by the protein.

[Fig. 7 about here.] Furthermore we have performed simulations of the Hes1 system on 3D spherical and ellipsoidal domains. In Fig. 8 we present snapshots of the Hes1 protein concentration in the cytoplasm obtained from 3-dimensional simulations of system ( 15) -( 18) on both spherical and ellipsoidal domains (all parameter values as per previous simulations). We found that using different domain geometries (of approximately the same volume) does not affect the solution qualitatively, i.e., oscillatory solutions were still observed. However, as is to be expected, the precise spatial distribution of concentrations at a given time varies between the different geometries.

[Fig. 8

about here.]

The simulation results from our system (15) -( 18) reflect qualitatively the dynamics of the Hes1 pathway, a simple example of control of protein expression by a negative feedback loop [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF].

The main advantage of using a system of PDEs to model intracellular reactions is that the PDEs enable spatial effects to be examined explicitly. To this end, we carried out a number of simulations on system (15) -( 18), where we varied the values of the diffusion coefficients (all other parameters remaining unchanged) of the mRNA and protein. By doing this we found a range of values for the diffusion coefficients D m n , D m c , D p c , D p n where the system exhibits oscillatory dynamics, i.e., for 1.67 × 10 -11 cm 2 s -1 ≤ D m , D p ≤ 9.72 × 10 -11 cm 2 s -1 oscillations were observed, while no oscillations were observed outside this range. Hence if diffusion of mRNA and protein is either too slow or too fast, oscillations do not occur. In a similar manner, we also calculated a range of mRNA degradation rates (μ m from 1.67 × 10 -4 to 1.17 × 10 -3 s -1 ), protein degradation rates (μ p from 1.94 × 10 -4 to 1.06 × 10 -3 s -1 ) and Hill coefficients (h ≥ 4), for which the system exhibits oscillations.

In order to examine further the influence of spatial effects on the oscillatory dynamics, in the next section we investigate the role of the function H 1 (x, y) which governs the production of the Hes1 protein in the cytoplasm (see equation ( 17)).

Varying the Location of Protein Synthesis in the Hes1 System

In this section we present a number of numerical simulations of the PDE system ( 15) -( 18) demonstrating the effect of varying the function, H 1 (x, y), defined in section 2.1 and shown graphically in the Appendix. This function controls the spatial location of Hes1 protein production in the cytoplasm which must take place away from the endoplasmic reticulum.

As can be seen from the simulation results shown in Figs. 9, 10 (top row), if protein production is located too close to the nucleus, the amplitude of the oscillations is much reduced and is not in line with experimental observations. In a similar manner, the plots in the bottom row of Figs. 9, 10 show that when protein synthesis is located too far from the nucleus, the oscillations are much reduced in amplitude or disappear altogether (with the concentrations tending to a steady state). The plots in the middle row demonstrate sustained oscillations in the mRNA and protein levels (in both the nucleus and cytoplasm) when protein synthesis is located at a "moderate" distance from the nucleus. The simulations therefore reveal an "optimum" distance outside the nucleus for protein production for which undamped oscillations of large amplitude are observed. Snapshots of the corresponding spatial plots of the Hes1 protein concentration are presented in

Inhibition of the Proteasome

Our previous simulation results have shown that oscillatory dynamics in the Hes1 system occur only for a suitable protein degradation rate μ p . Experiments have demonstrated that in the presence of the proteasome inhibitor MG132, hes1 mRNA is transiently induced by a serum treatment, but is then suppressed persistently thereafter [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF]. We now show the result of inhibiting the proteasome in the Hes1 model ( 15) -( 18) by reducing the decay rate for Hes1 protein μ p by a factor of 10.

The following parameter set is therefore used to simulate the effect of suppressing the proteasome:

D * m n = D * m c = D * p c = D * p n = 7.5 × 10 -4 , α * m = 1, (26) h = 5, p * = 1, μ * m = 0.03, μ * p = 0.003, α * p = 2.
Fig. 12 shows a plot of the total concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the nucleus over time, while Fig. 13 shows the corresponding concentrations in the cytoplasm. Finally, Figs. 14 and 15 show the spatio-temporal evolution of the mRNA and protein concentrations respectively over the same time period. As can be seen from all these plots, no oscillations in the concentration levels are observed, in line with the experimental results of [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF].

Having developed the PDE model ( 15) -( 18) for the Hes1 system and explored its spatio-temporal dynamics through a range of simulations, in the next section we turn our attention to the p53-Mdm2 system where a negative feedback loop also exists.

The p53-Mdm2 System

The pleiotropic p53 tumour suppressor protein is a well-established regulator of the cell cycle. In response to a variety of cellular stresses, such as DNA damage, ribosome biogenesis defects, oncogene activation, hypoxia and chemotherapeutic drugs, p53 is activated and induces a range of responses including cell cycle arrest, senescence or apoptosis (programmed cell death) [START_REF] Vousden | Blinded by the light: the growing complexity of p53[END_REF][START_REF] Vogelstein | Surfing the p53 network[END_REF]. The central role for p53 as a cell cycle regulator is highlighted in human cancers. Mutations that inactivate p53 function have been detected in more than 50% of human cancers [START_REF] Bennet | Molecular epidemiology of human cancer risk: gene environment interactions and p53 mutation spectrum in human lung cancer[END_REF]. Importantly, even tumours with wild type p53 have defects in upstream regulators or downstream effectors of p53. Therefore, inactivation of the p53 pathway is a common event in cancer development [START_REF] Zilfou | Tumor suppressive functions of p53[END_REF][START_REF] Toledo | Regulating the p53 pathway: in vitro hypotheses, in vivo veritas[END_REF].

In normal unstressed conditions, the levels and activity of p53 remain low, but in response to cellular stress, p53 levels are increased and the p53 pathway is activated. A vital negative regulator of p53 function in cells is the Mdm2 oncogene product. Mdm2 suppresses p53 function by at least two mechanisms. Firstly, Mdm2 interacts with the transactivation domain of p53 in the N-terminus inhibiting p53 transcriptional activity and also acts as a ubiquitin E3-ligase such that Mdm2 promotes p53 ubiquitination and proteasomal degradation. Secondly, Mdm2 is a target gene for p53. This creates a negative feedback loop which provides tight regulation of p53 function in cells [START_REF] Coutts | p53 ubiquitination by Mdm2: a never ending tail?[END_REF][START_REF] Carter | Modifications of p53: competing for the lysines[END_REF].

The importance of the p53-Mdm2 feedback loop was demonstrated in mouse animal model systems where deletion of Mdm2 caused embryonic lethality that was rescued by concomitant p53 deletion [START_REF] Jones | Rescue of embryonic lethality in Mdm2-deficient mice by absence of p53[END_REF][START_REF] Montes De Oca Luna | Rescue of early embryonic lethality in mdm2-deficient mice by deletion of p53[END_REF]. Mdm2 is overexpressed in tumours with wild type p53 function, which could account for suppression of p53 function [START_REF] Toledo | Regulating the p53 pathway: in vitro hypotheses, in vivo veritas[END_REF][START_REF] Marine | Mdmx and Mdm2: brothers in arms?[END_REF]. It has also been observed that Mdm2 protein levels dramatically decrease within the first 5 minutes after DNA damage, which allows for the accumulation of p53 [START_REF] Stommel | Accelerated MDM2 auto-degradation induced by DNA-damage kinases is required for p53 activation[END_REF]. Therefore, a key activity of Mdm2 in cells is to suppress p53 function. Given the importance of p53 in controlling cell cycle and tumour development, it is not surprising that the p53-Mdm2 feedback loop is very tightly controlled in cells. Experiments have been performed to measure the dynamics of fluorescently tagged p53 and Mdm2 over several days in individual living cells [START_REF] Geva-Zatorsky | Oscillations and variability in the p53 system[END_REF]. Some cells exhibited undamped oscillations for at least 3 days (more than 10 peaks).

Mathematical models of the p53-Mdm2 system have taken a variety of forms. One of the earliest models was that of [START_REF] Bar-Or | Generation of oscillations by the p53-Mdm2 feedback loop: A theoretical and experimental study[END_REF], which included an unknown intermediary component to the system representing the delayed synthesis of Mdm2 by p53 (despite extensive research into p53-Mdm2 interactions, no such intermediary has been identified to date). It was believed that p53 and Mdm2 concentrations exhibited damped oscillations in response to DNA damage. However, later experimental results in individual cells showed instead that p53 responded in digital pulses to DNA damage [START_REF] Geva-Zatorsky | Oscillations and variability in the p53 system[END_REF][START_REF] Lahav | Dynamics of the p53-Mdm2 feedback loop in individual cells[END_REF]. As mathematical models have evolved to capture the undamped oscillations revealed by experiments, many different approaches have been adopted, including combining positive feedback loops with negative feedback loops in ODE metapopulation-like models [START_REF] Ciliberto | Steady states and oscillations in the p53/Mdm2 network[END_REF][START_REF] Zhang | Exploring mechanisms of the DNA-damage response: p53 pulses and their possible relevance to apoptosis[END_REF]). These mod-els were the first to make the important distinction between nuclear and cytoplasmic concentrations. Some models have taken stochastic effects into account [START_REF] Puszyński | Oscillations and bistability in the stochastic model of p53 regulation[END_REF][START_REF] Proctor | Explaining oscillations and variability in the p53-Mdm2 system[END_REF][START_REF] Ouattara | From structure to dynamics: Frequency tuning in the p53-Mdm2 network. ii Differential and stochastic approaches[END_REF]) while others have used time delays [START_REF] Mihalas | A dynamic P53-MDM2 model with time delay[END_REF][START_REF] Ma | A plausible model for the digital response of p53 to DNA damage[END_REF][START_REF] Batchelor | Recurrent initiation: A mechanism for triggering p53 pulses in response to DNA damage[END_REF], in a manner similar to that discussed for the Hes1 system previously. An attempt was made to model the spatial aspect of the system by Gordon [START_REF] Gordon | Spatiotemporal modelling of the p53-mdm2 oscillatory system[END_REF]), but this model also relied on time delays to produce oscillations. In the model we present here, we show that by simply considering the spatial interactions explicitly, we can obtain the experimentally observed oscillatory dynamics.

[Fig. 16

about here.]

As a first approximation of the p53-Mdm2 pathway following [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF], we look at the main interactions (Fig. 16), setting aside post-translational modifications or other proteins that may have an effect on the system. We assume continuous stress on the system (reflecting some physiological stresses such as hypoxia) for simplicity, and as a result of this do not include a DNA damage term. p53 protein is assumed to be made at a constant rate [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF] in an annular region in the cytoplasm (see Appendix for details). p53 translocates to the nucleus where it acts as a transcription factor, upregulating production of Mdm2 mRNA. Newly synthesized Mdm2 mRNA is subsequently exported to the cytoplasm where it is translated into protein. Mdm2 protein enhances p53 degradation in both the nucleus and cytoplasm via ubiquitination [START_REF] Xirodimas | Cocompartmentalization of p53 and Mdm2 is a major determinant for Mdm2-mediated degradation of p53[END_REF]. In summary, p53 upregulates Mdm2, but Mdm2 downregulates p53 and so a negative feedback loop exists.

p53-Mdm2 Mathematical Model

We begin by looking at the fundamental reaction kinetics of the system. Denoting the concentrations of p53, Mdm2 and Mdm2 mRNA by [p53], [Mdm2] and [Mdm2m] respectively, the ODE system below is formulated to capture the interactions depicted in Fig. 16:

d[p53] dt = β - ⎛ ⎝ μ + ν ⎛ ⎝ [Mdm2] h 1 Mdm2 h 1 + [Mdm2] h 1 ⎞ ⎠ ⎞ ⎠ [p53], (27)
d[Mdm2m] dt = α + η [p53] h 2 p53 h 2 + [p53] h 2 -φ [Mdm2m], (28)
d[Mdm2] dt = γ[Mdm2m] -ρ[Mdm2]. (29)
The ODE describing p53 is composed of a production term β , followed by a natural degradation term of rate μ, and finally a degradation term dependent on the amount of Mdm2, with parameter ν. The second ODE, modelling Mdm2 mRNA, has a production term with basal rate α, followed by an enhanced production term dependent on the amount of p53, reflecting the activity of p53 as a transcription factor (with maximal initiation rate η), and finally a natural degradation term of rate φ . The final ODE is for Mdm2 protein, which simply has a production term dependent on the amount of Mdm2 mRNA, rate γ, and a natural degradation term, rate ρ. Mdm2 and p53 are activation thresholds, and h 1 and h 2 are Hill coefficients. As before with the Hes1 system, Monk [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF]) added a delay to account for transcript elongation, splicing, processing and export:

d[p53] dt = β - ⎛ ⎝ μ + ν ⎛ ⎝ [Mdm2] h 1 Mdm2 h 1 + [Mdm2] h 1 ⎞ ⎠ ⎞ ⎠ [p53], (30)
d[Mdm2m] dt = α + η [p53(t -τ)] h 2 p53 h 2 + [p53(t -τ)] h 2 -φ [Mdm2m], (31) d[Mdm2] dt = γ[Mdm2m] -ρ[Mdm2]. (32)
System ( 30), ( 31), (32) produces oscillations [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF], but does not distinguish between events taking place in the nucleus and cytoplasm. Therefore, we now consider the spatial interactions explicitly, allowing for diffusion within the cell, and arrive at the following system of PDEs:

∂ [p53 c ] ∂t = D c 1 ∇ 2 [p53 c ] + H 2 (x, y)β - ⎛ ⎝ μ + ν ⎛ ⎝ [Mdm2 c ] h 1 Mdm2 h 1 + [Mdm2 c ] h 1 ⎞ ⎠ ⎞ ⎠ [p53 c ], (33)
∂ [p53 n ] ∂t = D n 1 ∇ 2 [p53 n ] - ⎛ ⎝ μ + ν ⎛ ⎝ [Mdm2 n ] h 1 Mdm2 h 1 + [Mdm2 n ] h 1 ⎞ ⎠ ⎞ ⎠ [p53 n ], (34)
∂ [Mdm2m n ] ∂t = D n 2 ∇ 2 [Mdm2m n ] + α + η [p53 n ] h 2 p53 h 2 + [p53 n ] h 2 -φ [Mdm2m n ], (35)
∂ [Mdm2m c ] ∂t = D c 2 ∇ 2 [Mdm2m c ] -φ [Mdm2m c ], (36)
∂ [Mdm2 c ] ∂t = D c 3 ∇ 2 [Mdm2 c ] + H 1 (x, y)γ[Mdm2m c ] -ρ[Mdm2 c ], (37)
∂ [Mdm2 n ] ∂t = D n 3 ∇ 2 [Mdm2 n ] -ρ[Mdm2 n ], (38)
where a subscript "c" denotes a cytoplasmic variable and a subscript "n" denotes a nuclear variable. The parameters D i are the diffusion coefficients for each species and H 1 (x, y) and H 2 (x, y) are functions localising the protein production whose specific form will be given after the nondimensionalisation of the system. We apply continuity of flux boundary conditions across the (internal) nuclear membrane and zero-flux boundary conditions at the outer cell membrane:

D n 1 ∂ [p53 n ] ∂ n = D c 1 ∂ [p53 c ] ∂ n and [p53 n ] = [p53 c ] at nuclear membrane, ( 39
)
D n 2 ∂ [Mdm2m n ] ∂ n = D c 2 ∂ [Mdm2m c ] ∂ n and [Mdm2m n ] = [Mdm2m c ] at nuclear membrane, ( 40
)
D n 3 ∂ [Mdm2 n ] ∂ n = D c 3 ∂ [Mdm2 c ] ∂ n and [Mdm2 n ] = [Mdm2 c ] at nuclear membrane, (41) ∂ [p53 c ] ∂ n = 0 at cell membrane, (42)
∂ [Mdm2m c ] ∂ n = 0 at cell membrane, (43)
∂ [Mdm2 c ] ∂ n = 0 at cell membrane, ( 44
)
where n is a unit normal.

We nondimensionalise ( 33) -( 38) with appropriate reference values as follows:

[p53 n ] = [p53 n ] [p53 0 ] , [p53 c ] = [p53 c ] [p53 0 ] , [Mdm2m n ] = [Mdm2m n ] [Mdm2m 0 ] , [Mdm2m c ] = [Mdm2m c ] [Mdm2m 0 ] , [Mdm2 n ] = [Mdm2 n ] [Mdm2 0 ] , (45)
[Mdm2 c ] = [Mdm2 c ] [Mdm2 0 ] , t = t τ , x = x L , y = y L ,
where [p53 0 ], [Mdm2m 0 ], [Mdm2 0 ] are reference concentrations, τ is a reference time, and L is a reference length (10μm as with Hes1 system). Using this scaling, (33) -( 38) becomes:

∂ [p53 c ] ∂t = D * c 1 ∇ 2 [p53 c ] + H 2 (x, y)β * -μ * + ν * [Mdm2 c ] h 1 Mdm2 * + [Mdm2 c ] h 1 [p53 c ], (46)
∂ [p53 n ] ∂t = D * n 1 ∇ 2 [p53 n ] -μ * + ν * [Mdm2 n ] h 1 Mdm2 * + [Mdm2 n ] h 1 [p53 n ], (47)
∂ [Mdm2m n ] ∂t = D * n 2 ∇ 2 [Mdm2m n ] + α * + η * [p53 n ] h 2 p53 * + [p53 n ] h 2 -φ * [Mdm2m n ], (48)
∂ [Mdm2m c ] ∂t = D * c 2 ∇ 2 [Mdm2m c ] -φ * [Mdm2m c ], (49)
∂ [Mdm2 c ] ∂t = D * c 3 ∇ 2 [Mdm2 c ] + H 1 (x, y)γ * [Mdm2m c ] -ρ * [Mdm2 c ], (50)
∂ [Mdm2 n ] ∂t = D * n 3 ∇ 2 [Mdm2 n ] -ρ * [Mdm2 n ], (51)
where

D * c 1 = τD c 1 L 2 , β * = τβ [p53 0 ] , μ * = τμ, ν * = τν, Mdm2 * = Mdm2 h 1 [Mdm2 0 ] h 1 , D * n 1 = τD n 1 L 2 , D * n 2 = τD n 2 L 2 , α * = τα [Mdm2m 0 ] , η * = τη [Mdm2m 0 ] , ( 52
) p53 * = p53 h 2 [p53 0 ] h 2 , φ * = τφ , D * c 2 = τD c 2 L 2 , D * c 3 = τD c 3 L 2 , γ * = τγ[Mdm2m 0 ] [Mdm2 0 ] , ρ * = τρ, D * n 3 = τD n 3 L 2 ,
and

H 1 (x, y) = ⎧ ⎪ ⎨ ⎪ ⎩ 0, if x 2 2 + y 2 ≤ 0.25, 1, if x 2 2 + y 2 > 0.
25, and y) and H 2 (x, y) are illustrated graphically in the Appendix. We apply zero initial conditions, zero-flux boundary conditions at the cell membrane and flux continuity boundary conditions across the nuclear membrane:

H 2 (x, y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, if x 2 2 + y 2 ≤ 0.25, 1, if 0.25 < x 2 2 + y 2 < 0.375, 0, if x 2 2 + y 2 ≥ 0.375. H 1 (x,
[p53 n ] = [p53 c ] = [Mdm2m n ] = [Mdm2m c ] = [Mdm2 n ] = [Mdm2 c ] = 0, at t = 0, ( 53
) D * n 1 ∂ [p53 n ] ∂ n = D * c 1 ∂ [p53 c ] ∂ n and [p53 n ] = [p53 c ] at nuclear membrane, ( 54
)
D * n 2 ∂ [Mdm2m n ] ∂ n = D * c 2 ∂ [Mdm2m c ] ∂ n and [Mdm2m n ] = [Mdm2m c ] at nuclear membrane, ( 55
)
D * n 3 ∂ [Mdm2 n ] ∂ n = D * c 3 ∂ [Mdm2 c ] ∂ n and [Mdm2 n ] = [Mdm2 c ] at nuclear membrane, (56) ∂ [p53 c ] ∂ n = 0 at cell membrane, (57)
∂ [Mdm2m c ] ∂ n = 0 at cell membrane, (58)
∂ [Mdm2 c ] ∂ n = 0 at cell membrane, (59)
We found a reference concentration for [p53 0 ] of 0.5μM, and estimated reference concentrations for [Mdm2m 0 ] and [Mdm2 0 ] to be 0.05μM and 2μM respectively [START_REF] Ma | A plausible model for the digital response of p53 to DNA damage[END_REF]. In preliminary simulations of ( 46) -( 51) it was observed that a period of oscillation was approximately 400 time units. Knowing the period of oscillation of p53 roughly equals 3 hours [START_REF] Monk | Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays[END_REF], we can find an appropriate reference time: 400τ = 3hrs which implies that τ = 27s.

Parameter Estimation

For simulations of the nondimensional p53-Mdm2 system, the following parameter values were used: 60)

D * c 1 = D * n 1 = D * n 2 = D * c 2 = D * c 3 = D * n 3 = 9 × 10 -4 , β * = 0.5, μ * = 0.003, ν * = 1, h 1 = 2, Mdm2 * = 16, α * = 0.0175, η * = 1, (
h 2 = 4, p53 * = 5, φ * = 0.0175, γ * = 0.5, ρ * = 0.025.
We calculate estimates for parameter values of the dimensional p53-Mdm2 model ( 33) -( 38) using ( 52) and ( 60):

D c 1 = D n 1 = D n 2 = D c 2 = D c 3 = D n 3 = 3.33 × 10 -11 cm 2 s -1 , β = 9.26 × 10 -3 Ms -1 , μ = 1.11 × 10 -4 s -1 , ν = 0.04s -1 , h 1 = 2, Mdm2 = 8 × 10 -6 M, (61) α = 3.24 × 10 -11 Ms -1 , η = 1.85 × 10 -9 Ms -1 , h 2 = 4, p53 = 7.48 × 10 -7 M, φ = 6.48 × 10 -4 s -1 , γ = 0.74s -1 , ρ = 9.26 × 10 -4 s -1 .
Leaving all other parameter values unchanged, we have calculated a range of diffusion coefficients for all the variables, for which we have sustained oscillations: 7.41 × 10 -12 to 2.48 × 10 -10 cm 2 s -1 . In addition to this we calculated the following ranges of degradation rates and Hill coefficients for which, if all other parameters remain unaltered, oscillations are observed: μ = 0 to 5.19 × 10 -4 s -1 , ν = 2.67 × 10 -3 to 1.56s -1 , ρ = 2.96 × 10 -3 to 4.30 × 10 -3 s -1 , φ = 2.59 × 10 -4 to 1.59 × 10 -3 s -1 , h 1 ≥ 2, h 2 ≥ 2.

Once again, we note that the kinetic parameters we have used are in line with previous estimates [START_REF] Proctor | Explaining oscillations and variability in the p53-Mdm2 system[END_REF]. Regarding the values of our diffusion coefficients, we note that the results of [START_REF] Klonis | Fluorescence photobleaching analysis for the study of cellular dynamics[END_REF] show that diffusion rates of macromolecules in the cytoplasm and nucleus are up to 100-times slower than in aqueous buffers, which would then bring our estimated values of around 10 -12 -10 -10 cm 2 s -1 in line with the estimates of [START_REF] Seksek | Translational diffusion of macromolecule-sized solutes in cytoplasm and nucleus[END_REF] (10 -8 cm 2 s -1 ) and [START_REF] Matsuda | Direct measurement of protein dynamics inside cells using a rationally designed photoconvertible protein[END_REF] (10 -9 cm 2 s -1 ).

p53-Mdm2 Numerical Simulations

As in the case of the Hes1 system, we solved PDE system ( 46) -( 51) numerically using the COMSOL/FEMLAB package. For all our simulations we used the same 2-dimensional cell domain composed of two ellipses to represent the nucleus and cytoplasm. Figs. 17 and 18 show the concentrations of p53 and Mdm2 in the nucleus and cytoplasm obtained from computational simulations of the p53-Mdm2 system.

[Fig. 17 As with the Hes1 system simulations, we can see from these p53 simulations that the mRNA concentration is higher in the nucleus compared to the protein concentrations and vice versa for the cytoplasmic compartment. The relative ratio of nuclear and cytoplasmic concentrations of a protein can be easily obtained from our model but is difficult to measure experimentally. We could not find any existing experimental evidence in the literature to confirm this observation and therefore we leave it as a prediction of the model.

Figs. 19 and 20 show how the dynamics of the p53-Mdm2 system evolve in space as well as time. From Fig. 19, we see that p53 has accumulated in the cytoplasm at t = 45 minutes. The p53 then begins to diffuse across the nuclear boundary and enters the nucleus (t = 90 minutes). The presence of p53 in the nucleus upregulates the expression of Mdm2 which results in enhanced decay of p53 (t = 135 minutes). By t = 180 minutes, the p53 concentration begins to increase again, giving a period of oscillation of approximately 3 hours.

[Fig. 19 about here.] Fig. 20 shows the plots of Mdm2 protein concentration over time. Notice that Mdm2 protein is not produced in significant quantities until around t = 90 minutes, reflecting the time for Mdm2 mRNA production, export from the nucleus and translation in the cytoplasm. Mdm2 levels can then be observed rising in the cytoplasm, beginning to enter the nucleus at t = 135 minutes, enhancing the degradation of p53 which in turn causes the down-regulation of Mdm2 expression. This is shown at t = 225 minutes where Mdm2 levels have depleted considerably.

[Fig. 20

about here.]

Once again, in order to investigate the influence of spatial effects, we carried out a number of simulations on system (46) -( 51), where we varied the values of the diffusion coefficients (all other parameters remaining unchanged) of the mRNAs and proteins. By doing this we found a range of values for the diffusion coefficients D c 1 , D c 2 , D c 3 , D n 1 , D n 2 , D n 3 where the system exhibits oscillatory dynamics, i.e., for 7.41 × 10 -12 cm 2 s -1 ≤ D c , D n ≤ 2.48 × 10 -10 cm 2 s -1 oscillations were observed, while no oscillations were observed outside this range. Hence if diffusion of mRNA and protein is either too slow or too fast, oscillations do not occur (as per Hes1 results). In addition to this we calculated the following ranges of degradation rates and Hill coefficients for which, if all other parameters remain unaltered, oscillations are observed: μ = 0 to 5.19 × 10 -4 s -1 , ν = 2.67 × 10 -3 to 1.56s -1 , ρ = 2.96 × 10 -3 to 4.30 × 10 -3 s -1 , φ = 2.59 × 10 -4 to 1.59

× 10 -3 s -1 , h 1 ≥ 2, h 2 ≥ 2.
Finally, as with the Hes1 system, we carried out a number of numerical simulations of the PDE system ( 46) -( 51) where we investigated the effect of varying the two functions H 1 (x, y) and H 2 (x, y), controlling constant protein synthesis and protein translation via mRNA in the cytoplasm, respectively (see Appendix for full details). Once again the simulation results (similar to Figs. 9 -11 but not shown here) reveal an "optimum" distance outside the nucleus for protein production for which undamped oscillations of large amplitude are observed.

Discussion

Dissecting the mechanisms by which transcription factors are regulated within cells is critical to understanding cellular function in health and disease and the opportunities for therapeutic intervention. Results from previous mathematical models have reflected simplified experimental findings but have not distinguished explicitly between spatial compartments within the cell and have not considered (explicitly) spatial movement of molecules. Spatial localisation and considerations are particularly important when modelling transcription factors, which although produced in the cytoplasm must be translocated to the nucleus to function. For the p53 pathway, hindering p53 translocation to the nucleus alters the transcriptome of the cell and contributes to carcinogenesis [START_REF] Vousden | Blinded by the light: the growing complexity of p53[END_REF]. Using partial differential equations (PDEs), we can model these aspects of intracellular signalling pathways explicitly.

The simulation results of our models have demonstrated the existence of oscillatory dynamics in negative feedback systems both for relatively simple (Hes1) and more complex (p53-Mdm2) pathways and have been able to focus on reactions occurring both in the cell nucleus and in the cytoplasm. The main advantage of using systems of PDEs to model intracellular reactions is that the PDEs enable spatial effects to be examined explicitly.

In both the Hes1 and p53-Mdm2 system we varied the diffusion coefficients of the mRNAs and proteins and found a range of values for these diffusion coefficients where the system exhibits oscillatory dynamics, i.e., the results of the model have predicted a range of diffusion coefficients for the molecules involved so that oscillations can be observed. By varying the diffusion coefficients of the molecules, we can vary the flux rates across the nuclear membrane (equivalent to varying nuclear import and export rates), thus granting greater control and allowing a much more in depth analysis of the systems. Similar results were obtained by varying the mRNA degradation rates, protein degradation rates and Hill coefficients, further demonstrating that the oscillations are robust to parameter changes.

Exploiting the explicitly spatial nature of PDEs, we were also able to manipulate mathematically the location of the ribosomes, thus controlling where the proteins were synthesized within the cytoplasm. For both the Hes1 and the p53-Mdm2 systems, we carried out a number of numerical simulations where we investigated the effect of varying the two functions H 1 (x, y) and H 2 (x, y), controlling constant protein synthesis and protein translation via mRNA in the cytoplasm, respectively. For both model systems, the simulation results revealed an "optimum" distance outside the nucleus for protein production for which sustained (undamped) oscillations of large amplitude were observed.

Future work in this area will undoubtedly become more accurate and complex, but is likely to help elucidate the complex biochemical pathways involved in cellular decisions to undergo cell cycle arrest, senescence or apoptosis [START_REF] Vousden | Blinded by the light: the growing complexity of p53[END_REF][START_REF] Toledo | Regulating the p53 pathway: in vitro hypotheses, in vivo veritas[END_REF]. In particular, extensions of the current model will consider the active transport of proteins and mRNA within the cell as mechanisms of movement in addition to diffusion [START_REF] Cangiani | A spatial model of cellular molecular trafficking including active transport along microtubules[END_REF]. We also plan to model the nuclear membrane in more detail and take into account its thickness. This will allow us to model differences in the rate of transport of mRNA and protein across the nuclear membrane more accurately. Additional complexities of post-transcriptional mRNA and post-translational protein modifications, while not explicitly incorporated into the model can easily be done and will be included in future work [START_REF] Abou-Jaoudé | From structure to dynamics: Frequency tuning in the p53-Mdm2 network i. Logical approach[END_REF][START_REF] Zhang | Exploring mechanisms of the DNA-damage response: p53 pulses and their possible relevance to apoptosis[END_REF].

The spatial models presented here reflect experimental findings both in vitro [START_REF] Hirata | Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop[END_REF][START_REF] Geva-Zatorsky | Oscillations and variability in the p53 system[END_REF]) and in vivo [START_REF] Hamstra | Real-time evaluation of p53 oscillatory behaviour in vivo using bioluminescent imaging[END_REF]) and mark a conceptual advance in the modelling of intracellular processes. With the emergence of new imaging technologies, validation of spatial models will be possible, with dynamic molecular imaging of subcellular processes on the near horizon.

Future models should enable us to drill down into the fundamental differences between cancer cells and normal cells. As an exemplar, using the p53-Mdm2 pathway we will be able to model the effects of different therapeutic approaches, including the temporal and spatial distributions of targeted disruption of p53 or Mdm2 interactions by non-genotoxic mechanisms.

Appendix: Protein Translation and Synthesis in the Cytoplasm: Consideration of the Location of the Endoplasmic Reticulum

The endoplasmic reticulum (ER) is a network of flattened sacs and branching tubules surrounding the cell nucleus that extends throughout the cytoplasm in eukaryotic cells. These sacs and tubules are all interconnected by a single continuous membrane so that the organelle has only one large and intricately arranged lumen. Proteins made in the endoplasmic reticulum are mainly either exported to the exterior of the cell or are transported to other membrane structures such as the Golgi apparatus, lysosomes and endosomes [START_REF] Alberts | Molecular Biology of the Cell[END_REF]. Thus proteins made in the endoplasmic reticulum are unlikely to return to the nucleus.

In our models of the Hes1 and p53 pathways, we have made allowance for the endoplasmic reticulum by assuming that proteins made in the cytoplasm are translated a certain distance outside the nucleus. We now state this assumption explicitly using step functions in the relevant equations for the computational cell domain shown in Fig. 21. For the Hes1 system, the equation for Hes1 protein ( 17) is written as:

∂ [p c ] ∂t = D * p c ∇ 2 [p c ] + H 1 (x, y)α * p [m c ] -μ * p [p c ],
and in the p53-Mdm2 system, we can write the equations for p53 and Mdm2 concentrations in the cytoplasm (46) (50) as:

∂ [p53 c ] ∂t = D * c 1 ∇ 2 [p53 c ] + H 2 (x, y)β * -μ * + ν * [Mdm2 c ] h 1 Mdm2 * + [Mdm2 c ] h 1 [p53 c ], ∂ [Mdm2 c ] ∂t = D * c 3 ∇ 2 [Mdm2 c ] + H 1 (x, y)γ * [Mdm2m c ] -ρ * [Mdm2 c ],
where the functions H 1 (x, y) and H 2 (x, y) are defined as:

H 1 (x, y) = ⎧ ⎪ ⎨ ⎪ ⎩ 0, if x 2 2 + y 2 ≤ 0.25, 1, if x 2 2 + y 2 > 0.25,
and

H 2 (x, y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, if x 2 2 + y 2 ≤ 0.25, 1, if 0.25 < x 2 2 + y 2 < 0.375, 0, if x 2 2 + y 2 ≥ 0.375.
The function H 1 (x, y) is such that in a region close to the nucleus (representing the location of the ER), the function is zero, meaning there is no protein synthesis in this region. In a region further away from the nucleus (outside the ER) the function takes the value of one, modelling the translation of protein in this region of the cytoplasm. The function H 2 (x, y) is such that in a region close to the nucleus the function is zero, meaning there is no protein synthesis here. However, it is now assumed that the function takes the value of one in an annular region outside of the ER (again modelling the translation of protein). An annular region is chosen because we assume p53 is produced at a constant rate in the cytoplasm. This prevents p53 from being produced close to the plasma membrane, where mRNA is unlikely to reach in sufficient quantities. The two functions are illustrated graphically in Fig. 21.

[Fig. 21 x 2 /2 + y 2 > 0.45 Fig. 9: Plots showing the effect on the concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the nucleus over time by varying the function H 1 (x, y) which controls the spatial production of protein in the cytoplasm (all other parameters as per (25)). It can be observed from the plots that there exists a range of distances for protein production to begin outside the nucleus in order to see oscillatory dynamics. 25)). It can be observed from the plots that there exists a range of distances for protein production to begin outside the nucleus in order to see oscillatory dynamics. ,y) and H 2 (x, y) are non-zero. The blue region of the cytoplasm depicts where we allow constant protein synthesis to occur, i.e., this represents the rectangular function H 2 (x, y). The blue and red regions together depict where we allow protein translation via mRNA, i.e., this represents the function H 1 (x, y). In the white region representing the ER and nucleus, no protein synthesis takes place.

[

Fig

[

  Fig. 12 about here.] [Fig. 13 about here.] [Fig. 14 about here.] [Fig. 15 about here.]

  about here.] [Fig. 18 about here.]

Fig. 3 :Fig. 4 :

 34 Fig.3: Schematic diagram showing how the spatial interactions between hes1 mRNA and Hes1 protein are modelled. hes1 mRNA is produced in the nucleus (transcription), then exported across the nuclear membrane into the cytoplasm where it is translated into protein, i.e., transcription occurs exclusively in the nucleus and translation/synthesis occurs exclusively in the cytoplasm. Hes1 protein is then imported back across the nuclear membrane to the nucleus where it inhibits the production of its own mRNA, i.e., a negative feedback loop exists.

Fig. 5 :

 5 Fig. 5: Plot of the concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the cytoplasm over time. The period of oscillations is approximately 120 minutes. Parameter values as per (25).

Fig. 6 :

 6 Fig. 6: Plots showing the spatio-temporal evolution of hes1 mRNA concentration within the cell from times t = 0 to t = 480 minutes at 60 minute intervals. The concentration oscillates in both time and space. Parameter values as per (25).

Fig. 7 :

 7 Fig. 7: Plots showing the spatio-temporal evolution of Hes1 protein concentration within the cell from times t = 0 to t = 480 minutes. The concentration oscillates in both time and space. Parameter values as per (25).

  Fig. 8: Plot showing a snapshot of the Hes1 protein concentration at t = 60 minutes in a 3D spherical (a) and ellipsoidal (b) domain. Axes are presented in nondimensional units. Parameter values as per (25).

Fig. 10 :

 10 Fig.10: Plots showing the effect on the concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the cytoplasm over time by varying the function H 1 (x, y) which controls the spatial production of protein in the cytoplasm (all other parameters as per (25)). It can be observed from the plots that there exists a range of distances for protein production to begin outside the nucleus in order to see oscillatory dynamics.

Fig. 11 :Fig. 12 :

 1112 Fig. 11: Plots showing the spatial distribution of Hes1 protein concentration at t = 60 minutes for various functions H 1 (x, y) which controls the spatial production of protein in the cytoplasm (all other parameter values as per (25)). It can be observed from the plots that the protein concentration throughout the cell varies with the function H 1 (x, y).

Fig. 13 :

 13 Fig. 13: Plot of the total concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the cytoplasm over time when the proteasome is inhibited. No oscillations are observed. Parameter values as per (26).

Fig. 14 :

 14 Fig. 14: Plots showing the spatio-temporal evolution of hes1 mRNA concentration within the cell from times t = 0 to 300 minutes when the proteasome is inhibited. No oscillations are observed. hes1 mRNA levels peak at t = 60 minutes and then decrease by t = 120 minutes and remain low until the end of the simulation. Parameter values as per (26).

Fig. 15 :

 15 Fig. 15: Plots showing the spatio-temporal evolution of Hes1 protein concentration within the cell from times t = 0 to 300 minutes when the proteasome is inhibited. No oscillations are observed. As expected by decreasing the protein degradation rate, Hes1 protein rises to a high level. It reaches a steady state level by t = 300. Parameter values as per (26).

Fig. 16 :

 16 Fig.16: A schematic representation of the p53-Mdm2 pathway. p53 is synthesised in the cytoplasm at a constant rate. It then translocates into the nucleus where it acts as a transcription factor, upregulating Mdm2 mRNA transcription. Mdm2 mRNA is then exported to the cytoplasm where it is translated into Mdm2 protein. Mdm2 protein ubiquitinates p53 in both the nucleus and cytoplasm which results in an increased decay rate for p53.

Fig. 18 :

 18 Fig. 18: Plots showing the concentrations of p53 (blue), Mdm2 mRNA (green) and Mdm2 (red) in the cytoplasm. The period of oscillations is approximately 180 minutes. Parameter values as per (60).

Fig. 19 :

 19 Fig. 19: Plots showing the spatio-temporal evolution of p53 protein concentration within the cell from times t = 0 to t = 360 minutes. The concentration oscillates in both time and space. The patterns of oscillations bear a striking resemblance to those obtained in individual cells in Geva-Zatorsky et al. (2006). Parameter values as per (60).

Fig. 20 :

 20 Fig. 20: Plots showing the spatio-temporal evolution of Mdm2 protein concentration within the cell from times t = 0 to t = 360 minutes. The concentration oscillates in both time and space. Parameter values as per (60).

Fig. 21 :

 21 Fig. 21: Schematic diagram showing the regions where the two functions H 1 (x, y) and H 2 (x, y) are non-zero. The blue region of the cytoplasm depicts where we allow constant protein synthesis to occur, i.e., this represents the rectangular function H 2 (x, y). The blue and red regions together depict where we allow protein translation via mRNA, i.e., this represents the function H 1 (x, y). In the white region representing the ER and nucleus, no protein synthesis takes place.

  about here.] It is then exported to the cytoplasm where translation into Hes1 protein occurs. hes1 mRNA is then inhibited in the nucleus by its own protein. This is one of the simplest examples of a negative feedback loop. . . . . . . . . . . 34 3 Schematic diagram showing how the spatial interactions between hes1 mRNA and Hes1 protein are modelled. hes1 mRNA is produced in the nucleus (transcription), then exported across the nuclear membrane into the cytoplasm where it is translated into protein, i.e., transcription occurs exclusively in the nucleus and translation/synthesis occurs exclusively in the cytoplasm. Hes1 protein is then imported back across the nuclear membrane to the nucleus where it inhibits the production of its own mRNA, i.e., a negative feedback loop exists. . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Plot of the concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the nucleus over time. The period of oscillations is approximately 120 minutes. Parameter values as per (25). . . . . . Parameter values as per (26). . . . . . . . . . . . . . . . . . . . . 47 16 A schematic representation of the p53-Mdm2 pathway. p53 is synthesised in the cytoplasm at a constant rate. It then translocates into the nucleus where it acts as a transcription factor, upregulating Mdm2 mRNA transcription. Mdm2 mRNA is then exported to the cytoplasm where it is translated into Mdm2 protein. Mdm2 protein ubiquitinates p53 in both the nucleus and cytoplasm which results in an increased decay rate for p53. . . . . . . . . . . . . . 48 17 Plots showing the concentrations of p53 (blue), Mdm2 mRNA (green) and Mdm2 (red) in the nucleus. The period of oscillations is approximately 180 minutes. Parameter values as per (60). . . . 49 18 Plots showing the concentrations of p53 (blue), Mdm2 mRNA (green) and Mdm2 (red) in the cytoplasm. The period of oscillations is approximately 180 minutes. Parameter values as per (60). 50 19 Plots showing the spatio-temporal evolution of p53 protein concentration within the cell from times t = 0 to t = 360 minutes. The concentration oscillates in both time and space. The patterns of oscillations bear a striking resemblance to those obtained in individual cells in Geva-Zatorsky et al. (2006). Parameter values as per (60). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 20 Plots showing the spatio-temporal evolution of Mdm2 protein concentration within the cell from times t = 0 to t = 360 minutes. The concentration oscillates in both time and space. Parameter values as per (60). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 21 Schematic diagram showing the regions where the two functions H 1 (x, y) and H 2 (x, y) are non-zero. The blue region of the cytoplasm depicts where we allow constant protein synthesis to occur, i.e., this represents the rectangular function H 2 (x, y). The blue and red regions together depict where we allow protein translation via mRNA, i.e., this represents the function H 1 (x, y). In the white region representing the ER and nucleus, no protein synthesis takes place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Fig. 1: A generic negative feedback loop. Fig. 2: A schematic representation of the Hes1 model. hes1 mRNA is transcribed in the nucleus. It is then exported to the cytoplasm where translation into Hes1 protein occurs. hes1 mRNA is then inhibited in the nucleus by its own protein. This is one of the simplest examples of a negative feedback loop.

	List of Figures 15 Plots showing the spatio-temporal evolution of Hes1 protein con-
	1 2	centration within the cell from times t = 0 to 300 minutes when A generic negative feedback loop. . . . . . . . . . . . . . . . . . 33 the proteasome is inhibited. No oscillations are observed. As ex-A schematic representation of the Hes1 model. hes1 mRNA is pected by decreasing the protein degradation rate, Hes1 protein transcribed in the nucleus. . 36 rises to a high level. It reaches a steady state level by t = 300.
	5	Plot of the concentrations of hes1 mRNA (red) and Hes1 protein
		(blue) in the cytoplasm over time. The period of oscillations is
		approximately 120 minutes. Parameter values as per (25). . . . . 37
	6	Plots showing the spatio-temporal evolution of hes1 mRNA con-
		centration within the cell from times t = 0 to t = 480 minutes at
		60 minute intervals. The concentration oscillates in both time and
		space. Parameter values as per (25). . . . . . . . . . . . . . . . . 38
	7	Plots showing the spatio-temporal evolution of Hes1 protein con-
		centration within the cell from times t = 0 to t = 480 minutes. The
		concentration oscillates in both time and space. Parameter values
		as per (25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
	8	Plot showing a snapshot of the Hes1 protein concentration at t =
		60 minutes in a 3D spherical (a) and ellipsoidal (b) domain. Axes
		are presented in nondimensional units. Parameter values as per (25). 40

Preprint submitted to Journal of Theoretical BiologyDecember 8, 2010

Acknowledgments

The authors gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, "M5CGS -From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread".

Plots showing the effect on the concentrations of hes1 mRNA (red) and Hes1 protein (blue) in the nucleus over time by varying the function H 1 (x, y) which controls the spatial production of protein in the cytoplasm (all other parameters as per ( 25)). It can be observed from the plots that there exists a range of distances for protein production to begin outside the nucleus in order to see oscillatory dynamics. .

Graphical Abstract

The components of the Hes1 partial differential equation (PDE) system are shown in the schematic diagram above. hes1 mRNA (m n ) is produced in the nucleus, from where it is exported to the cytoplasm (m c ). Here the process of translation takes place, creating Hes1 protein (p c ). Hes1 protein then translocates to the nucleus (p n ) where it inhibits the production of its own mRNA.

Research Highlights

• PDE models of feedback inhibition loops mark a conceptual advance in modelling

• The precise spatial localisation of transcription factors can be determined

• The models yield oscillatory dynamics with a period in line with experimental data

• Computational simulations predict a range for the protein diffusion coefficients


C De Runz

E Desjardin
email: eric.desjardin@univ-reims.fr

H Akdag

Study of Spatial Fusion of Geographical Entities and Quantitative Information in Accordance with their Imprecision Application to agricultural information in Observox

Keywords: fusion, fuzzy representation, imprecise geographical entities, fuzzy spatial object storage, mutualized GIS I

The article deals with imprecise geographical entities modelled according to the fuzzy set theory for both spatial and quantitative information. It presents the issues of fusion of fuzzy geographical objects according to its storing modes (raster, vector) in a mutualised geographical information system (GIS). We study the aggregation of imprecise spatial entities and its impact in the imprecise description of quantities associated to space. It exposes the current choices in the Observox Project managing multiple sources of information.

INTRODUCTION

In the sustainable development context, the AQUAL project (a State-Region Project in the Champagne-Ardenne, France) highlights the need of a monitoring environment for the study of agricultural practices and of their pressure on the water resources in the Vesle basin. It is called Observox and it exploits data coming from heterogeneous sources: satellite images, land registry, statistical data, Corine Land Cover and other European data. Those data have different qualities; they are often uncertain, imprecise and/or incomplete. They do not have the same spatial representation (vector or raster). Thus, their combinations or fusions imply many problems. In this article, we focus on the fusion of imprecise spatial entities modeled as fuzzy entities.

The multiple sources give information on the same space but do not split it in the same partition. Most of the time, the boundaries of the same spatial entity will differ according to the source. Thus, spatial entities provided by the combination of all the sources will better be modeled as imprecise spatial entities such as fuzzy entities. In fact, according to [START_REF] Fisher | Models of uncertainty in spatial data[END_REF], vagueness is well represented by fuzzy sets.

The first goal of our project is to store multivariate data into a GIS. According to this goal, it is fundamental to store merging results in a unique spatial representation mode. Our project deals with vector and raster spatial representation mode. Thus, the choice of a unique spatial representation mode is essential. [START_REF] Fonte | Conversion between the vector and raster data structures using Fuzzy Geographical Entities[END_REF] proposes an approach for the conversion between raster and vector using fuzzy geographical entities.

In fact, if the vector representation is selected, then raster information will be over-interpreted, but if it is the raster representation, the vector information will be underinterpreted. For instance, the choice of raster representation mode does not allow unrealistic interpretation and facilitates the merging, but the generalization and the study at a larger scale will be more difficult and the storage of raster entities will be more complex. In vector mode, objects can be stored as UGML files [START_REF] Morris | UGML: an extension of GML to provide support for geographic objects with uncertain boundaries[END_REF] or using multiple representations [START_REF] Dilo | Definition and implementation of vague objects[END_REF]). In the two hypotheses, the conversion (raster-vector) and the combination of the spatial entities produce fuzzy spatial entities.

However, the resulting fuzzy spatial entities may partially overlap. Moreover, quantitative information describing each entity should be studied depending on the local region of interest, and so the propagation of quantitative information should be considered. Therefore, an objective is to merge them in order to obtain an appropriate spatial vision for the study of the phytosanitary product diffusion pressure on the state of water in the Vesle basin. The merging operation should consider the spatial imprecision but also the quantitative information propagation. In this article, we present a study of choices (raster/vector, merging function) needed for the fusion of imprecise geographical entities and quantitative information.

This paper is a preliminary study for the building of the observatory. It would present our choice at the beginning of the project. This paper is organized as follows. In the section 2, the modeling approaches for geographical entities are presented and their storing in a GIS. The section 3 illustrates implications in their fusion. The last section is devoted to the conclusion.

II. SPATIAL IMPRECISION AND REPRESENTATION

This section is devoted to imprecise spatial data representation and their storage. Using spatial fuzzy sets is an usual way to represent imprecision [START_REF] Zadeh | Fuzzy Sets[END_REF][START_REF] Klir | Fuzzy sets and fuzzy logic: Theory and applications[END_REF]. This section highlights the modeling of fuzzy objects and its storage in a GIS.

A. Fuzzy geographical objects

Fisher (1996) presents a comparative study between crisp sets and fuzzy sets in order to model landscape. The firsts simplify the modeling but could amplify errors. The others make complex the models and treatments. In [START_REF] Dilo | Definition and implementation of vague objects[END_REF], the concept of vague objects is introduced. Fig. 1 illustrates fuzzy regions: the grey scale of a point corresponds to its membership value ("black" is used for "inside", "white" for "outside" and intensity of any other color corresponds to the value of the membership degree of the location to the region).

Quantitative data attributed to a fuzzy region could also be imprecise and represented by a fuzzy set or a confidence index value.

There are two representation modes for storing spatial data: raster and vector. In order to take into consideration the imprecision, those two modes are adapted in order to record fuzzy spatial data.

B. Fuzzy spatial object storage as raster

A classic in image processing for representing a fuzzy region after a fuzzy segmentation is to affect to each pixel the membership value computed for this pixel and the studied region. An example of fuzzy segmentation approach is presented in (Philipp-Foliguet, Bernardes [START_REF] Philipp-Foliguet | Fuzzy segmentation of color images and indexing of fuzzy regions[END_REF].

According to this representation, a raster representation of a fuzzy spatial object could be a matrix for which each cell value is computed using the membership function associated to the studied region. Guesguen and Hertzberg (2001) propose some algorithms in order to define fuzzy raster map. [START_REF] Duff | An evaluation of buffering algorithms in fuzzy GISs[END_REF] develop the approach.

C. Fuzzy spatial object storage as vector

There are two possible ways to store fuzzy data as vectors but both are based on multi-representation principle. For Dilo, de [START_REF] Dilo | Definition and implementation of vague objects[END_REF], "to store a primitive vague region it suffices to store the boundary of its core, together with the boundary of its support set, as simple vague lines.

[They] assume the membership value between the two boundaries decreases linearly". We should remark that this approach propagates, by the linear transformation, not only the imprecision but also the possible over-interpretation according to scale. The Figure 2 illustrates the storage of vague region according to Dilo de By and Stein. ( 2004). [START_REF] Morris | UGML: an extension of GML to provide support for geographic objects with uncertain boundaries[END_REF] use the alpha-cuts principle (definition of crisp subsets on a fuzzy set). They consider a number of alpha-cuts for each fuzzy spatial object (only one if the object has a crisp/precise boundary), and each alphacut of that object will represent the boundary of that object with a certain degree of membership. The system may store any number of alpha-cuts for each fuzzy spatial object in accordance with the requiring degrees of precision as illustrated in Figure 3.

Morris and Petry proposed to store fuzzy spatial entities into UGML (Uncertain GML) files that allow us to describe multiple alpha-cut framework within the GML specification. In order to do that, UGML is based on the multiple spatial representations proposed to record spatial configuration according to temporal series.

As for UGML, this approach could simply be adapted in a classical GIS conception by replacing time associated to a spatial configuration by a degree of precision as presented in figure 3. It induces to store many spatial objects in the case of temporal series and fuzzy spatial entities. We should remark that this representation reduces the impact of overinterpretation according to scale.

According to each representation mode, the issue of object combination in the overlap surface arises. The merging of fuzzy geographical entities is more complex than classical topological approach of fusion. Taking imprecision into consideration implies the use of fuzzy combination operators (t-norm, t-conorm, etc.) for relations between overlapped regions.

There are eight topological relations between simple (classical or crisp) regions. As illustrated in figure 4, the fusion of fuzzy spatial entities is more complex. Tang (2004) proposes to define more topological relations between fuzzy spatial objects. [START_REF] Alboody | Fuzzy Intersection and Difference Model for Topological Relations[END_REF] define 152 topological relations between fuzzy regions reduced as kernel, support and boundary. In this paper, we focus on the overlap of fuzzy spatial entities.

A. Fusion of fuzzy geographical entities 1) A proposition of fuzzy geographical entity formalization

Let us define a fuzzy geographical entity A as an object described by:  A label or concept L A member of an ontology.

 A fuzzy set FS A describing its spatial representation.

The membership function μ A of FS A is defined on ℝ 2 .

 At least, a quantity (precise or not) valid in the region (e.g. a quantity Q P,A of a phytosanitary product P using by spatial unit measure).

A quantity can be precise and be stored as a positive real number q PA . It can also be imprecise and be stored as a fuzzy set FQ A (with a membsership function γ P,A defined on ℝ +) .

During the merging, even if the quantity is precisely defined, the obtained result may be imprecise. Thus, precise quantity are represented by a singleton in the fuzzy set theory having (γ P,A (q)=1 if y=q P,A , else γ P,A (q)=0, q belongs to ℝ + ).

2) Aggregation in an overlap region

Now, let us consider two fuzzy geographical A and B entities that overlap in A⨂B (a set of x). x is a location where μ A (x) (and μ B (x)) takes a unique value in ℝ + .

The fusion A⨂B could be view as the aggregation of their spatial representation membership degree, the aggregation of their quantity for each present product and the affectation of a label corresponding to A and to B.

The label affected to x could simply be the lowest parent in the ontology shared by A and B (if the labels of A and B are different).

The spatial membership function FS A⨂B associated for A⨂B to the affected label could be either a t-norm (as for example the minimum (1)) or a t-conorm (as for example the maximum (2)).

⨂ ( ) ( ( ) ( ))

(1)

⨂ ( ) ( ( ) ( )) (2)
If ( 1) is chosen, it could mean that we consider the intersection of A and B, if it is (2) it may mean that the union of A and B is used. In order to consider only the overlap region, (1) will be preferred in our application.

The aggregation of a quantity Q P,A⨂B (of phytosanitary product in Observox project) of a product P (with a membership function γ P,A⨂B ) should add the fuzzy sets FQ A and FQ B . In order to conserve all information about quantities, we use the traditional operator of addition in fuzzy set (3).

γ ⨂ ( ) ( (γ ( ) γ ( ))) (3)
The adding aspect of (3) allows cumulating the membership degrees if the supports of FQ A and FQ B are overlapped. It is our first choice to model fusion of agricultural information.

In general, more aggregation operators could be chosen as [START_REF] Truck | A tool for aggregation with words[END_REF]. Detyniecki proposes in (2000) a review of aggregation operators.

3) On the point of view of a specific location x

If the objective is to obtain a fuzzy quantity Q P,X (with γ P,X ) of P for each location x of space in the overlap region between A and B, we have to consider both the membership of x to A and B and also FQ A and FQ B .

The first option could be to consider the previous approach and thus to obtain Q P,X using Q P,A⨂B and FS A⨂B as propose in (4).

γ ( ) ⨂ ( ) γ ⨂ ( ) (4)
The second possibility is to consider that the confidence in Q P,A should be relativized by the membership degree ( ) in order to define Q P,A(X) with its membership function γ ( ) ( ) as proposed in (5).

γ ( ) ( ) ( ) γ ( ) (5)
Using this hypothesis, we define Q P,X using (6) for the definition of its membership function.

γ ( ) ( (γ ( )( ) γ ( )( ))) (6)
Those two possibilities can be viewed as upper and lower limits of Q P,X . In our approach, the first objective is to give indices with a degree of relativity, and we will give the two degrees.

B. Fusion according to Storage Mode

In order to reduce the time of analyses and computing process, our system has to minimize the number of merging calculi.

The intersection of fuzzy geographical entities stored as raster will be computed on each raster cell. In accordance with this operator, a misinterpretation (over and lower) of data is possible. Furthermore, the storage of those regions is space expensive and thus time expensive in analyses and distant access.

In [START_REF] Dilo | Definition and implementation of vague objects[END_REF] representation mode, the combination should be computed in accordance to the linear diffusion of precision. The overlap region may simply be obtained using a topological division. This approach minimizes the space cost but may increase the time of analyses.

Using the multiple alpha-cut framework, the intersection of two fuzzy region could be viewed as a set of all intersections between the crisp set corresponding to the two fuzzy regions alpha-cuts. It will efficiency be obtained using a topological division. In this approach, the overinterpretation is limited by the combination of imprecision and the pre-minimization due to alpha-cut storage. Furthermore, the access time is reduced in comparison to other modes and the post computing cost is limited. The space cost is intermediary. This is the approach chosen in our project. We prefer it because it presents an intermediary between the two others and it limits the interpretation impact of vector representation mode.

IV. CONCLUSION

In this paper, we propose a study of the fusion between imprecise geographical entities in accordance with the storage representation mode.

We highlight the impact of the representation mode for data interpretation and fusion (we focus on the intersection/overlap). Thus, we give a reading grid of the link between data storage, data fusion and data analysis.

The thread of this paper is the future construction of an agricultural practice observatory. In our future work, we would use this study for data storage but we also want to study the propagation of quantitative imprecise information into other topological relations between fuzzy spatial objects. This paper is a preliminary study for the building of the observatory. It presents our choice at the beginning of the project. In our future work, we would validate our approach and modeling choices by practice and indices construction in the observatory.

Figure 2 .

 2 Figure 2. Storage of vague region according to[START_REF] Dilo | Definition and implementation of vague objects[END_REF]

Figure 4 .

 4 Figure 4. Two overlapped fuzzy regions

This work is funded by the Champagne-Ardenne Region Council, France and Europe (FEDER).