Spatio-Temporal Modelling of Intracellular Signalling Pathways: Transcription Factors, Negative Feedback Systems and Oscillations

Abstract

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. One such class of pathways is those involving transcription factors (e.g. Hes 1, p53-Mdm2, NF-κ B, heat-shock proteins) which often exhibit oscillations in both space and time. In this chapter we present a partial differential equation model of the transcription factor, Hes 1. Our model considers the dynamics of Hes 1 in a 2-dimensional cellular domain including a nucleus, cytoplasm and microtubule-organising centre (MTOC). Spatial movement of the molecules (protein, mRNA) is assumed to be by diffusion, and also convection along microtubules. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. In order to bridge the gap between in vivo and in silico experiments we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain.

Non-dimensionalisation of Hes1 model

We summarise our non-dimensionalisation of the extended Hesl model (described in Sect. 2.2). The original Hesl model (described in Sect. 2.l above) is non-dimensiona- lised in a similar way - for details, see [47].

To non-dimensionalise the extended Hesl model given by Eqs. (1)-4 and (15), subject to the conditions in Eqs. 8-(14), we first define re-scaled variables by dividing each variable by a reference value. Re-scaled variables are given overlines to distinguish them from variables that are not re-scaled. Thus we can write:where the right hand side of each equation is a dimensional variable divided by its reference value. From Eq. (19), we can write variables in terms of re-scaled variables and then substitute these expressions into Eqs. (1)-4 and (14), and into the conditions in Eqs. 8-(14). This gives a model defined in terms of re-scaled variables which has the same form as the dimensional model but now the parameters are all non-dimensional. Denoting the non-dimensional parameters with an asterisk, they are related to dimensional parameters as follows:

Fig. 19

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Fig. 20

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We solve the non-dimensional model using the method described in Sect. 2.1. We simulate the model in COMSOL 3.5a, finding non-dimensional parameter values that yield oscillatory dynamics. We chose the same values as in Eq. 25 in [47] except for those parameters which were new because of our extension to the model. These latter values were chosen as follows: ${D}_m^{*}={D}_{i_j}^{*}/5,{D}_p^{*}={D}_{i_j}^{*}/15,{d}^{*}=0.01,{a}^{*}=0.03,{l}^{*}=0.63.$

Finally, we calculated the dimensional parameter values. To do this, we needed to estimate the reference values. Since Herl in zebrafish and Hesl in mice are both pathways connected with somitogenesis, we used the reference concentrations for Herl protein and herl mRNA in [50] as our reference concentrations for Hesl protein and hesl mRNA. Thus, we chose [m₀] = l.5 × l0^-9M and [p₀] = l0^-9M. We assumed a cell to be of width 30μm. But from Fig. 2 and Fig. 4, the cell width is equal to 3 non-dimensional spatial units or 3L dimensional units (using (l9)). Hence we set 3L = 30μm, so that L = l0μm. The experimentally observed period of oscillations of Hesl is approximately 2 hours [l6]. Our simulations of the non-dimensionalised model gave oscillations with a period of approximately 300 non-dimensional time units or 300τ dimensional units (using (l9)). Hence we set 300τ = 2hrs = 7200s, so that τ = 24s. Using our references values and non-dimensional parameter values, we found dimensional parameter values from 20.

Note that we chose our reference time τ = 24s based on simulations of the extended Hesl model since this was our most realistic Hesl model. For the original Hesl model and for all special cases of the Hesl model (for example, setting active transport rates to zero), we retained the reference time τ = 24s.