Abstract
We discuss our approach to the detailed computational modeling of the molecular processes involved in early signaling of membrane-bound receptors, typically exemplified by members of the receptor tyrosine kinase (RTK) family. This includes receptors whose mutations are associated with increased risk of cancers (ErbB2) or are involved in the survival of nascent tumors (Kdr). Current imaging methods can visualize individual molecules in the context of the living cell, allowing the direct observation of molecular movement and transformations. Modeling and simulation are necessary to connect these observations to the cell-level kinetics of signaling and to help reveal connections between molecular properties and cell signaling, under both normal and pathological conditions. We describe the relevant methods and provide a minimal mathematical justification for the reader interested in understanding or applying them. The chapter builds up from the simplest modeling approach to the fully spatial, agent-based simulation that is currently used by our group. This should be useful from a tutorial perspective and also to provide the proper connections between models at different levels of granularity.
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Notes
- 1.
In a “fundamentalist” version of reductionism (motivated by nineteenth century classical Physics, but long since abandoned in the physical sciences [40]), the properties of a system must be a sum of the properties of its parts, negating the possibility of emerging system properties.
- 2.
There is a distinction between processes that are intrinsically random and deterministic chaotic behaviors that can be treated as random.
- 3.
We implicitly assume here that the reactions take place between substances in the same solution, and therefore the stoichiometric coefficients carry over from the reactions to the concentrations.
- 4.
Such as concentration, mass, etc.
- 5.
Note that for an odd number of trials the number of heads and tails cannot be equal, increasing the likelihood of a given relative difference.
- 6.
Strictly speaking, only the limit version of Eq. (6) is exact; the second form is for \(\varDelta p \ll 1\).
- 7.
- 8.
If the reacting particles are identical, we have a combinatorial factor: \(Z_\mathrm {A+A\rightarrow C} = \frac{1}{2} n_A\cdot (n_A - 1) \).
- 9.
Note that (20) approaches a Dirac delta at \(t=0\): \(\lim \limits _{t\rightarrow 0} p(x,y;\,t)=\delta (x)\delta (y)\).
- 10.
This is justified at low overall densities, and also when the density of the particle species that are explicitly modeled is sufficiently low.
- 11.
When the same trajectory is sampled for intervals corresponding to multiple frames, the intervals must not overlap in order to avoid oversampling.
- 12.
To obtain (26) we need to first change variables from (x, y) to \((r,\alpha ) \rightarrow \text {d}x \text {d}y = r \text {d}r \text {d}\alpha \); integrate out the angle \(\alpha \), then change variable to \(w=r^2 \rightarrow \text {d}w = 2 r \text {d}r\).
- 13.
With the one possible exception of nonspatial, but “rule based” simulations that keep track of individual molecule parts [12].
- 14.
A potentially confusing practice is to characterize the amount of a membrane bound species in terms of a (volume) concentration that reflects the copy number per cell multiplied by the concentration of cells in a suspension.
- 15.
We will discuss intermolecular collisions in the context of reaction–diffusion models.
- 16.
For a proper rendering of the Michaelis–Menten mechanism, one should include an intermediate, reversible step.
- 17.
All four reaction instances can be encoded as changes to the states of two receptor molecules: Receptor 1 forms a dimer with Receptor 2 which is initially liganded; then Receptor 1 is phosphorylated; after that, Receptor 2 loses its ligand; finally, the dimer breaks up.
- 18.
In the case of membrane-bound species, the volume is replaced by area, and the conversion between the per-pair propensity and the physical (effective) rate constant has to account for the units used for concentration. As we pointed out, sometimes molar concentrations are used for receptors, representing a molar concentration based on a density in suspension of a specific type of cell, and the average number of receptors per cell.
- 19.
If nothing else, the collision may be elastic.
- 20.
Alternatives include using a reaction probability that controls whether a reaction occurs upon collision, as well as manipulating the unbinding radius discussed in the next paragraph.
- 21.
An alternative would be to use a larger binding radius and trigger reactions with probability \({<}1\).
- 22.
In our current EGF related work [17], all membrane bound species are either monomer or dimer receptors; which may only participate in dimerization or dissociation reactions, respectively.
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Acknowledgements
This work was supported by NIH CA119232 (BSW), NIH P50GM085273 (BSW), R01GM104973 (JSE and ÁMH), and NIH K25CA131558 (ÁMH). MMP was supported in part by the U.S. Department of Energy through the LANL/LDRD Program.
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Halász, Á.M., Pryor, M.M., Wilson, B.S., Edwards, J.S. (2017). Spatiotemporal Modeling of Membrane Receptors. In: Graw, F., Matthäus, F., Pahle, J. (eds) Modeling Cellular Systems. Contributions in Mathematical and Computational Sciences, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-45833-5_1
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