Stochastic spatial discrete-event systems

Considering the above notations we define stochastic spatial discrete-event systems (SSpDES) as an extension of SDES with a set of spatial state variables SpSV and

a spatial value assignment function SpV .

Definition 5 Stochastic spatial discrete-event system (SSpDES) An SSpDES M is a 7-tuple hS, T , µ, NSV , SpSV , NV , SpV i where:

• hS, T , µ, NSV , NV i is a SDES (see Subsection 2.4.2.1, Definition 4); • SpSV is the set of spatial state variables;

• SpV is the spatial value assignment function.

The set SpSV contains all spatial state variables i.e. the variables recording the configuration of the discretised space in the current system state. The value of these variables is computed using the spatial value assignment function SpV :

SpV : E × S × SpSV → Rm×n+ ,

where E denotes the set of all possible model executions/simulations, S the set of states, SpSV the set of spatial state variables, and m and n the dimensions of the discretised space. Given a model simulation σ at state s and a spatial state variable spsv, SpV (σ, s, spsv) = sv such that sv ∈ Rm×n+ returns a m × n matrix of real non-negative values, where each element of the matrix corresponds to a position in the discretised space. For explanatory purposes an illustrative example of a simple SSpDES is provided below.

Example 8 Illustrative example of an SSpDES encoding the growth of a population of cells

Let us assume that we would like to model the growth of a population of cells in a fixed size environment. For simplicity purposes let us consider that the environment comprises 2 × 2 spatial compartments where each compartment can hold at most one cell. Cells can be of two types, wild type (A) or mutant (B). The probability of obtaining a type A/B offspring cell when a parent cell of type A/B divides is 70%, respectively 30% if the parent cell is of type B/A. Since each compartment can be occupied by at most one cell, whenever a parent cell divides the offspring cell is displaced to a neighbouring compartment. Two compartments are neighbouring if the Manhattan distance between them is at most 1. Finally the cell population survival condition is that the concentration of O2 in the environment is greater or equal to 50%. Each new type A cell reduces the O2 concentration with 20%, and type B cell by 15%.

Although the above described scenario is not realistic for practical applications, it is sufficient to illustrate how an SSpDES model can be constructed for a biological system which evolves in time and space. The reason for strongly constraining the size of the environment, the neighbourhood relation between different compartments and the behaviour of the cells was to limit the number of possible system states such that they can all be explicitly enumerated.

The behaviour of this simple system is characterised at each moment in time by a set of state variables. Spatial state variables of interest are Cells A and Cells B representing the number of type A, respectively type B cells in the environment. Conversely the numeric state variable O 2 is used to record the concentration of O2 in the environment. Considering these spatial and numeric state variables the initial state/configuration S0 of the system is depicted in Figure 3.2.

State S₀

Cells_A Cells_B O_2

80% 0 0 1 0 0 0 0 0

Figure 3.2: Initial state of SSpDES encoding the growth of a population of cells in a fixed size environment. Cells A and Cells B are the spatial state variables representing the number of type A, respectively type B cells in the environment. O 2 represents the current concentration of O2 in the

environment.

Starting from S0 the system probabilistically transitions from one state to the next until it reaches its final configuration; see Figure 3.3 for all possible states which can be reached starting from the initial state.

Considering the initial state S0 the system can transition to four possible states described by the following behaviours: the type A cell from the lower right corner either divides and the offspring is of the same type (S1, S3) or of type B

State S₀

Cells_A Cells_B O_2

80% 0 0 1 0 0 0 0 0 State S₁

Cells_A Cells_B O_2

60% 0 1 1 0 0 0 0 0 State S₃

Cells_A Cells_B O_2

60% 0 0 1 1 0 0 0 0 State S₂

Cells_A Cells_B O_2

65% 0 0 1 0 0 1 0 0 State S₄

Cells_A Cells_B O_2

65% 0 0 1 0 0 0 1 0 State S₆

Cells_A Cells_B O_2

50% 0 0 1 0 0 1 1 0 State S₅

Cells_A Cells_B O_2

50% 0 0 1 0 1 1 0 0 State S₇

Cells_A Cells_B O_2

50% 0 0 1 0 1 0 1 0 35% 15% 35% 15% 30% 30% 70% 70%

Figure 3.3: The state space of SSpDES encoding the growth of a population of cells in a fixed size environment i.e. all possible states which can be reached from the initial state S0. Cells A

and Cells B are the spatial state variables representing the number of type A, respectively type B cells in the environment. O 2 represents the current concentration of O2 in the environment. The

percentages associated with the arrows connecting each pair of states represent the probability of transitioning between states.

(S2, S4). In both cases the offspring can be either displaced above the parent (S3, S4) or to its left (S1, S2). Given that the overall probability of a cell to produce offspring of the same type is 70% and in our case there are 2 relevant state transitions (S0 → S1, S0 → S3), the probability associated with each of these state transitions is 70% / 2 = 35%. Analogously the probability associated with each state transition where the offspring cell is of different type (S0 → S2, S0 → S4) is equal to 30% / 2 = 15%. The concentration of O2 has been decreased by 20% in states S1 and S3 due to a new type A cell, respectively by 15% in states S2 and S4 due to a new type B cell. Therefore the O2 level is 80% - 20% = 60% in states S1 and S3, and 80% - 15% = 65% in states S2 and S4. Since the birth of a new cell reduces the O2 concentration by at least 15%, and the minimal O2 concentration required by the cell population to survive is 50%, no further cellular division can occur starting from states S1 and S3 (60% - 15% < 50%). Conversely starting from states S2 and S4 at most one new type B cell can be created (65% - 15% ≥ 50%). Given state S2 a type B cell can be produced either from the existing type A (S2 → S6, probability 30%) or type B (S2 → S5, probability 70%) cell. Similarly given state S4 a type B cell can be produced either from the existing type A (S4 → S6, probability 30%) or type B (S4 → S7, probability 70%) cell.

Using the above descriptions the formal SSpDES M = hS, T , µ, NSV , SpSV , NV , SpV i corresponding to the system is defined as follows:

• S = {S0, S1, S2, S3, S4, S5, S6, S7}. • T = S0 S1 S2 S3 S4 S5 S6 S7                               S0 0 35% 15% 35% 15% 0 0 0 S1 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 70% 30% 0 S3 0 0 0 0 0 0 0 0 S4 0 0 0 0 0 0 30% 70% S5 0 0 0 0 0 0 0 0 S6 0 0 0 0 0 0 0 0 S7 0 0 0 0 0 0 0 0 .

• µ is the function used to compute probabilities associated with cylinder sets C(σfinite) defined over finite computation path prefixes σfinite. The probability value associated with C(σfinite) is computed by multiplying the probabilities of the state transitions encoded by σfinite. For instance, if σfinite = {S0, S2, S6} then µ(C(σfinite)) = P (S0, S2) · P (S2, S6) = T [S0, S2] · T [S2, S6] = 15% · 30% = 4.5%.

• NSV = {O 2}, and NV is the function used to compute the value of O 2 in the current system state.

• SpSV = {Cells A, Cells B}, and SpV is the function used to evaluate Cells A and Cells B in the current system state.

Although only a simple example was considered here the same modelling principles are employed to construct SSpDES models of more complex (realistic) systems. One of the main differences is that due to the high complexity associated with some real systems the number of possible system states is very large, even potentially infinite. Therefore in such cases explicitly enumerating all possible paths starting from the initial state is not feasible in reasonable time.

Remark 3. The probabilities employed in Example 8 were chosen for explanatory purposes and were not derived from experimental data or the literature.

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The size of the discretised space and the semantics of the values stored for each spatial compartment depend on the addressed biological problem. In general the granularity of the discretised spatial domain should be sufficiently fine to enable answering the biological problem of interest but not finer than that. The main reason for this is that increasing the resolution of the discretised space through

fine-graining leads to a potential increase in the size of the state space and/or model simulation time. Conversely reducing the resolution of the discretised space too much may lead to large approximation errors and/or biologically irrelevant conclusions.

Finally one of the main advantages of defining SSpDESs as an extension of SDESs is backwards compatibility i.e. existing SDES models can be interpreted as SSpDESs having an empty set of spatial state variables SpSV . Moreover SSpDESs enable scaling up the development of computational models by extending existing non-spatial models, typical for subcellular scales (e.g. intracellular networks), with spatial information relevant to potentially higher scales (e.g. cellular/tissue level).