Multi-scale stochastic organization-oriented coarse-graining exemplified on the human mitotic checkpoint

The complexity of biological models makes methods for their analysis and understanding highly desirable. Here, we demonstrate the orchestration of various novel coarse-graining methods by applying them to the mitotic spindle assembly checkpoint. We begin with a detailed fine-grained spatial model in which individual molecules are simulated moving and reacting in a three-dimensional space. A sequence of manual and automatic coarse-grainings finally leads to the coarsest deterministic and stochastic models containing only four molecular species and four states for each kinetochore, respectively. We are able to relate each more coarse-grained level to a finer one, which allows us to relate model parameters between coarse-grainings and which provides a more precise meaning for the elements of the more abstract models. Furthermore, we discuss how organizational coarse-graining can be applied to spatial dynamics by showing spatial organizations during mitotic checkpoint inactivation. We demonstrate how these models lead to insights if the model has different “meaningful” behaviors that differ in the set of (molecular) species. We conclude that understanding, modeling and analyzing complex bio-molecular systems can greatly benefit from a set of coarse-graining methods that, ideally, can be automatically applied and that allow the different levels of abstraction to be related.


Reaction Network Models (Model -Model 3)
Model 1 as rea-file format. For the modified "short timescale" version (Model1b) we remove two reactions: "KinU -> KinA" and "O-Mad2 ->". The modified version represents a short timescale at which no kinetochore gets attached and O-Mad2 does not decay. Model 2 as rea-file format. For the modified "short timescale" version (Model2b) we remove reaction: "KinU -> KinA".

4/9
In this section we recall a method 24 for finding approximate aggregations of an ODE systemẋ = A(x) where A : R n → R n is a quadratic polynomial. This case is important as any set of interesting biochemical reaction will usually incorporate rules for producing one type of particles by combining two different types of reactants. This kind of dynamics translates to A being a polynomial map of degree two. Oddly enough, reactions involving terms of order 3 and higher are rarely encountered as they can, at least conceptually, be simulated as successive reactions of order two.
Firstly, recall that finding aggregations of the systemẋ = A(x) is equivalent to finding aggregations of the map x → A(x) 43 . It can be checked that any quadratic polynomial A : R n → R n can be written as where DA denotes the differential of A. Note that the second order term in the equation above is indeed linear as the other power of x is hidden within the differential. It is possible to write our function in this convenient form only because we assumed that A(x) is a quadratic polynomial and this sort of argument does not straightforwardly generalize to polynomials of higher degrees. Since entries of DA are affine maps we can further write where e i for 1 ≤ i ≤ n are unit vectors of the canonical basis in R n and x is a vector with entries x = (x 1 , . . . , x n ). Thus, Then applying Ξ to A(x) we get: where the last expression is clearly a function of Ξx which further means that Ξ is compatible with map A(x). The converse to this fact is also known to be true 45 . Together, this allows us to reduce our problem of finding aggregations for A to that of finding aggregations simultaneously compatible with each of the matrices DA(0), DA(e 1 ), . . . , DA(e n ).
In fact, rather than looking for aggregations compatible simultaneously with DA(0), DA(e 1 ), . . . , DA(e n ) which may not even exist, it seems more reasonable to look for a set of matrices DA(0), DA(e 1 ), . . . , DA(e n ) which approximate them in some matrix norm while at the same time being simultaneously compatible with a (preferably large) set of aggregations S. The first step in doing so would be to do this for just one n-by-n matrix M.
Recall that an m-by-n (m ≤ n) 0-1 matrix Ξ with exactly one entry equal to 1 in each column is called an aggregation matrix and these are in 1-1 correspondence with the partitions of the set {1, 2, . . . , n}. For each such Ξ let V Ξ be the set of all n-by-n matrices coarse grained by Ξ. One can show that this set is a linear subspace of the set of all matrices of order n which we denote by M n (R) 24 . Thus, after fixing some matrix norm, one can orthogonally project matrix M onto V Ξ in order to obtain the best approximation M which is coarse grained by Ξ.
But this only ensures that M is coarse grained by Ξ, whereas one would like to get as many coarse-grainings as possible. Note however that if Ξ 1 and Ξ 2 are two aggregation matrices, then V Ξ 1 ∩V Ξ 2 is again a linear subspace of M n (R) consisting precisely of matrices which are coarse grained by both Ξ 1 and Ξ 2 . Projecting orthogonally onto this subspace will yield an approximation of M which has at least those two valid reductions.
This idea extends to an arbitrary number of aggregation matrices, and one is inclined to ask how large a subset of aggregation matrices {Ξ 1 , . . . , Ξ r } ⊆ {Ξ p | p is a partition of {1, 2, . . . , n}} can be while ensuring that the distance from A to its orthogonal projection onto V Ξ 1 ∩ · · · ∩V Ξ r is kept within a given error threshold. Note that the number of partitions of 5/9 a set {1, . . . , n} is already super-exponential in n, and going through all subsets of those would yield a growth rate greater than doubly exponential, thus making any brute force approach ineffective.
Instead, in Algorithm 1 below we propose running through all aggregations and retaining in a set S only those for which induced projections produce an approximation matrix within the error threshold ε. The final approximation is then obtained by projecting onto p∈S V Ξ p . The reasoning behind this is that a matrix M that is ε-close to a subspace V ∩W cannot be further than ε from either V or W . This however is not a guarantee that the converse, which is needed here, is true.
Algorithm 1 Finding a nearby matrix that can be coarse grained It is not hard to modify Algorithm 1 so that it works with a set of matrices. We give the modification below as Algorithm 2. Again, there is no guarantee that the resulting matrices are ε close to the initial ones, but the same argument as before justifies this approach. Going back to our original problem, assume that we used this algorithm to obtain matrices DA(0), DA(e 1 ), . . . , DA(e n ) and the set of aggregations S that simultaneously coarse grains each of them. Then the approximate quadratic map A(x) is simply given by x i ( DA(e i ) − DA(0))x and the ODE systemẋ = A(x) can be aggregated using any of the coarse grainings from the set S.

Relation of this method to other model reduction techniques
The problem that standard model reduction techniques are attempting to solve usually consists of finding the evolution of a quantity of interest whose dynamics is modeled by a high-dimensional ODE system. The goal is then to find a low-dimensional model in which the evolution of the said quantity will match its actual evolution as accurately as possible. 46,47 In our approach we focus our attention to the high dimensional model itself and from there we attempt to derive quantities whose evolution can be exactly computed using reduced models. The guiding idea here being that it should be possible to automatically derive certain "conservation laws" that govern the dynamics of our system. The goal of our 6/9 approach is learning a hierarchical structure of the system under consideration and seeing how its organizational structure fits together. The model reduction part then comes as a by-product of this process.
Consequently, the approximate aggregation method discussed above favors models with rich structure, models with abundance of coarse grainings. This is perhaps best seen on an extremely simple example. The following two-dimensional linear ODE model is a modification of an example by Rowe  x y Our method on the contrary produces a system close to the original ẋ y = .015 .985 .985 .015 x y but in which variables x and y can be aggregated together to give a reduced model z = z where z = x + y.