Modeling breast cancer progression to bone: how driver mutation order and metabolism matter

Background Not all the mutations are equally important for the development of metastasis. What about their order? The survival of cancer cells from the primary tumour site to the secondary seeding sites depends on the occurrence of very few driver mutations promoting oncogenic cell behaviours. Usually these driver mutations are among the most effective clinically actionable target markers. The quantitative evaluation of the effects of a mutation across primary and secondary sites is an important challenging problem that can lead to better predictability of cancer progression trajectory. Results We introduce a quantitative model in the framework of Cellular Automata to investigate the effects of metabolic mutations and mutation order on cancer stemness and tumour cell migration from breast, blood to bone metastasised sites. Our approach models three types of mutations: driver, the order of which is relevant for the dynamics, metabolic which support cancer growth and are estimated from existing databases, and non–driver mutations. We integrate the model with bioinformatics analysis on a cancer mutation database that shows metabolism-modifying alterations constitute an important class of key cancer mutations. Conclusions Our work provides a quantitative basis of how the order of driver mutations and the number of mutations altering metabolic processis matter for different cancer clones through their progression in breast, blood and bone compartments. This work is innovative because of multi compartment analysis and could impact proliferation of therapy-resistant clonal populations and patient survival. Mathematical modelling of the order of mutations is presented in terms of operators in an accessible way to the broad community of researchers in cancer models so to inspire further developments of this useful (and underused in biomedical models) methodology. We believe our results and the theoretical framework could also suggest experiments to measure the overall personalised cancer mutational signature. Electronic supplementary material The online version of this article (10.1186/s12920-019-0541-4) contains supplementary material, which is available to authorized users.

C is the matrix in [0, 1] k×m d describing which mutated gene is considered necessary in a given compartment where the rows η k represent the genes and the columns C j identify a gene in the compartments. One or more genes may be important in various compartments and C may not be a full rank matrix.
Let us define a state vector m j ∈ R 2 corresponding to the gene g j in the mutations representation so that, in the standard basis B mj = 1 0 0 1 , we can introduce the mutation observable as a diagonal matrix operator which acts on the respective eigenvectors as follows: M j (m j ) = λ mj m j . The mutation operator has two eigenvalues λ mj : 1 and -1 representing the case gene g j is mutated or not mutated, respectively. The corresponding eigenvectors are equal to the basis vectors. Generally, there are multiple genes involved in the dysfunction of a cell or in the appearance of a new oncogenic behaviour. Hence, the vector space of the genes' multitude can be represented as the direct product of single gene vectors: and, similarly, the mutation operator for multiple genes is given by: where, formally, the symbol ⊗ can be interpreted as the Kronecker product of the operators M j . The mutations operator applied to an eigenvector of genes returns the eigenvector times the product of all the genes' eigenstates: From the mutation operator M j and the elements c kj of the matrix C of necessary genes in a compartment, we derive the operator for a single gene: which measures the deregulation introduced by a gene in a set of important alterable genes like those belonging to a genetic pathway. D kj is diagonal in the mutations representation (B mj ), and the two eigenvalues λ D kj are degenerate with value 1 if the gene g j is unimportant in compartment k, while they are non degenerate if the gene is important. In the latter case, λ D kj is 0 when g j mutated and 1 otherwise. As shown before, we can use the direct product over all the genes to define the deregulated pathway operator referring to compartment k (or to a genetic pathway) for all the involved genes: If the state vector m is an eigenvector of M , it is also an eigenvector of the deregulated pathway operator: and the eigenvalue λ D k ∈ {0, 1} determines if all the genes in compartment k are mutated. In case we want to compare the state of two different cells or the states of a cell at two different times given by m1 and m2, we define the inner product m2, m1 = δ m1,m2 so that: It is straightforward noticing that the deregulated pathway operator allows us to properly address the problem of inter compartmental unordered cumulated mutations and the intra-compartmental order of mutation through the index k just by imposing the orthogonality between elements belonging to different compartments.
In fact, if k1 and k2 are two generic compartments, and m k1 and m k2 are the mutation vectors referring to the multiple genes in the respective compartments, then: To easily deal with the order of mutation among compartments in terms of operators, we need to extend the notation by defining the level state vector ∈ L where the vector space L is given by the direct sum of k identical M space vector, which for convenience we rename as M k , such that: Let us call k the gene state eigenvectors of a cell in compartment k such that the components from 2 m d +k + 1 to 2 m d +k+1 are identical to the respective eigenvectors m k ∈ M k in the k-th compartment. In this representation, we can redefine and unify the set of deregulated pathway operators in a simpler form: It is important to stress all the state vectors defined above are all factorizable because genes and compartments are all considered independent.

Additional File 1..2 Ladder operators and effective driver mutations
The deregulated pathways operator D is diagonal in the representation of the orthonormal level basis B k = { 1 , 2 , . . . , k }, and it only has two degenerate eigenvalues λ D equal to 0 and 1; the null eigenvalue means not all the required genes involved in the regulation/compensation of a specific pathway are mutated so as to not result in further oncogenic activity, and the latter eigenvalue means the prooncogenic behavior is present and the cell can use it when needed. Let us introduce the ladder operators for the level of the path L + and L − . The two operators act on the basis by increasing or decreasing the level respectively. Therefore, the creation operator gives L + ( k ) = k+1 for any integer 1 ≤ k < k, and it returns 0 when applied on k . In the opposite way, the action of the annihilation operator is L − ( k ) = k−1 for any integer 1 < k ≤ k, and it gives 0 when applied on 1 .
The ladder operators are the transpose of one another: L + = L − T . For extended discussions on ladder operators and their applications see [1,2,3], Coupling the degenerate operator and the ladder operators such that B + = L + D and B − = D L − is useful to determine which one of a succession of ordered events in the cell path represents a barrier. The barrier operators B + and B − applied to the elements of the basis vectors B k give In order to determine the value of the transition rate of each cell to jump from one compartment to the next, it is necessary to determine how many driver mutations are effective for the specific cell. In a system with a finite number of levels, meaning a finite number of compartments, this computation can be achieved by using the effective ladder operators E ± = 1 1− B ± . The result of the effective ladder operators on the basis vector k is: where the factors j i=0 λ D k±i are different from zero only when all the ordered sequence of driver mutations previous to (in case of E − ) and afterwards (in case of E + ) the unknown value m e in a given compartment k have already been acquired. The number of effective driver mutations m e for a cell in compartment k is finally retrieved by the square of the norm of E − ( k ) as follows: This result can be applied to find the value of r pass of each cell in each compartment and at each simulation step by knowing the cell mutated genes.