SciCode / 44.2 / Let r_(i) denote the concentration of all chains ending with monomer i, and…

Background
- The concentration of chains ending with i can be counted by r_(i)(t)=c_(i)−∑_(k)d_(ik)(t)
- The concentration of chains starting with i can be counted by l_(i)(t)=c_(i)−∑_(k)d_(ki)(t)

Background
- The concentration of chains ending with i can be counted by $r_i(t)=c_i-\sum_k d_{i k}(t)$
- The concentration of chains starting with i can be counted by $l_i(t)=c_i-\sum_k d_{k i}(t)$

Let r_(i) denote the concentration of all chains ending with monomer i, and l_(j) the concentration of all chains starting with monomer j. They can be calculated from the concentration of each type of monomer c_(i) (which is a constant throughout the process) and the 2-mer concetrations d_(ij). In the night phase, when 2 ends i and j of such chains meet due to hybridization with a complementary template j'i', they are ligated at rate λ_(ij) and form a new 2-mer ij; in the day phase, 2-mers break up spontaneously at a given rate. Let's describe this process by the master euqation $\dot{d}_{ij}(t)=\lambda_{i j} \cdot r_i(t) \cdot l_j(t) \cdot d_{j' i'}(t)-d_{i j}(t)$, where all the breakage rates are set to 1 for simplicity. To integrate this ODE method, write a function that returns the change of d_(ij) (flattened to 1d) during timestep δt. The inputs are the following: the time t; the flattened d_(ij) matrix, noted as y; the concentration of each type of monomer c_(i), and the ligation rate matrix λ_(ij).

Let $r_i$ denote the concentration of all chains ending with monomer i, and $l_j$ the concentration of all chains starting with monomer j. They can be calculated from the concentration of each type of monomer $c_i$ (which is a constant throughout the process) and the 2-mer concetrations $d_{ij}$. In the night phase, when 2 ends i and j of such chains meet due to hybridization with a complementary template j'i', they are ligated at rate $\lambda_{ij}$ and form a new 2-mer ij; in the day phase, 2-mers break up spontaneously at a given rate. Let's describe this process by the master euqation $\dot{d}_{ij}(t)=\lambda_{i j} \cdot r_i(t) \cdot l_j(t) \cdot d_{j' i'}(t)-d_{i j}(t)$, where all the breakage rates are set to 1 for simplicity. To integrate this ODE method, write a function that returns the change of $d_{ij}$ (flattened to 1d) during timestep $\delta t$. The inputs are the following: the time t; the flattened $d_{ij}$ matrix, noted as y; the concentration of each type of monomer $c_i$, and the ligation rate matrix $\lambda_{ij}$.