Supplementary Material: Advection, diffusion and delivery over a network. Luke L.M. Heaton 1,2 , Eduardo L´ opez 2,3 , Philip K. Maini 3,4,5 , Mark D. Fricker 3,6 , Nick S. Jones 2,3,5,7 1 LSI DTC, Wolfson Building, University of Oxford, Parks Road, Oxford, OX1 3QD 2 Physics Department, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU 3 CABDyN Complexity Centre, Sa¨ ıd Business School, University of Oxford, Park End Street, Oxford, OX1 1HP 4 Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB 5 Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, Oxford, OX1 3QU 6 Department of Plant Sciences, University of Oxford, South Parks Road, Oxford, OX1 3RB 7 Department of Mathematics, Imperial College London, SW7 2AZ ∗ In the Supplementary Material we detail the mathematical machinery involved in solving the advection, diffusion and delivery equation over a network. In Section A we describe how to solve the particular case of stepwise constant initial conditions. In Section B we show how to calculate the concentration of resource that leaves its initial edge over the time step in question. In Section C we show how to calculate the concentration of resource that remains in the edge in which it started, and in Section D we describe how to calculate the total quantity of resource in each section of the network. Finally, in Section E we describe the Gaver-Stehfest algorithm for inverting our solutions from Laplace space into the time domain. SOLVING ADVECTION, DIFFUSION AND DELIVERY IN LAPLACE SPACE We are interested in calculating how the quantity of resource in a network changes over time, given that the resource decays or is delivered out of the network at a given rate, and is subject to advection and diffusion. In other words, we wish to solve a system of equations de- fined over a network, where the resource in edge ij of the network is governed by an equation of the form ∂q ij ∂t + R ij q ij + u ij ∂q ij ∂x − D ij ∂ 2 q ij ∂x 2 =0, (1) where q ij is the quantity of resource per unit length, u ij is the mean velocity, D ij is the dispersion coeffi- cient and R ij is the rate at which a unit of resource is lost, or delivered out of the network. As we are in- terested in the case where the advective velocities u ij may vary over several orders of magnitude, it is conve- nient to operate in Laplace space, and invert our solutions back into the time domain by using the Gaver-Stehfest algorithm. Note that after taking Laplace transforms L ( q ij (x, t) ) =  ∞ 0 q ij (x, t)e −st dt = Q ij (x, s), the funda- mental Equation (1) becomes (s + R ij )Q ij + u ij ∂Q ij ∂x − D ij ∂ 2 Q ij ∂x 2 = q ij (x, 0). (2) Also note that as in the Main Text, for each edge ij and every s> 0 we let α ij (s)=  u 2 ij +4D ij (s + R ij ), * nick.jones@imperial.ac.uk g ij = u ij l ij 2D ij and h ij (s)= α ij (s)l ij 2D ij . A. Stepwise constant initial conditions We are interested in calculating how the quantity of resource in a network changes over time, given that the resource is subject to the fundamental Equation (1). In particular, it is convenient to consider a stepwise constant initial condition, as we can then calculate how the total quantity of resource in each segment of the network has changed by time t. The first step in this calculation is to find the Laplace transform of the concentrations at each node ¯ C (s). As we have seen, to calculate ¯ C (s) we must first find M ij (s) and ¯ Υ(s), which do not depend on the initial condition. For each sample point s and each edge ij we must also calculate β ij (s) and β ji (s), which capture the effect of the initial condition q ij (x, 0). In particular, we start this subsection by considering the case where the initial condition is q ij (x, 0) =  k if n−1 N l ij ≤ x< n N l ij 0 otherwise, where n ≤ N , before moving on to consider the more general case of stepwise constant initial conditions. For the sake of clarity we drop the subscripts ij from l ij , N ij , R ij , α ij , g ij and h ij , and ignore the dependence on s of the terms α ij and h ij . Now, to find a particular solution to the fundamental Equation (2) we use the method of