Cancer-immune stochastic model. A Cancer cells (red) stochastically divide and die with rates b and d, respectively (here Ø represents no cell). B During a division event, daughter cells inherit all of their mother cell’s antigenic and neutral mutations, depicted by numbers (where underlined numbers are antigenic). Cells carrying antigenic mutations have probability pe to escape and become neutral, as shown in the lower daughter cell with mutation 2. Each daughter cell also acquires a random number (drawn from a Poisson distribution with mean λ) of new mutations, where each mutation is antigenic with probability pa and neutral with probability 1 − pa. C For each antigenic mutation i present in the system, a corresponding effector cell population Ei exists (blue), which grows with constant rate B and shrinks with per-capita death rate D. D Antigenic mutations in cancer cells (such as i and k) display unique neoantigens at the cell surface, whereas neutral mutations (such as j) do not. The neoantigens can be identified by specialised effector cells, which can only interact with the corresponding cancer cells. E When a cancer cell carrying the antigenic mutation i meets an effector cell of type i, three outcomes are possible: active recruitment of another effector cell of type i with rate αi, killing of the cancer cell with rate βi and inhibition/exhaustion of the effector cell with rate γi. The expressions for the rates are found in equation (1).

Single stochastic realisations for λ = 1 (A-B) and λ = 10 (C-D). A & C Population dynamics, where red and blue lines depict total cancer and effector cell populations. B & D Average immunogenicity (yellow line) and antigenicity (orange line) in the effector population for the corresponding realisation. Interaction parameters: α0 = 0.03 and β0 = 0.3 (A-B) and α0 = 0.005 and β0 = 0.01 (C-D).

Outcome heat maps for λ = 1 (A-C) and λ = 10 (D-F) for tumours interacting with an immune system characterised by cancer cell-effector cell interaction parameters α0 (active recruitment of effectors) and β0 (killing of cancer cells). A & D Proportion of suppressed tumours. B & E Proportion of extinct tumours. C & F Proportion of slow-growing tumours: tumours that are suppressed but do not go extinct. Points a and b (c and d for λ = 10) label parameter sets of low and high immune effectiveness, respectively. The red dashed line denotes γ0 (here γ0 = 10−3), the rate of inhibition/exhaustion (see Section A4 for further discussion); see the Discussion section for details of point e. All parameter values not specified here are listed in Table 1.

Notation and baseline parameter values, which are used for all simulations unless otherwise specified.

Average number of antigenic mutations per cell (solid oranges lines) for several representative realisations when λ = 1. Theoretical prediction for the accumulation of neutral mutations per cell in an exponentially-growing population shown in grey dashed line. A Low immune effect: α0 = 0.002 and β0 = 0.001. B High immune effect: α0 = 0.03 and β0 = 0.3. (Parameter sets chosen as points a and b from Figure 3A.)

Genetic markers of selection and cancer-immune interactions for λ = 1. We present the SFS of neutral (A & E) and antigenic mutations (B & F) and the MBD of neutral (C & G) and antigenic mutations (D & H). Left-column panels depict low immune effectiveness (α0 = 0.002 and β0 = 0.001) and right-column panels depict high immune effectiveness (α0 = 0.03 and β0 = 0.3). Black dashed lines are the theoretical predictions in the absence of an immune response. The dashed vertical lines represent the means of the MBDs in panels C, D, G & H. Results are averaged over 100 realisations, and all parameter values not specified here are listed in Table 1.

The net difference Δ1 (solid pink line) between active recruitment α1 (dotted blue line) and inhibition/exhaustion γ1 (dashed red line) interactions between effector and cancer cells splits the cancer cell population C1 into three natural regions: [0, C*), [C*, C**) and [C**, ∞). Note that the active recruitment α1 is a type-II functional response (reaching a maximum of as C1 → ∞), whose shape more resembles the dotted pale blue line than our piecewise linear approximation shown with a thicker line; this also translates to the thinner solid pink line for Δ1 = α1 − γ1.

Neoantigen burden distribution (NBD) (yellow lines) and the corresponding distribution for immunogenicity (purple lines) averaged over 100 realisations, when λ = 1 (A-B) and λ = 10 (C-D), for both low (dotted lines) and high (dashed lines) immune effectiveness. All parameter values not specified here are listed in Table 1 of the main text.

Genetic evidence of selection for a faster-acting immune system. A-B Average number of antigenic mutations per cell in solid oranges lines for several representative realisations when λ = 1. Theoretical prediction for the accumulation of neutral mutations per cell in an exponentially-growing population shown in grey dashed line. C-D Simulated SFS for antigenic mutations averaged over 100 realisations when λ = 10 (orange points), along with the theoretical predictions (black dashed lines) in the absence of an immune response. A & C Low immune effect: α0 = 0.002 and β0 = 0.001. B & C High immune effect: α0 = 0.03 and β0 = 0.3. (Parameter sets chosen as points a and b from Figure 3A.)

Cancer-effector population dynamics (first row) and tumour antigenicity (second row) for five different sets of parameters. A-B λ = 1, α0 = 0.002 and β0 = 0.001. C-D λ = 1, α0 = 0.03 and β0 = 0.3. E-F λ = 10, α0 = 0.002 and β0 = 0.001. G-H λ = 10, α0 = 0.005 and β0 = 0.01. I-J λ = 10, α0 = 0.03 and β0 = 0.3. In the first row, red lines depict cancer cell populations for several representative realisations. Blue histograms above show the final time TK where the population size reached K = 3 × 104 for all non-extinct realisations, with a mean given by a dashed vertical black line. In the second row, red lines with increasing paleness, labelled by i = 0, 1, 3 or 5, depict the proportion of cancer cells contain i or more antigenic mutations. All parameter values not specified here are listed in Table 1 of the main text.

Extinction time heat maps for λ = 1 (A) and λ = 10 (B). All parameter values not specified here are listed in Table 1 of the main text.

Heat maps depicting the tumour composition (that is, the proportion of tumour cells that are antigenic) for λ = 1 (A-C) and λ = 10 (D-F). The first row is where one antigenic mutation held by a cell makes the cell antigenic; the second and third rows allow for one and two antigenic mutations (respectively) to be possessed by a cancer cell while still considering it neutral. Points a and c (respectively b and d) label parameter sets of low (respectively high) immune effectiveness. All parameter values not specified here are listed in Table 1 of the main text.

Average number of antigenic mutations in solid oranges lines for several representative realisations when λ = 10. Theoretical prediction for the accumulation of neutral mutations in an exponentially-growing population shown in grey dashed line. A Low immune effect: α0 = 0.002 and β0 = 0.001. B Middling immune effect: α0 = 0.005 and β0 = 0.01. C High immune effect: α0 = 0.03 and β0 = 0.3. (Parameter sets for A and C chosen as points c and d from Figure 3C, respectively.)

Genetic markers of selection: SFS (A-B & E-F) and MBD (C-D & G-H) for low (left column; α0 = 0.002 and β0 = 0.001) and high (right column; α0 = 0.005 and β0 = 0.01) immune effectiveness averaged over 100 realisations when λ = 10, along with the theoretical predictions (black dashed lines) in the absence of an immune response. Green data represents neutral mutations (A, C, E & G) and orange data represents antigenic mutations (B, D, F & H), with dashed vertical lines representing the means of the distributions for MBDs in panels C, D, G & H. All parameter values not specified here are listed in Table 1 of the main text.