The ecology of the cancer-immune system interaction pervades several complexity levels, from a vast antigenome [23] to multilayer cellular competition dynamics [24], and a first step towards modeling such ecology lies in dissecting which specific ingredients are key drivers in the phenomena we aim to understand.

Recent research points out that there might be up to 28 immune cell types with both antitumor and immunosupressive roles infiltrated within a tumor [25]. Focusing on the immuno-surveillance mechanism of tumor growth inhibition following immune system recognition (early introduced in [26]), a minimal modelling approach recalls at least considering a population of tumor cells growing in competition with immune cells. It is commonly accepted that the immune response to cancer is mostly driven by an adaptive cohort of cytotoxic immune cells, such as CD8 ⁺ T cells, together with a cellular compartment of the innate immune system such as NK cells [27, 28]. Despite this work focuses on the adaptive response to neoantigen presentation, including an innate effector response will allow for understanding relevant non-antigenic immune effects.

Even if further models have been useful at depicting very advanced properties of the immune system [29], we have chosen to keep a minimal scenario able to describe the competition dynamics at play. We apply a well characterised model (see e.g. [30]) that has been used to account for experimental results in cancer immunology such as tumor-immune equilibrium [31]. This model has been studied using parameter ranges measured from experimental setups consistent with several tumor types (Table 1, see [20, 32]).Tab1

Parameter values for the cancer-immune ecology model, estimated from experimental data of BCL ₁ lymphoma in the spleen of chimeric mice (see [20])

The cellular interactions considered here involve a commonly used well-mixed (mean-field) model [20, 22] where the population of cancer cells c follows a logistic growth (at effective replication rate r=b−d and carrying capacity K) and immune-cell mediated death (at rate δ _c ). This saturating growth model captures several tumor microenviroment effects of malignant cell competition and death, such as spatial constraints or nutrient availability [33]. 1 dcdt=rc1−cK−δccE.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{dc}{dt} = rc \left(1-{c \over K} \right) - \delta_{c}c E. $$ \end{document}

The effector immune population includes both NK and T cell compartments. Despite further modeling has been able to capture specific dynamics of T cell activation by cancer-NK cell encounter [27], activation of both cell types by malignancy can be described in a similar form [22], here described by 2 dEdt=m+ρcg+cE−δEcE−dE,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{dE}{dt} = m + \rho \left({c \over g+c} \right)E - \delta_{E} c E - d E, $$ \end{document}

In this framework, the innate and adaptive immune populations are encapsulated into a single Effector compartment that grows due to a constant migration of cells and a predation term ρ that is commonly acknowledged to obey a Michaelis-Menten-like saturation due to limitations in immune cell circulation through the tissue [20] and penetration within the solid tumor [32, 34]. The peculiarity of the model lies in considering this predation term different for both NK and T cells. As discussed below, ρ is split into a constant rate refering to innate NK predation (see [27] and references therein) together with a variable part that will relate to antigen recognition by T cells, so that ρ=ρ _NK +ρ _T . Effector cells also have a natural decay rate, d, and die when competing with tumor cells at a rate −δ _E c. The complete set of interactions described by (1) and (2) is schematically shown in Fig. 1.Fig1Fig. 1

A schematic summary of the basic cancer-immune cell-cell interactions. The two key components are (a) a tumor population driven by genetic instability and (b–c) interactions associated to tumor cell recognition and attack by T and NK cells. The strength T cell attack depends on the number of surface neoantigens (c), while NK killing is constant [27]. In (d) the population-level interaction diagram is displayed based on the model in [20]. Here c and E indicate the number of cancer and T and NK cells, respectively. Cancer cells grow at a rate r (and have a limited carrying capacity) while immune cells enter the system at a constant production rate m and react at malignant cells at a rate ρ that will be different for NK cells and instability-dependent T cell recognition. A constant average death rate d is associated with their removal. Two constant cross-interactions rates are also indicated as δ _T and δ _c associated to the removal efficiency of cancer cells and the death of immune cells resulting from the same process, respectively

As discussed above, genetic instability plays a key role in tumor evolution, acting as the driving mechanism towards phenotypic variation and adaptation. Within our model, this can be translated as the replication rate being a function of its level of genetic instability μ. On the other hand ρ _T , the rate of cancer cell recognition by T cells, is also μ-dependent because of neoantigen production. Below we propose a minimal characterization of r and ρ able to describe how genetic instability modulates such trade-off.

Once the proper role of genetic instability on cancer adaptation and immune response is defined, the original model is reinterpreted as a pair of coupled populations with instability-dependent rates, i.e. 5 dcdt=r(μ)c1−cK−δ(ρNK+ρT(μ))cE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ {\frac{dc}{dt} = r(\mu)c\left(1-{c \over K}\right) - \delta(\rho_{NK} + \rho_{T}(\mu))c E} $$ \end{document}

6 dEdt=(ρNK+ρT(μ))g+ccE+m−δEcE−dE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ {\frac{dE}{dt} = \frac{(\rho_{NK} + \rho_{T}(\mu))}{g+c}cE + m - \delta_{E} c E - d E } $$ \end{document}

A global picture for the behavior of the system is obtained by studying its possible attractor states taking into account the variability of the mutational load. Together with the cancer free attractor (c ^∗,E ^∗)=(0,m/d), other attractors can be inferred from the intersections between nullclines 7 E1(c)=r(μ)δ(ρNK+ρT(μ))1−cKE2(c)=mδEc+d−(ρNK+ρT(μ))cg+c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \begin{aligned} E_{1}(c) &= \frac{r(\mu)}{\delta(\rho_{NK} + \rho_{T}(\mu))}\left(1-{c \over K}\right) \\ E_{2}(c) &= \frac{m}{\left(\delta_{E}c + d - \frac{(\rho_{NK} + \rho_{T}(\mu)) c}{g + c} \right)} \end{aligned} $$ \end{document}

Nullcline 1 is a simple line with a negative slope controlled by the inverse of the carrying capacity of cancer cells. On the other hand, nullcline 2 is a peaked curve, with a height controlled by immune cell migration and a denominator that might eventually produce divergences. Through their crossings we will find which steady states coexist under which parameter domains (See Results section and Fig. 4).Fig4Fig. 4

Cancer-Immune response attractors driven by instability. In (a–d) we display the nullclines as we increase mutation probability values. Arrows indicate the system flow towards the small and large tumor attractors. Two transitions can be seen. a At low genetic instability levels of 10 ⁻⁵ mutations per gene per division, such as those common in mutator tumors, only a large cancer attractor coexists with the unstable tumor-free equilibrium left from the graph at c=0. b Beyond μ ^∗∼1.6×10⁻⁵, two new attractors are created, which correspond to a stable microtumor attractor and an unstable twin [30]. c At μ ^∗=2.0×10⁻⁵, the microtumor attractor becomes smaller; until eventually the attractor of uncontrolled tumor growth is eliminated (d) at mutational levels similar to those attained after Mismatch-Repair knockout [40]. In (e) and (f) we summarise the bifurcation diagrams for the possible scenarios as a function of μ and m. For standard immue migration rates (e, black region in f), mutational increases drive the system across the two transitions observed in (a–d) and towards the controlled tumor state. However, by increasing both μ and m through combining Mismatch Repair knockout with adoptive cell therapy, the total cancer clearance state can be accessed

Before engaging into a full analysis of the complete model, we can study the behavior of the system for initial phases of progression. This corresponds to a small tumor of size c<<K=2×10⁹ cells. Under this assumption, the population dynamics of c(t) simplifies to 9 dcdt=cr(μ)−δ(ρNK+ρT(μ))E−dc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \displaystyle{\frac{dc}{dt} = c\Big(r(\mu) - \delta(\rho_{NK} + \rho_{T}(\mu)) E - d_{c}}\Big) $$ \end{document}

where we have now included a natural death rate −d _c that accounts for growth barriers of initial malignant cells if away from the microenviroment carrying capacity [33]. From (9) we can isolate a condition for tumor control, i.e.: 10 dcdt<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{dc}{dt} < 0 $$ \end{document}

which leads to a crude estimate of the amount of effector immune cells required to counterbalance tumor growth, namely 11 E(μ)>r(μ)−dcδC(ρNK+ρT(μ)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ E(\mu)>\frac{r(\mu)-d_{c}}{\delta_{C}(\rho_{NK} + \rho_{T}(\mu))}. $$ \end{document}

The inequality consistently shows that E(μ) will be proportional to the instability landscape of cancer growth rate divided by both NK and immune-mediated death. This acknowledges that both NK or T cells can play crucial roles in cancer surveillance. To understand the role of the adaptive compartment and genetic instability in controling a growing cancer population, we use validated data from [20] (Table 1) and consider a healthy adaptive immune population of T∼10⁷ cells ([29] and following sections), to obtain that the immune control condition is fulfilled for μ>5.75×10⁻⁵ mutations per gene and replication. This can be understood as the minimal mutation rate required to generate a critical neoantigen load for T-cell immune attack, not considering here NK or other innate components away from the scope of the work. The estimated value is consistend within the range of genetic instability levels associated to MMR knockout [47], indicating a connection between mutagenic therapies enhancing genetic instability and a threshold level to activate the immune response.

For well-formed tumors, no similar approach can be performed, but we can study the effects of changes in genetic instability in the sytem defined by equations (4) and (5) by picturing the intersections between nullclines described in the Methods section. As we are interested in the specific role of genetic instability and neoantigen presentation, we will focus here on the adaptive part of immune recognition, ρ(μ). It is straightforward to see how several transitions regarding creation and anihilation of steady states are governed by mutational probability μ (Fig. 4a-d).

As expected from [30] and previous discussions, we know that the cancer-free attractor will always be present, but local stability will be ensured if r(μ)/(ρ _NK +ρ _T (μ))<m δ/d (depicted in Fig. 4f). Without an innate component, the condition is only fulfilled at very high instability levels above 10 ⁻⁴ mutations per gene per division. This implies that no complete tumor clearance solely by neoantigen recognition seems possible at realistic mutation rates for fixed m, meaning that an innate response might also play a role in complete respondant patients, as many therapies do elicit total tumor eradication [45]. Additionally, we can see that a large-tumor solution c _L is also present at low instabilities (Fig. 4a), and it is globally asymptotically stable. Interestingly, a transition seems to occur as the value for μ becomes larger: before E ₂(c) diverges, a smaller stable attractor c _S is created together with its unstable twin (Fig. 4b), which is often described as a microtumor controlled by the immune system. Furthermore, nullcline 2 diverges at μ∼1.75×10⁻⁵ (Fig. 4c), and, as the two values for divergence of E ₂(c) grow further appart, the large cancer attractor disappears and only the controlled microtumor coexists with the cancer free attractor and is globally asymptotically stable (Fig. 4d). These results are consistent with those of [30], where such solution is considered a microtumor controlled by the immune system. However, both transitions of microtumor creation and large tumor elimination being a function of the mutational levels of the tumor population are new to the present work.

At this point it is clear that understanding at what instability levels these transitions happen is key to the possible outcomes of the tumor-immune interaction. For the given parameter region and in the absence of a strong innate response, a basic computational approach lets us see that the first transition happens around μ∼1.65×10⁻⁵ (Fig. 4b), whereas another transition where the large tumor attractor disappears happens at higher μ values of about μ∼4×10⁻⁵ (Fig. 4d).

Following extensive data, unleashed genetic instability after Mlh1 knockout in mice accounts for increasing mutation frequencies ranging from 10 ⁻⁶∼10⁻⁵ up to 10 ⁻⁴ mutations per gene per division (values assessed for transgenic mice containing supFG1 or cII from [47]). Interestingly, instability levels before MMR knockout put our system within a region where the large cancer attractor is stable and no controlled microtumor exists. However, the increase after Mlh1 knockout might be pushing cancer cells to a region beyond μ1∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu ^{*}_{1}$\end{document} , where the stable microtumor attractor appears, or even μ2∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu ^{*}_{2}$\end{document} , where the stable large cancer attractor has disappeared (Fig. 4e).

The resemblance between the model and experiments linking genetic instability to adaptive immune surveillance seems intuitive enough. Following [17], we think that there is a connection between the observed phenomenon of immune reactivity and tumor collapse after Mismatch Repair knockout and the qualitative behavior of our model, which depicts a transition of this kind at high μ values. Furthermore, we have taken advantage of recent research in order to use quantitative data to build our model. The fact that our model predicts the range for which immune surveillance reacts at increased cancer instability levels emphasizes the possible existence of transitions like the ones studied here.

Assessing if these two transitions are in fact well defined in vitro or if genetic instability can modulate tumor evolution towards controlled states can shed new light into the precise nature of mutagenic therapy as a mechanism towards increasing tumor immunogeneicity. Such therapies have produced key results in the field of virology [48], but, within the context of cancer, recent insight seems to indicate that increasing the immunogeneicity of a tumor preludes evolution of subclonal neoantigen heterogeneity [49–51].

Besides the possible implications for mutagenic therapy as a facilitator of immunotherapy effectiveness, the fact that genetic instability shapes the landscape of the cancer-immune interaction has further implications on the fate of tumor growth. Tumor size has been shown to be associated with response to immunotherapies [52], but several scenarios, from surveillance to evasion, are known to occur [31, 53, 54]. Is genetic instability related to the polymorphic nature of immunotherapy prognosis?

From Fig. 4a we know that, in conditions of low genetic instability, the large tumor equilibrium is globally asymptotically stable (GAS), and insufficient presentation of antigens implies that even small tumors can evade immune surveillance in the absence of a strong innate response through NK cells or macrophages. This could be the case of both initial microsatellite-stable malignancies or clones that have evolved low antigenicity through genome editing [45].

Increases in genetic instability result in a phase transition that creates a micro-tumor attractor (Fig. 4b-c). This state has been previously related to dormancy, where the adaptive immune system is able to control cancer growth [31]. However, the large cancer attractor is still present, and local asymptotic stability ensures that tumor sizes within its basin of attraction will stil grow towards it. The implications are revelant to therapy: small tumors of medium antigenicity can be controled, but large tumors will still grow towards larger disease. This result is consistent with the notion that therapy reducing tumor mass is often effective prior to immunotherapy [20, 55].

The second transition, consistent with experiments of immune surveillance after Mismatch-Repair Knockout [17], indicates the disappearance of the large cancer attractor (Fig. 4d). This implies that highly immunogenic tumors will always elicit a sufficiently effective immune response that will drive them towards microtumor control [31], no matter their initial size. However, the fact that there is no complete remission implies that evolutionary pressures still act on the remaining rogue population, and the small clone can eventually evolve immune evasion [45].

Mutagenic therapy remains a relevant actor on the cancer-immune ecology. However, without the cooperative effects of an innate response, through the constant recognition rate ρ _NK , or the buffering of immune migration m, the cancer-free equilibrium is only stable at very high genetic instability levels that do not seem attainable through mutagenic agents. What are the cooperative dynamics of genetic instability with these immune agents?

Beyond the relevance of genetic instability as a driver of tumor antigenicity, the fact that the cancer free attractor becomes stable at very high mutational levels above 10⁻⁴ mutations per gene and division (at least for the data on adaptive immunity from [20]) implies that further considerations on therapy need to be taken into account. The overall condition for total disease eradication is 12 r(μ)ρNK+ρT(μ)<mδd.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{r(\mu)}{\rho_{NK} + \rho_{T}(\mu)} < \frac{m\delta}{d}. $$ \end{document}

If genetic instability alone does not suffice to fulfill this condition, what other therapeutic schemes are of relevance to our model? A first notion lies on understanding how does μ alter the minimal innate recognition ρ _NK necessary for complete disease remission, as defined by the condition 13 ρNK∗>r(μ)dmδ−ρ(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \rho_{NK}^{*} > \frac{r(\mu)d}{m\delta} - \rho(\mu) $$ \end{document}

For microsatellite stable tumors with μ<<10⁻⁵, the necessary recruitment rate of NK cells is within the 10⁻¹ day−1 range, an order of magnitude larger than that measured in [27]. However, increasing genetic instability decreases ρNK∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{NK}^{*}$\end{document} in a quasi-linear way, so that after a possible MMR knockout, a recruitment rate within 10⁻² day ⁻¹ would suffice for cancer clearance, indicating the possibility of a combination therapy enhancing both mutagenesis and NK cell activation [28].

Together with the role of innate immunity, another key observation is considering the rate of immune migration (m) as a measure of immune activation. The necessary flow of immune cells to the tumor to achieve complete remission is 14 m∗>r(μ)dδ(ρNK+ρT(μ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ m^{*}> \frac{r(\mu)d}{\delta(\rho_{NK} + \rho_{T}(\mu))} $$ \end{document}

Interestingly enough, the migration rate necessary for cancer clearance does not decay linearly with genome instability, as for ρNK∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{NK}^{*}$\end{document} , but in an exponential way, meaning that increases in genetic instability within the MMR knockout range rapidly decrease the condition for immune migration rate (Fig. 4f). This indicates a strong synergy between mutagenesis and immune activation therapies such as Adoptive Cell Therapy (ACT) [56], consistent with recent discussion on combination therapies [7, 19].

Moreover, by picturing the bifurcation diagram in standard μ and m regions as described in the Methods section (Fig. 4e), it is interesting to see how the first transition towards microtumor creation, μ1∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu ^{*}_{1}$\end{document} , has a weak dependency on m, since the appearance of the intermediate attractors depends mostly on the denominator of nullcline 2 becoming null, so that E ₂(c) diverges at 15 δEc+d−((ρNK+ρT(μ))c/(g+c))=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \delta_{E}c + d - ((\rho_{NK} + \rho_{T}(\mu)) c/(g + c))=0, $$ \end{document}

which is not a function of m. On the other hand, the transition to disappearance of the large-cancer attractor does depend on m, since m affects the width of E ₂(c), so that for higher m values E ₂(c) will go faster towards infinity and not cross E ₁(c). However, it seems intuitive from Fig. 4f that the role of genetic instability in increasing neoantigen production might be crucial even in the presence of high immune activation.

Mathematical work previous to our instability-driven model developed interesting considerations on derivation of cancer vaccines (see e.g. [57]), and introduced time dependent treatments [58] or time-delays in the immune response [59] based on the immune migration parameter, despite mathematical considerations remained somehow distant from clinical immunology and not many of the described behaviors after mathematically designed therapies have been observed in vivo [22].

Recent research has highlighted the importance of genetic instability as a marker for good prognosis in immune checkpoint inhibition therapies [14–16]. Its role in neoantigen production is acknowledged as crucial [10]. Our results describing μ as another driver towards surveillance complementing m and ρ _NK reinforce the relevance of genetic instability in the tumor-immune dynamics, further supporting the possibility of increasing tumor immunogeneicity by promoting T cell antigen presentation [7, 9].