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Guim Aguadé-Gorgorió, R. Solé (2017)
Adaptive dynamics of unstable cancer populations: The canonical equationEvolutionary Applications, 11
Background Genetic instability is known to relate with carcinogenesis by providing tumors with a mechanism for fast adaptation. However, mounting evidence also indicates causal relation between genetic instability and improved cancer prognosis resulting from efficient immune response. Highly unstable tumors seem to accumulate mutational burdens that result in dynamical landscapes of neoantigen production, eventually inducing acute immune recognition. How are tumor instability and enhanced immune response related? An important step towards future developments involving combined therapies would benefit from unraveling this connection. Methods In this paper we present a minimal mathematical model to describe the ecological interactions that couple tumor adaptation and immune recognition while making use of available experimental estimates of relevant parameters. The possible evolutionary trade-offs associated to both cancer replication and T cell response are analysed, and the roles of mutational load and immune activation in governing prognosis are studied. Results Modeling and available data indicate that cancer-clearance states become attainable when both mutational load and immune migration are enhanced. Furthermore, the model predicts the presence of well-defined transitions towards tumor control and eradication after increases in genetic instability numerically consistent with recent experiments of tumor control after Mismatch Repair knockout in mice. Conclusions These two main results indicate a potential role of genetic instability as a driver of transitions towards immune control of tumors, as well as the effectiveness of increasing mutational loads prior to adoptive cell therapies. This mathematical framework is therefore a quantitative step towards predicting the outcomes of combined therapies where genetic instability might play a key role. Keywords: Genetic instability, Neoantigen load, Mismatch repair, Immune surveillance, Combination therapies Background instability sustain a very diverse population [4], and intra- Cancer is a disease resulting from Darwinian evolution tumor heterogeneity lies at the very core of why cancer is in cellular tissues[1]. Following depletion of a vast set still difficult to define, characterize and cure [5]. of genetic insults altering normal multicellularity pheno- In this paper we aim at understanding an impor- types, roguecells areabletoadapt andevade selection tant relationship between the effectiveness of cancer barriers leading to uncontrolled proliferation. In this con- immunotherapy and genetic instability. The relevance of text, genomic instability plays a key role as a driver of such link needs to be found in the challenges faced by the genetic novelties required for tumor progression and immunotherapies based on immune checkpoint inhibi- rapidly adapting phenotypes [2, 3]. High levels of evolving tion or adoptive cell transfer [6], where mutational burden seems to play a key role. Due to the underlying complexity of cancer immunology, interdisciplinary efforts towards novel immunotherapies are much required [7–9]. As dis- *Correspondence: guimaguade@gmail.com; ricard.sole@upf.edu cussed below, the crucible of the problem might be to ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain the nonlinear dynamics associated to cancer neoantigen Institut de Biologia Evolutiva (CSIC-UPF), Psg Maritim Barceloneta, 37, 08003 production and the consequent enhancement of immune Barcelona, Spain surveillance. Full list of author information is available at the end of the article © The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 2 of 13 A key point in cancer immunotherapy lies on the of exceptional importance in our context: they indicate mechanisms by which T cells actually recognize cancer- the existence of well defined conditions (and perhaps ous from healthy tissue [10] and eventually attack tumor therapeutic strategies) allowing an all-or-none response. cells expressing tumor-specific antigens [11]. On a general However, a mathematical description of the specific role basis, such antigens can be common proteins for which of genetic instability in cancer immunology has not yet T cell acceptance is incomplete, or more importantly, been developed. Below we provide a first approach to such novel peptides [10, 12]. Except for specific tumor types goal, based on considering both cancer adaptation and of viral etiology, these so-called neoantigens arise after immune surveillance as influenced by mutational burden, DNA damage resulting in the production of novel pro- and we analyze how genetic instability can account for teins. Recent advances highlight the importance of under- transitions towards states of cancer control and elimina- standing neoantigen generation as a consequence of the tion. The implications of these transitions on combination tumor mutational load and dissecting specific neoantigen therapies are discussed, pointing towards possible cross- immunogeneicity [10, 11, 13]. Furthermore, direct corre- therapies activating neoantigen production and immune lations have been suggested between neoantigen produc- stimulation. tion at high microsatellite instability, eventual immune surveillance and clinical response to immunotherapies Methods [14–16]. Population dynamics of the tumor-immune interaction Several experimental and clinical sources are point- The ecology of the cancer-immune system interac- ing towards a causal relation, including tumor growth tion pervades several complexity levels, from a vast impairment after inactivation of MLH1 [17], or the posi- antigenome [23] to multilayer cellular competition tive response to PD-1 blockade across different mismatch dynamics [24], and a first step towards modeling such repair (MMR) defficient cancer types [18]. The inacti- ecology lies in dissecting which specific ingredients are vation of MMR results in increased mutational burden key drivers in the phenomena we aim to understand. of cancer cells, promoting the generation of neoanti- Recent research points out that there might be up to gens which improve immune surveillance and eventual 28 immune cell types with both antitumor and immuno- tumor arrest. These obxservations suggest a novel view supressive roles infiltrated within a tumor [25]. Focusing on immunotherapy, where targeting mutagenic pathways on the immuno-surveillance mechanism of tumor growth can result in an alternative mechanism to unleash immune inhibition following immune system recognition (early responses [9, 19]. introduced in [26]), a minimal modelling approach recalls All in all, genetic instability seems to play a conflictive at least considering a population of tumor cells growing in role in cancer evolution and proliferation. It appears that competition with immune cells. It is commonly accepted the same genome alterations that activate cancer progres- that theimmuneresponsetocancerismostlydrivenbyan sion can trigger T cell recognition and immune attack. adaptive cohort of cytotoxic immune cells,suchasCD8 The extent of such trade-off and its application to ther- T cells, together with a cellular compartment of the innate apy, however, is not clear. On the one hand, mutagenic immune system such as NK cells [27, 28]. Despite this therapies coexist with an intrisic risk, as increased genetic work focuses on the adaptive response to neoantigen pre- instability on heterogeneous populations might activate sentation, including an innate effector response will allow oncogenic outgrowth in previously stable cells. Moreover, for understanding relevant non-antigenic immune effects. areactiveimmunesystemmight pose aselective pres- Even if further models have been useful at depicting sure for immune editing, leading to selection for T cell very advanced properties of the immune system [29], evading tumor subclones. How do these two components we have chosen to keep a minimal scenario able to -instability and immune response- interact and what are describe the competition dynamics at play. We apply a the consequences? Is it possible to provide useful insight well characterised model (see e.g. [30]) that has been used from mathematical models without a detailed picture of to account for experimental results in cancer immunol- the immune landscape of cancer?. ogy such as tumor-immune equilibrium [31]. This model Nonlinear responses associated to cancer-immune sys- has been studied using parameter ranges measured from tem interactions have been known from the early days experimental setups consistent with several tumor types of cancer modelling, from more classical approaches (Table 1,see [20, 32]). [20] to recent perspectives based on neoantigen recog- The cellular interactions considered here involve a com- nition fitness [21]. These studies have revealed a num- monly used well-mixed (mean-field) model [20, 22]where ber of interesting properties exhibited by toy models, the population of cancer cells c follows a logistic growth including in particular the existence of shifts and break- (at effective replication rate r = b − d and carry- points separating cancer progression from its extinc- ing capacity K) and immune-cell mediated death (at tion (see [22] and references therein). Such shifts are rate δ ). This saturating growth model captures several c Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 3 of 13 Table 1 Parameter values for the cancer-immune ecology Ecological trade-offs in genetic instability model, estimated from experimental data of BCL lymphoma in As discussed above, genetic instability plays a key role in the spleen of chimeric mice (see [20]) tumor evolution, acting as the driving mechanism towards Parameter Meaning Kuznetsov et al. (1994) phenotypic variation and adaptation. Within our model, estimate this can be translated as the replication rate being a func- −1 r Cancer cell replication rate 0.18day tion of its level of genetic instability μ. On the other hand K Tumor carrying capacity 2 × 10 cells ρ , the rate of cancer cell recognition by T cells, is also μ- −7 −1 −1 dependent because of neoantigen production. Below we δ Immune-mediated cancer 1.101 × 10 day cells cell death rate propose a minimal characterization of r and ρ able to −10 −1 −1 δ Cancer-mediated immune 3.422×10 day cells describe how genetic instability modulates such trade-off. cell death rate 4 −1 Cancer adaptation as a function of genetic instability m Rate of T cell migration 1.3 × 10 cells day towards tumor site Cancer adaptation, here summarized to modulations in its replication rate, stems from the phenotypic plasticity g Tumor size limit for 2.019 × 10 cells effective T cell infiltration resulting from mutations and copy-number alterations. −1 ρ Rate of cancer cell 0.1245day On a general basis, enhanced tumor replication fol- recognition by the lows from mutations affecting oncogenic pathways, which immune system poses a trade-off on genetic instability as it can, as well, −1 d Intrinsic T cell death rate 0.0412day damage any of the necessary machinery for cell viability. Following previous research [35, 36], an adaptive land- scape is build on several assumptions based on the proba- bilities of mutating oncogenic and house-keeping genes. tumor microenviroment effects of malignant cell compe- Genetic instability has a twofold impact on cell fitness. tition and death, such as spatial constraints or nutrient Specifically, replication rate r will be considered a func- availability [33]. tion of mutation probability μ.Alandscape r(μ) is now in place [35, 37], and follows from considering that muta- dc c = rc 1 − − δ cE.(1) tions on oncogenes can translate into a linear increase in dt K replication rate. This follows from assuming that repro- ductive effects of oncogenes, as for advantageous muta- The effector immune population includes both NK and tions on many systems, are exponentially distributed [38], T cell compartments. Despite further modeling has been so that their sum is gamma distributed with average able to capture specific dynamics of T cell activation by increasing with the number of mutated oncogenes. This cancer-NK cell encounter [27], activation of both cell will be expressed as R (μ) = r + N δ μ with r being types by malignancy can be described in a similar form 1 0 R R 0 the basal replication rate of normal cells, N the number [22], here described by R of oncogenes responsible for increased replication and δ the mean effect on replication rate when mutating one of dE c = m + ρ E − δ cE − dE,(2) such genes. dt g + c To account for cell viability, the number of house- keeping genes N is taken into account so that mutations HK In this framework, the innate and adaptive immune affecting them result in null replication [39]. This intro- populations are encapsulated into a single Effector com- duces the constraint of not having any of them mutated, partment that grows due to a constant migration of cells HK R (μ) = (1 − μ) . Grouping both considerations and a predation term ρ that is commonly acknowledged together we obtain an analytical description of the cou- to obey a Michaelis-Menten-like saturation due to limi- pling between replication rate and mutation probability tations in immune cell circulation through the tissue [20] r(μ) = R (μ)R (μ) which reads: 1 2 and penetration within the solid tumor [32, 34]. The pecu- liarity of the model lies in considering this predation term different for both NK and T cells. As discussed below, ρ is N HK r(μ) = (r + N δ μ)(1 − μ) (3) 0 R R 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 This adaptive landscapeisofcourseofqualitative that ρ = ρ + ρ . Effector cells also have a natural decay nature, and realistic fitness landscapes for unstable tumor NK T rate, d, and die when competing with tumor cells at a rate environments are still far from our knowledge. However, −δ c. The complete set of interactions described by (1) certain points can be made if we give values within realis- and (2) is schematically shown in Fig. 1. tic parameter ranges to our function. The number of both Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 4 of 13 Fig. 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 and δ associated to the removal efficiency of cancer cells and the death of immune cells resulting from the same process, respectively oncogenes and house-keeping genes have been widely first step is describing immune reactivity as proportional assessed, and we take them to be about N ≈ 140 [40] to the adaptive compartment of cancer cell recognition and N ≈ 3804 [39] respectively. Interestingly, consid- ρ , a rate that itself depends on the dynamics of neoanti- HK T ering small replication effects for δ , such experimental gen expression. Under our assumptions, since adaptive values produce an adaptive landscape that has an optimal immune response follows from neoantigen detection we −5 −4 region for tumor replication at about μ ≈ 10 − 10 , expect ρ being a function of the overall mutational land- which is in accordance with the point-mutation proba- scape of a tumor, μt, which is eventually responsible for bility levels experimentally measured for unstable tumor such neoantigen dynamics. Following recognition proba- cells [41]. bility distributions from [21], we expect the average dom- inance to initially increase with mutations as more and Immune recognition of malignancy as a function of genetic more neoantigens are generated and eventually saturate as instability very dominant neoantigens are rare. Building a mathematical description of how the immune The mathematical shape of this dependency ρ (μt) system reacts at the mutational burden of cancer cells is could stem from purely stochastic dynamics, but recent research gives better insight into the shape of this cor- not straightforward. This stems from the fact that such relation. Rooney and colleagues provided an enlighting behavior is yet starting to be understood at the molecular perspective in this direction by comparing a measure of level and it probably builds upon many layers of com- plexity [10]. In our minimal mathematical approach, the immune response from the transcript levels of two key Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 5 of 13 −5 cytolytic effectors with the total mutation count for eight rates of about 10 mutations per gene per cell division tumor types [42]. [44], which account for the accumulation of about 10 Cytolytic response strengths in [42] seem to indicate a somatic mutations per tumor life [42], so that average dependency on tissue and tumor microenviroment, which tumor divisions lies at about t ∼ 10 . Using this approx- we have not included in our study since our model is not imation we obtain our preliminar expression for how the tumor type-specific. For each tumor type, a least-squares immune reactivity rate depends on the mutation levels, linear regression is used (Melanoma in Fig. 2). When ρ (μ) = 4.35 × 10 μ. comparing across tumor types the shape of the immune In this first correlation measure from [42], however, response seems to obey a common pattern across many immune recognition grows constantly with mutational cancers, once cytolytic response values are normalized load. This growth should not be indefinite, and many fac- (Table 2). A linear relation can be found for which nor- tors counteract the cytolytic effect of antigen-producing malized cytolytic activity scales with mutational load as mutations. As an example, increases in genetic instability −4 CYT∼ 4.35 × 10 μt when averaged across the range of can also account for antigen silencing and immune edit- tumor types explored here. However, we expect a func- ing, which itself would reduce cytolytic activity [45]. All tion depending only on mutation probability. The variable in all, it seems plausible to consider that antigenic and t in this expression refers to the evolutionary life history immune-suppressing mutations could balance beyond of mutations accumulation of the tumor. This time scale is certain mutational threshold. Following data from [42]it much larger than the faster ecological dynamics that gov- seems that the tumor-immune cytolytic interaction is far ern the cancer-immune system interactions, so that we from saturation, with an estimated saturation behavior −4 can consider it an average measure of tumor age at the to happen beyond μ ∼ 10 , a mutational level higher time of detection, and consider it constant when intro- than those of most tumors measured by recent method- ducing ρ in the ecological dynamics. From these facts, ologies (see e.g. [42]). This saturating function follows the only variable governing immune recognition at the the same trend of the data-based linear relationship and cancer-immune competition level is the point mutation reads probability μ. A very rough estimate for t could be either inferred from 2 5 ρ (μ) = 4.35 × 10 μ ∼ − ,(4) average cell replication data or from the fact that values T 14000μ 1.4 + e 6 for the mean mutation rate and the absolute mutational and can be compared with tumor adaptability r(μ) (Fig. 3) load are known for many tumors [43]. For example, we can use the notion that mutator tumors have mutation to obtain a full mutational landscape for tumor progres- Fig. 2 Measuring immune reactivity as a function of the mutational load. Melanoma is plotted as an example, where a linear regression (black line, scale=3.36E-4) between total mutation count and relative cytolytic activity is evaluated. Results for 12 cancer types in Table 2. Data is obtained from th th [42]. As in the original work, the correlation spans the 5 to 95 percentile of the mutation count variable Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 6 of 13 Table 2 Linear regressions for ρ(μt) across 12 cancer types, killing (δ). Since saturating dynamics are already present −4 resulting in ρ(μt) = 4.35 × 10 μt in the mathematical shape of ρ ,thislastrate δ is con- Cancer Type Gradient of ρ(μt) sidered constant, which is consistent with other recent −4 GBM 1.38×10 modeling efforts [46]. −4 LUAD 4.73×10 Cancer-Immune system attractor states −4 LUSC 6.98×10 Once the proper role of genetic instability on cancer adap- −3 BRCA 2.17×10 tation and immune response is defined, the original model −4 UCEC 2.30×10 is reinterpreted as a pair of coupled populations with −4 CRC 2.88×10 instability-dependent rates, i.e. −4 STAD 3.29×10 dc c −4 = r(μ)c 1 − − δ(ρ + ρ (μ))cE (5) NK T HNSC 8.37×10 dt K −4 SKCM 3.36×10 −4 dE (ρ + ρ (μ)) CESC 9.25×10 NK T = cE + m − δ cE − dE (6) −4 dt g + c BLCA 3.98×10 −3 LGG 2.98×10 A global picture for the behavior of the system is obtained by studying its possible attractor states tak- Data is obtained from [42] with linear regressions performed as in Fig. 2 ing into account the variability of the mutational load. ∗ ∗ Together with the cancer free attractor (c , E ) = sion in the presence of T cells. Assumptions on immune (0, m/d), other attractors can be inferred from the inter- response saturation at high genetic instabilities do not sections between nullclines affect theoutcome of themodel.Finally,the deathrateof r(μ) c E (c) = 1 − cancer cells increases as they become immunogenic and δ(ρ + ρ (μ)) K NK T detectable byTcells [10, 46]. This is translated in the (7) E (c) = model as cancer cells dying at rate δ = (ρ + ρ (μ))δ, c NK T (ρ +ρ (μ))c NK T δ c + d − g+c therateofimmunedetection (ρ)times therateofTcell Fig. 3 Functional forms for cancer replication r(μ) and the adaptive compartment of immune recognition ρ (μ) related to neoantigen presentation. The first (black curve) provides a representation of the cancer instability landscape, as predicted from our theoretical approach (see Methods section) and calibrated by available data. It reveals a very slow increase (in this log-linear diagram) at low instability levels followed by an increase associated to favourable mutations allowing for faster replication and a marked decay at high instabilities due to mutations on viability −4 genes. The immune reactivity to genetic instability function ρ(μ) (in red, obtained from [42]) rises from zero to saturation beyond μ ∼ 10 .The relevant domain of common cancer instability levels is highlighted. The innate response, ρ , is not depicted as is not a function of genetic NK instability and lies in a smaller order of magnitude of around ρ = 2.5 × 10 2[27] NK Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 7 of 13 Nullcline 1 is a simple line with a negative slope con- ρ for simplicity. The previous identity leads to a cubic 3 2 trolled by the inverse of the carrying capacity of cancer expression of the form Ac + Bc + Cc + D = 0, with cells. On the other hand, nullcline 2 is a peaked curve, δ r(μ)b with a height controlled by immune cell migration and a A =− δρ denominator that might eventually produce divergences. r(μ) Through their crossings we will find which steady states B =− b(d − ρ) + δ (bg − 1) coexist under which parameter domains (See Results δρ (8) section and Fig. 4). r(μ) C = d + gδ − ρ − bgd − m Along with genetic instability, another parameter is key δρ to the dynamics of the system. Regarding the second null- r(μ)gd D = − mg. cline, we can see its size is linearly affected by the influx δρ m of immune cells arriving at the tumor site. It is there- fore interesting to understand how μ and m are related to The sign of the discrimant = 18ABCD − 4B D + 2 2 3 2 2 cancer-immune scenarios, since this will open the door to B C −4AC −27A D will define of which combinations further discussion on mutagenic and immune activation of m and μ belong to which scenarios of Fig. 4. Knowing therapies. that three real roots exists for > 0 and only one for By solving E (c) = E (c), we can understand how the < 0, the transitions between attractor scenarios happen 1 2 values of m and μ affect the nature and number of possible to occur at = 0. This condition can be used to easily solutions of the system. We here write (ρ + ρ (μ)) = NK T describe the whole bifurcation space as seen in the results Fig. 4 Cancer-Immune response attractors driven by instability. In (a–d) we display the nullclines as we increase mutation probability values. −5 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 mutations per gene per division, such as those common in mutator tumors, only a large cancer attractor coexists with the unstable tumor-free ∗ −5 equilibrium left from the graph at c = 0. b Beyond μ ∼ 1.6 × 10 , two new attractors are created, which correspond to a stable microtumor ∗ −5 attractor and an unstable twin [30]. c At μ = 2.0 × 10 , 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 Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 8 of 13 and Fig. 4e and f, showing how mutation frequencies and in the Methods section. As we are interested in the specific immune stimulation affect the possible outcomes of the role of genetic instability and neoantigen presentation, we system. will focus here on the adaptive part of immune recogni- tion, ρ(μ). It is straightforward to see how several tran- Results sitions regarding creation and anihilation of steady states Minimal mutation rate for an efficient immune response are governed by mutational probability μ (Fig. 4a-d). As expected from [30] and previous discussions, we Before engaging into a full analysis of the complete model, know that the cancer-free attractor will always be present, we can study the behavior of the system for initial phases of progression. This corresponds to a small tumor of size but local stability will be ensured if r(μ)/(ρ + ρ (μ)) < NK T c << K = 2 × 10 cells. Under this assumption, the mδ/d (depicted in Fig. 4f). Without an innate component, population dynamics of c(t) simplifies to the condition is only fulfilled at very high instability levels −4 above 10 mutations per gene per division. This implies dc that no complete tumor clearance solely by neoantigen = c r(μ) − δ(ρ + ρ (μ))E − d (9) NK T c dt recognition seems possible at realistic mutation rates for where we have now included a natural death rate −d that fixed m, meaning that an innate response might also play accounts for growth barriers of initial malignant cells if a role in complete respondant patients, as many therapies away from the microenviroment carrying capacity [33]. do elicit total tumor eradication [45]. Additionally, we can From (9) we can isolate a condition for tumor control, i.e.: see that a large-tumor solution c is also present at low instabilities (Fig. 4a), and it is globally asymptotically sta- dc < 0 (10) ble. Interestingly, a transition seems to occur as the value dt for μ becomes larger: before E (c) diverges, a smaller sta- which leads to a crude estimate of the amount of effector ble attractor c is created together with its unstable twin immune cells required to counterbalance tumor growth, (Fig. 4b), which is often described as a microtumor con- namely trolled by the immune system. Furthermore, nullcline 2 −5 diverges at μ ∼ 1.75 × 10 (Fig. 4c), and, as the two r(μ) − d E(μ) > . (11) values for divergence of E (c) grow further appart, the δ (ρ + ρ (μ)) C NK T large cancer attractor disappears and only the controlled microtumor coexists with the cancer free attractor and is The inequality consistently shows that E(μ) will be pro- globally asymptotically stable (Fig. 4d). These results are portional to the instability landscape of cancer growth consistent with those of [30], where such solution is con- rate divided by both NK and immune-mediated death. sidered a microtumor controlled by the immune system. This acknowledges that both NK or T cells can play cru- However, both transitions of microtumor creation and cial roles in cancer surveillance. To understand the role large tumor elimination being a function of the mutational of the adaptive compartment and genetic instability in levels of the tumor population are new to the present controling a growing cancer population, we use validated data from [20](Table 1) and consider a healthy adaptive work. immune population of T ∼ 10 cells ([29] and follow- At this point it is clear that understanding at what ing sections), to obtain that the immune control condition instability levels these transitions happen is key to the pos- −5 is fulfilled for μ> 5.75 × 10 mutations per gene sible outcomes of the tumor-immune interaction. For the and replication. This can be understood as the minimal given parameter region and in the absence of a strong mutation rate required to generate a critical neoantigen innate response, a basic computational approach lets us load for T-cell immune attack, not considering here NK see that the first transition happens around μ ∼ 1.65 × −5 or other innate components away from the scope of the 10 (Fig. 4b), whereas another transition where the large work. The estimated value is consistend within the range tumor attractor disappears happens at higher μ values of −5 of genetic instability levels associated to MMR knockout about μ ∼ 4 × 10 (Fig. 4d). Following extensive data, unleashed genetic instabil- [47], indicating a connection between mutagenic thera- ity after Mlh1 knockout in mice accounts for increasing pies enhancing genetic instability and a threshold level to −6 −5 mutation frequencies ranging from 10 ∼ 10 up to activate the immune response. −4 10 mutations per gene per division (values assessed for transgenic mice containing supFG1 or cII from [47]). Transitions to tumor control and eradication at genetic Interestingly, instability levels before MMR knockout put instabilities within the mMR-knockout range our system within a region where the large cancer attrac- For well-formed tumors, no similar approach can be per- tor is stable and no controlled microtumor exists. How- formed, but we can study the effects of changes in genetic ever, the increase after Mlh1 knockout might be pushing instability in the sytem defined by equations (4) and (5) by cancer cells to a region beyond μ ,where thestable picturing the intersections between nullclines described 1 Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 9 of 13 microtumor attractor appears, or even μ ,where the will still grow towards larger disease. This result is consis- stable large cancer attractor has disappeared (Fig. 4e). tent with the notion that therapy reducing tumor mass is The resemblance between the model and experiments often effective prior to immunotherapy [20, 55]. linking genetic instability to adaptive immune surveil- The second transition, consistent with experiments lance seems intuitive enough. Following [17], we think of immune surveillance after Mismatch-Repair Knock- that there is a connection between the observed phe- out [17], indicates the disappearance of the large cancer nomenon of immune reactivity and tumor collapse after attractor (Fig. 4d). This implies that highly immunogenic Mismatch Repair knockout and the qualitative behavior tumors will always elicit a sufficiently effective immune of our model, which depicts a transition of this kind at response that will drive them towards microtumor con- high μ values. Furthermore, we have taken advantage of trol [31], no matter their initial size. However, the fact recent research in order to use quantitative data to build that there is no complete remission implies that evolu- our model. The fact that our model predicts the range tionary pressures still act on the remaining rogue popu- for which immune surveillance reacts at increased can- lation, and the small clone can eventually evolve immune cer instability levels emphasizes the possible existence of evasion [45]. transitions like the ones studied here. Mutagenic therapy remains a relevant actor on the Assessing if these two transitions are in fact well defined cancer-immune ecology. However, without the coopera- in vitro or if genetic instability can modulate tumor evo- tive effects of an innate response, through the constant lution towards controlled states can shed new light into recognition rate ρ , or the buffering of immune migra- NK the precise nature of mutagenic therapy as a mechanism tion m, the cancer-free equilibrium is only stable at very towards increasing tumor immunogeneicity. Such thera- high genetic instability levels that do not seem attain- pies have produced key results in the field of virology [48], able through mutagenic agents. What are the cooperative but, within the context of cancer, recent insight seems to dynamics of genetic instability with these immune agents? indicate that increasing the immunogeneicity of a tumor preludes evolution of subclonal neoantigen heterogeneity Effects of modulating immune migration and the innate [49–51]. response Beyond the relevance of genetic instability as a driver of Implications on immune surveillance: the role of tumor size tumor antigenicity, the fact that the cancer free attractor −4 Besides the possible implications for mutagenic therapy becomes stable at very high mutational levels above 10 as a facilitator of immunotherapy effectiveness, the fact mutations per gene and division (at least for the data on that genetic instability shapes the landscape of the cancer- adaptive immunity from [20]) implies that further consid- immune interaction has further implications on the fate of erations on therapy need to be taken into account. The tumorgrowth. Tumorsizehas been showntobeassoci- overall condition for total disease eradication is ated with response to immunotherapies [52], but several r(μ) mδ scenarios, from surveillance to evasion, are known to < . (12) ρ + ρ (μ) d NK T occur [31, 53, 54]. Is genetic instability related to the polymorphic nature of immunotherapy prognosis? If genetic instability alone does not suffice to fulfill this From Fig. 4a we know that, in conditions of low condition, what other therapeutic schemes are of rele- genetic instability, the large tumor equilibrium is globally vance to our model? A first notion lies on understanding asymptotically stable (GAS), and insufficient presenta- how does μ alter the minimal innate recognition ρ nec- NK tion of antigens implies that even small tumors can evade essary for complete disease remission, as defined by the immune surveillance in the absence of a strong innate condition response through NK cells or macrophages. This could be the case of both initial microsatellite-stable malignan- r(μ)d ρ > − ρ(μ) (13) NK cies or clones that have evolved low antigenicity through mδ genome editing [45]. −5 Increases in genetic instability result in a phase transi- For microsatellite stable tumors with μ<< 10 , tion that creates a micro-tumor attractor (Fig. 4b-c). This the necessary recruitment rate of NK cells is within the −1 state has been previously related to dormancy, where the 10 day−1 range, an order of magnitude larger than that adaptive immune system is able to control cancer growth measured in [27]. However, increasing genetic instability [31]. However, the large cancer attractor is still present, decreases ρ in a quasi-linear way, so that after a possi- NK −2 −1 and local asymptotic stability ensures that tumor sizes ble MMR knockout, a recruitment rate within 10 day within its basin of attraction will stil grow towards it. would suffice for cancer clearance, indicating the possibil- The implications are revelant to therapy: small tumors of ity of a combination therapy enhancing both mutagenesis medium antigenicity can be controled, but large tumors and NK cell activation [28]. Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 10 of 13 Together with the role of innate immunity, another key Discussion observation is considering the rate of immune migration In the present work we have studied a minimal math- (m) as a measure of immune activation. The necessary ematical scenario describing how genetic instability, flow of immune cells to the tumor to achieve complete by means of enhancing tumor adaptation along with remission is neoantigen production and immune recognition, can trigger sharp transitions towards tumor control and r(μ)d m > (14) eradication. δ(ρ + ρ (μ)) NK T Starting from basic considerations, we have asked our- Interestingly enough, the migration rate necessary for selves about the ecological interactions between malig- cancer clearance does not decay linearly with genome nant cells and, in particular, effector immune cells able instability, as for ρ , but in an exponential way, to respond after neoantigen recognition. Specifically, NK meaning that increases in genetic instability within the we consider how genetic instability, here as a muta- MMR knockout range rapidly decrease the condition tion probability, shapes tumor adaptability and immune for immune migration rate (Fig. 4f). This indicates a response. strong synergy between mutagenesis and immune acti- Interestingly, genetic instability governs the possible vation therapies such as Adoptive Cell Therapy (ACT) outcomes of the system. Increasing mutational levels drive [56], consistent with recent discussion on combination the system across two phase transitions. In the first one, therapies [7, 19]. two attractors are created involving smaller tumors coex- Moreover, by picturing the bifurcation diagram in stan- isting with a larger population of T cells. This state dard μ and m regions as described in the Methods section has been characterized as a controlled, but not totally (Fig. 4e), it is interesting to see how the first transition eliminated microtumor [30, 31]. The second transition towards microtumor creation, μ ,has aweakdependency accounts for the disappearence of the cancer-wins sce- on m, since the appearance of the intermediate attrac- nario, so that only solutions of immune control are present tors depends mostly on the denominator of nullcline 2 at large genetic instability levels. becoming null, so that E (c) diverges at Recent advances in the field of cancer immunology have proven that genetic instability is a key ingredient δ c + d − ((ρ + ρ (μ))c/(g + c)) = 0, (15) E NK T of the immune response [14–16], and particular research which is not a function of m. On the other hand, the tran- claims immune surveillance after MMR knockout fol- sition to disappearance of the large-cancer attractor does lows from this causal relation between high mutational loads and neoepitope production [17]. In the context depend on m,since m affects the width of E (c),sothatfor of this research, our model provides a conceptual and higher m values E (c) will go faster towards infinity and numerical description for how a transition between can- not cross E (c). However, it seems intuitive from Fig. 4f cer growth and arrest can follow only from damaging that the role of genetic instability in increasing neoantigen DNA repair mechanisms. More generally, the fact that production might be crucial even in the presence of high microsatellite instability levels govern transitions sepa- immune activation. rating cancer growth from immune surveillance might Mathematical work previous to our instability-driven be indicative of why highly unstable tumors are better model developed interesting considerations on derivation respondants to immunotherapy [10]. Furthermore, we of cancer vaccines (see e.g. [57]), and introduced time have used available data to calibrate the model parameters dependent treatments [58] or time-delays in the immune and to construct the immune recognition function. Using response [59] based on the immune migration parameter, this information, we consistently explain phase transi- despite mathematical considerations remained somehow distant from clinical immunology and not many of the tions happening at microsatellite instability levels that described behaviors after mathematically designed thera- resemble those of MMR knockout. However, even if these pies have been observed in vivo [22]. transitions could exist in the laboratory, we have dis- Recent research has highlighted the importance of cussed further aspects that need to be accounted when genetic instability as a marker for good prognosis in dealing with increasing tumor immunogeneicity through immune checkpoint inhibition therapies [14–16]. Its role mutagenesis [49, 50]. in neoantigen production is acknowledged as crucial We have also studied the roles of ρ ,the recruit- NK [10]. Our results describing μ as another driver towards ment of NK cells, and m, a parameter refering to surveillance complementing m and ρ reinforce the immune migration or an eventual immune therapy. The NK relevance of genetic instability in the tumor-immune model indicates a cooperative effect between thera- pies affecting mutagenesis together with NK or migra- dynamics, further supporting the possibility of increas- tion buffering. The strength of this cooperative effect ing tumor immunogeneicity by promoting T cell antigen is linear for genetic instability and innate immune cell presentation [7, 9]. Aguadé-Gorgorió and Solé Journal for ImmunoTherapy of Cancer (2019) 7:345 Page 11 of 13 recruitment, but the model also predicts that, when Conclusions an innate response and T cell recognition alone can- This work provides a first effort towards modeling the not control tumor growth, cross-therapies modulating double-edged effect of genetic instability in both can- both m and μ might be exponentially effective in driv- cer adaptation and immune surveillance with the goal of ing the tumor-immune interaction into a state of total understanding the specific role of mutational load as a disease eradication, thus indicating a mathematical vali- driver of immune attack. Two main results stem from dation for recent insight into combined immunotherapies the model. First, transitions towards tumor control follow [7]. We further suggest that the relevance of m in pro- from increases in mutational levels similar to those after ducing transitions to tumor arrest is low, while minor MMR knockout. Second, genetic instability and immune increases in genetic instability seem much more effective activation have a cooperative effect in driving tumor elim- against large tumors. This indicates that cross therapies ination, indicating that combination therapies enhancing inducing DNA damage prior to immunotherapy might both might be key in the future. drivetumorstoneoantigen-rich states [18, 19] before immune editing processes enter at play [45, 60]. We Acknowledgements therefore postulate a possible mathematical description of The authors thank Elisa Beltran and Jordi Piñero for fruitful discussions and Blai recent discussions for novel perspectives on combination Vidiella for detailed revisions, as well as the rest of Complex Systems Lab members and Pierre Menard for their inspiring ideas. G.A. thanks Noemi Andor immunotherapy [7]. and the team at the Evolutionary Biology and Ecology of Cancer Summer School All the previous conclusions stem from a very mini- 2018 for interesting comments on the cancer-immune system interaction. mal mathematical model, whereas the immune system is Authors’ contributions knowntobecomplex [45, 61] Additionally, other inter- GA and RS developed the mathematical model, wrote the manuscript and actions between immunotherapies and conventional ther- created the figures. Both authors read and approved the final manuscript. apies need to be taken into account [19]. In particular, several cooperative mechanisms between immune popu- Funding This work has been supported by the Botín-Foundation by Banco Santander lations might play a role in non-antigenic T cell activation through its Santander Universities Global Division, a MINECO grant [27]. Further research should consider the possible non- FIS2015-67616-P (MINECO/FEDER, UE) fellowship, an AGAUR grant 2018 by the linear dynamics stemming from T cell sensitization after Universities and Research Secretariat of the Ministry of Business and Knowledge of the Generalitat de Catalunya and the European Social Fund and cancer-NK cellular interactions. by the Santa Fe Institute. Finally, as a result from the lack of heterogeneity, our model does not yet capture immune editing, a phe- Availability of data and materials Not applicable nomenom at the core of immunotherapy failure, in which the tumor might develope immune resistance by means of Ethics approval and consent to participate either buffering the growth of immunosilent cells or edit- Not applicable ing its genome to express fewer neoantigens [60]. Within Consent for publication this view, current research claims that tumor mutational Not applicable burden might not be a sufficient biomarker [46, 50]. In the presence of an effective immune response, anti- Competing interests The authors declare that they have no competing interests. genic subclones can be negatively selected, giving rise to immuno-silent tumors despite its possibly high muta- Author details tional load. Together with immune editing, recent studies 1 ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain. Institut de Biologia Evolutiva (CSIC-UPF), Psg Maritim Barceloneta, 37, highlight heterogeneity itself as a source for failure of 08003 Barcelona, Spain. Santa Fe Institute, 399 Hyde Park Road, 87501 Santa theimmuneresponse[49, 51] as it directly affects the Fe, NM, USA. spatial and clonal distribution of neoantigens. Further Received: 12 June 2019 Accepted: 30 October 2019 modeling of the tumor-immune ecology could benefit from considering heterogeneous populations where anti- gen frecuencies are taken into account. 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Published: Dec 11, 2019
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