Indian Academy of Sciences Conference Series (2020) 3:1 © Indian Academy of Sciences DOI: 10.29195/iascs.03.01.0031

The Warburg effect: A computational modeling approach

UGO H. P. SILVA1,2,∗ and FERNANDO F. FERREIRA2,3

1 Department of Informatics and Tourism, Federal Institute of Education, Science and Technology of São Paulo, 01109-010 São Paulo, SP, Brazil 2 Center for Interdisciplinary Research on Complex Systems, University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil 3 Department of Physics, School of Philosophy, Sciences and Letters, University of São Paulo, 14040-901 Ribeirão Preto, SP, Brazil ∗Corresponding author. E-mail: [email protected]

Abstract. The approach of complex systems has been increasingly used by the different areas of knowledge, such as physics, biology, social sciences, and economy. In studies started in the 1920s that resulted in the pub- lication of a paper in 1956, Warburg was one of the first to study the origin of tumorigenic cells, changing the metabolic process of oxidative phosphorylation to in cellular respiration, and its implications in the control and proliferation of tumor cells. In this report, we propose a computational method to study the Warburg effect; the competitive advantage of different strains; and cooperation mechanisms in the uptake of nutrients used for energy generation. The objective here is to discuss the trade-off between high-yielding organ- isms and high reproduction rate. We propose a model to study the competition between normal and tumor cells exposed to hypoxia, or low presence of oxygen, and a more acidic environment. We qualitatively predict the invasion of tumor cells for a broad range of parameter values. This is useful to calibrate a model with real data. Keywords. The Warburg effect; equilibrium dynamics of cancer; stability.

PACS Nos 02.70.-c; 05.45.-a; 64.60.Ht; 87.19.xj

1. Introduction cell must undergo multiple mutations. Later, mathe- matical modeling of cancer began with a statistical Cancer is the result of a dynamic evolutionary process analysis of age patterns of incidence, in which the that can lead to improvements or innovations. The evo- emergence of cancer requires multiple probabilistic lutionary process that leads to cancer is different from events [1]. other processes because many genes can be inactivated In this context, changes in play a fun- or modified without any loss of fitness for the cells, and damental role in the progression and maintenance sometimes a gain (in this way their progression is seen of tumor cells, which causes the scientific commu- as ‘destructive evolution’) as Nowak explains [1]. nity to investigate the actions of oncogenes, proto- Ernest Tyzzer was the first to use the term ‘somatic oncogenes and tumor suppressor genes, transcription mutation’ meaning that cancer is a genetic disease factors that can explain the altered state, metabolism, caused by somatic developments, approximately one and possible therapies that can be applied to fight the hundred years ago. Then, other experts, like David disease. von Hansemann and Theodor Boveri, already had stud- In this section, we briefly review and discuss the pro- ied abnormal cell division and observed that there cesses associated with the Warburg effect, which has was something wrong with the cancer cells chromo- already been extensively studied but are not satisfacto- somes [2]. His studies showed that ionizing radiation, rily and precisely understood in terms of determining besides being carcinogenic, was also mutagenic. This the causes or effects of this change in the state of cells corroborates the idea that for cancer to occur a single with tumor. 54 Indian Academy of Sciences Conference Series (2020) 3:1

2. Cancer as a complex system The release of O2, as a result of photosynthe- sis, altered the environment in which cells developed 2.1 Evolution of metabolism and possibly led to the development of an oxidative According to [3], metabolism is the set of chemi- metabolic model that currently occurs in cells for energy cal processes by which living organisms use energy generation [4, 5]. from food, air, or sunlight to maintain all the func- Cellular respiration is a characteristic process of tions necessary to live. These chemical processes occur both normal and tumor cells. The mechanisms that largely within cells, through currents of chemical reac- make tumor cells the preferred choice of a less effi- tions called metabolic pathways. In an organism’s body cient metabolic pathway in ATP generation are widely the transformation of nutrients is processed to produce studied by the scientific community. energy-generating molecules and cellular components 2.2 Cell cycle and tumor progression are constructed through parallel metabolic pathways. In general, metabolic pathways are complex sequences of In general, cancer is caused by mutations in genes that chemical reactions controlled by self-regulatory feed- regulate the cell cycle. For a better understanding of back. Metabolism is the heart of cell biology, and under- tumor formation, a basic understanding of this process standing how cancer cells deal with metabolic needs has is necessary. been the focus of cancer research for many years. Under normal conditions, within a cooperative model, Since the molecules were originated from a set of cells undergo selective pressures and occasional muta- organic molecules, they were able to obtain food and tions that do not affect the functioning of the organ- energy directly from the environment. However, this ism, because the function of the cell is to maintain is very limiting, which led them to develop their own the machinery as a whole, and, when necessary, the mechanisms to generate energy and synthesize the nec- apoptosis-induced death of some defective individuals essary molecules for their replication. The generation occur to keep the system in balance. and controlled use of metabolic energy is essential for Cancer is the result of a system failure in a cellular all cellular activities, and the main pathways of energy society in which a single cell (due to a mutation or set metabolism are largely conserved in today’s cells. All of mutations) begins to exhibit uncontrolled growth, the cells use (ATP) as a source of cooperation that maintains the integrity of a multicel- metabolic energy to control the synthesis of cellular lular organism is then interrupted. Abnormal cells can constituents and perform other activities that require be identified through interactions with other cells by energy. cellular signaling mechanisms [5, 6]. It is assumed that the mechanism used by cells for ATP generation evolves in three stages, correspond- 2.3 Cancer cycle ing to the evolution of glycolysis, photosynthesis, and Dated from 1911, the first recognition that the tumor oxidative metabolism (figure 1)[4]. develops in stages is attributed to Haaland by Foulds in The first reactions of energy generation involved the studies of breast and lung tumors in mice. Three stages breakdown of organic molecules in the absence of oxy- were considered to be virtually axiomatic, but standard gen. They were probably very similar to current gly- terminology was introduced by Friedewald and Rous in colysis, the anaerobic breakdown of glucose to produce 1944: lactic acid with a net gain of two molecules of ATP and • Initiation. State of neoplasia, a form of prolifera- its use as an energy source intracellular chemistry. tion not controlled by the organism. The different types of genetic alterations are mutations that activate function gain, gene amplification, point mutations, and promoter mutations that change an allele from a proto-oncogene to an oncogene. • Promotion. Tumor cells are stimulated to prolif- erate to form a visible tumor. • Progression. Development of the malignant tumor from tissues bearing a benign tumor, through the accumulation of additional genetic damage, through mutations or epigenetic silenc- ing of the genes that shaved the DNA and the altered expression of genes that promote the vas- Figure 1. Generation of energy metabolism. Adapted cularization and spread of the tumor through local from [4]. invasion or distant metastases. Indian Academy of Sciences Conference Series (2020) 3:1 55

From the point of view of cell biology, a tumor needs Not only can the tumor’s micro-environment select to acquire the following properties to develop: a metabolism more appropriate to its conditions, but • uncontrolled proliferation, the status of the oncogene can also lead to metabolic • insensitivity to antiproliferative signs, changes. The signaling molecule of Ras – rat sarcoma • evasion of programmed cell death, virus, a potent oncogene that when mutated pro- • motes glycolysis, and the Akt kinase – family of unlimited replicative capacity, 2 • angiogenesis, serine/threonine protein kinases, an effector down- • handling. stream well characterized by insulin signaling, resumes its role in glucose uptake and use in the cancer 2.4 Oncogenes environment [8]. Consider a compartment of replicating cells. During The discovery that tumor-causing oncogenes are related each cell division there is a small probability that an to normal genes raised several questions about the role error will be made during DNA replication, a mutant of these genes in growth and development (differentia- daughter cell with a better adaptive advantage in the tion) of normal and tumor cells. existing function or even the appearance of a new func- It seems certain to say that stages of tumor initia- tion will be produced. Alternatively, the mutation can tion and promotion and the existence of a malignant impair important cellular functions with a negative apti- itself depends on the increased expression tude for the cell, causing it to proliferate more slowly or (manifestation of the effect) of oncogenes, caused by die more quickly than its neighbors [1]. The dynamics of amplification (increased number of copies of the gene), a specific mutation within a compartment is discussed. by altered expressions of repressor genes or by critical Initially, all cells are not mutated and it questions the mutations in areas of a given oncogene. probability that a single mutated cell appeared in the Stimulation of proliferation of normal cells is almost time t measured in cell cycles. If the relevant cells divide always triggered by growth factors that bind recep- once a day, then the unit of time will be one day. tors on cell membranes. The signal received by them N is the number of cells within the compartment and is transmitted to the cytoplasm and, finally, to the u is the mutation rate per gene per cell division. The nucleus. probability that at least one mutated cell appeared in An oncogene is a mutant gene whose altered func- time t is given by tion or expression results in abnormal stimulation of cell division and proliferation, which has a dominant P(t)=1 − exp(−Nut). (1) effect at the cellular level, i.e., when activated or over- expressed, it contributes to initiate a change in the What is the fate of a single mutated cell? In this scenario, phenotype of a cell and to the progression of the there is a constant probability q that this cell will not cell tumor if an allele is mutated or inappropriately die, but it will start a neoplasm or uncontrolled tissue expressed. growth. Thus, the probability that a compartment started Many tissues of multicellular organisms are subdi- a neoplasm in time t is given by vided into compartments subject to homeostatic mech- P(t)=1 − exp(−Nuqt). (2) anisms1 which ensure that the number of cells remains constant over time. Cancer appears if the balance In an alternative scenario, cells that have mutated have between the birth and death of the cells is targeted for a relative aptitude r compared to the non-mutated cells uncontrolled proliferation [1, 7]. whose aptitude is 1. If r > 1 the mutation is advanta- Oncogenes, when activated, are associated with pro- geous, if r < 1 the mutation is not advantageous, and teins that act in many stages of the cell growth control neutral if r = 1. Mutations in oncogenes are expected to pathway, including: cause an increase in the growth rate, r > 1. However, a mutation in an oncogene can be kept in check by apop- • Growth factors that stimulate cell division totic defense mechanisms, and therefore r can be less • The receptors and cytoplasmic proteins that than one. translate these signals Consider the Moran process that represents a simple • The transcription factors that respond to the and possible stochastic model for studying selection in translated signals a finite population, in which, at each step of time, two • The proteins that prevent programmed cell death individuals are chosen: one to reproduce and the other (apoptosis) to be eliminated. The offspring of the first individual

2Is said to be cell, tissue, or organ that exerts an action or activity in response to a 1 Balance or stability of the system. stimulus. 56 Indian Academy of Sciences Conference Series (2020) 3:1 will replace the second one. Note that the two random sparked a debate about the role of glycolysis in normal choices can fall on the same individual, and in this case and cancer cells. an individual will be replaced by his own offspring and Normal cells depend on oxidative phosphorylation the total population size is strictly constant. According to generate energy, but many cells with cancer resort to to this model, the probability that a mutation takes over aerobic glycolysis or the fermentation process, which is the compartment, given the probability of fixing a single an inefficient way of generating ATP (essential for cell mutation with relative aptitude r,isgivenby proliferation). According to [13], in single-celled organ- isms, such as microbes, there is an evolutionary pressure 1 − 1/r ρ = . (3) to reproduce quickly when nutrients are available. 1 − 1/rN In multicellular organisms, cells are exposed to a con- stant supply of nutrients, which require a control system For neutral mutation r =1,wehaveρ = 1 . An advan- N that stops too much proliferation in order to support cell tageous mutation has a higher probability of fixation division. For [8], it is difficult to start a discussion about than a neutral mutation, which has a higher probability the metabolism of cancer cells without first mentioning of fixation than a deleterious mutation. Small compart- Otto Warburg, a pioneer in the study of respiration. He ment events, however, are dominated by random drift. was responsible for discovering a remarkable effect in If N is small, then a deleterious mutation can reach the 1920s that was named after him. a relatively high probability of fixation due to chance Clinical studies for tumor detection using the tech- events [1]. nique (fluorodeoxyglucose positron emission tomogra- The probability that a mutation has been fixed in time phy) show an increase in glucose absorption. Although t is given by aerobic glycolysis is accepted as a metabolic feature of P(t)=1 − exp(−Nuρt). (4) cancer, its causal relationship with cancer progression is still uncertain for the authors. Any mutation is more likely to fix, ρ in a small com- A convincing idea to explain the Warburg effect is partment than in a large compartment, but P(t)isan that the altered metabolism of cancer cells confers a increasing function from N to r > 1 and decreas- selective advantage for survival and proliferation in the ing to r < 1. Thus, large compartments accelerate microenvironment of the original tumor. This alteration the accumulation of advantageous mutations, but delay is doubly advantageous for the cancer cells since one the accumulation of harmful mutations, whereas, small of the consequences of this glucose metabolization is compartments delay the accumulation of advantageous the accumulation of lactic and carbonic acid forming a mutations, but accelerate the accumulation of harmful microenvironment with low pH that favors tumor inva- mutations. Therefore, according to [1], the size of the sion and suppression of the anticancer effectors of the compartment is important in determining the types of immune system. mutations that may occur. As the tumor expands very early, it exceeds the diffusion limits of its local blood supply leading to 3. The Warburg effect hypoxia and stabilization of the hypoxia-inducible tran- scription factor (HIF). Decreased dependence on aer- While normal differentiated cells maximize ATP pro- obic respiration is advantageous. Cellular metabolism duction by mitochondrial oxidative phosphorylation of is switched to glycolysis by increasing the expres- glucose under normoxic conditions, cancer cells gen- sion of glycolytic , glucose transporters, and erate much less ATP from glucose by aerobic glycol- mitochondrial metabolism inhibitors. ysis, a particular type of metabolism. They have all In the tricarboxylic acid cycle, when oxygen is the enzymes needed for the most part of the central present, many of the differentiated cells primarily metabolic pathways; however, they show an anomaly in metabolize glucose to carbon dioxide by oxidizing the integration of the glycolytic sequence with the tricar- pyruvate in a reaction that produces the reduced form boxylic cycle. They consume less oxygen than normal of the coenzyme nicotinamide adenine dinucleotide, cells; however, they tend to use about 5–10 times more then maximizes ATP production by phosphorylation glucose, converting much of it into lactate, forming ATP oxidative with minimal lactate production (figure 2). in the extra-mitochondrial compartment by glycolysis Along with the glycolytic pathway, intermediate [9, 10]. metabolites can be channeled to synthesize amino acids, In Warburg’s observation, see [11, 12], cancer cells , and lipids, if the flow rate through the path- exhibit a high rate of glycolysis, necessary to meet the way is controlled. Normal differentiated cells produce a increased energy for rapid tumor progression even in large amount of lactate only under anaerobic conditions, the presence of oxygen (aerobic glycolysis). This has whereas tumor cells produce a large amount regardless Indian Academy of Sciences Conference Series (2020) 3:1 57

the blood. Another possible explanation is that prolifer- ating cells have important metabolic requirements that go beyond ATP.

3.1 Agents related to the alteration of the metabolic state of cancer Warburg’s work marked the beginning of an era of study on tumor metabolism focused on the relation- Figure 2. Differences between oxidative phosphorylation and ship between glycolysis and cellular bioenergetics. His aerobic and anaerobic glycolysis. Adapted from [13]. work has been reviewed and expanded by generations of tumor biologists. The valorization of the general- ity of the Warburg effect stimulated the broader con- of oxygen supply. Glycolysis first requires the conver- cept that a ‘metabolic transformation’ is necessary sion of glucose to pyruvate (figure 2) and then to lactic for tumorigenesis. The permanent challenge in tumor acid. cell metabolism is to understand how individual path- In most mammalian cells, glycolysis is inhibited by ways fit into the overall metabolic phenotype of cell the presence of oxygen, which allows mitochondria growth. to oxidize pyruvate by CO2 and H2O. Afferent blood Research over the past few years reinforces the (hemoglobin) provides glucose and oxygen to the tis- idea, revealing the conservation of metabolic activities sues, where it reaches cells by diffusion. Glucose among different types of tumors and proving that onco- is collected by specific carriers, in which glucose-6- genic mutations can promote metabolic autonomy by phosphate is converted first by and then to driving the absorption of nutrients to levels that often pyruvate, generating two molecules of ATP per glu- exceed those required for cell growth and proliferation cose. In the presence of oxygen, pyruvate is oxidized [14, 15]. to HCO3, generating 36 additional ATP molecules per At the center of discussions related to the altered state glucose. of metabolism are oncogenes, tumor suppressor genes, In the absence of oxygen, pyruvate is reduced to and transcription factors. Here we choose some of these lactate, which is exported from the cell. Both pro- agents that have been studied extensively in the search cesses produce hydrogen ions (H+), which cause acidi- of understanding the Warburg effect [13, 16, 17]. fication of the extracellular space, HbO2, oxygenated hemoglobin, a mechanism that can be observed in 3.2 complex figure 3 [12]. If glycolytic aerobic metabolism can provide enough Studies of pyruvate kinase (PKM2) regulators in yeast energy for cell proliferation then as [13] reveal that the central metabolism is self-adapting to “ Why is a less efficient metabolism at least in terms synchronize redox activities when breathing is activated of ATP generation preferentially used by tumor [16, 18]. cells?” It is difficult to determine precisely what causes tumor The authors give a possible explanation that this cells to change their metabolism for the Warburg effect, would be a problem only in the scarcity of resources, that but studies in [10, 19, 20] suggest a strong correlation is not the case in mammalian cells exposed to glucose with the dimeric shape M2 of the isoenzyme PKM2 as and constant supply of nutrients that circulate through a central point in the regulation of cancer metabolism. While dimeric PKM2 shifts glucose metabolism to anabolism through aerobic glycolysis, and catalyzes the final reaction and also a rate-limiting reaction in the glycolytic pathway, tetrameric PKM2 promotes the flow of glucose-derived carbon atoms for the produc- tion of ATP through oxidative phosphorylation. The balance of dimers and tetramers is critical for tumori- genesis and is controlled by several factors. The PKM2 dimer also promotes aerobic glycolysis, modulating the regulation of transcription. It is subjected to complex regulation by oncogenes and tumor suppressors, which allows fine regulation of its activity and the Warburg Figure 3. Glycolysis metabolism. Adapted from [12]. effect. 58 Indian Academy of Sciences Conference Series (2020) 3:1

Tumor cells have to survive with variations in the sup- 3.2.2 hypoxia-induced factor-1 (HIF-1): As mentioned ply of oxygen and nutrients, depending on their distance in [24, 25] experimental works show that the War- from blood vessels, which request special demands burg effect is not a universal phenomenon. It uses an on the metabolism of tumor cells. This metabolic approach to quantify the plasticity conferred to can- phenotype has been coined as a tumor metabolome. cer cells through gene regulatory networks and consider The increase in the rate of lactate production, even in that, contrary to the Warburg effect, oxidative mitochon- the presence of oxygen cell proliferation is not always drial phosphorylation plays a crucial role in the progres- associated with a high rate of conversion of glucose to sion of cancer. The epithelial–mesenchymal transition lactate. The PKM2 key glycolytic enzyme is constantly is a phenomenon common to all epithelial tumors and altered during tumorigenesis, within a sequencing, it is it is part of the way in which neoplastic cells behave responsible for the ATP production network and energy to invade the adjacent stroma and result in the acqui- production. In contrast to mitochondrial respiration it is sition of a mesenchymal cell phenotype, whose capac- independent of oxygen and allows cells to survive under ity for migration, invasion, and resistance to apoptosis low oxygen supply conditions [12]. increases. Hypoxia occurs when the available oxygen It seems paradoxical that tumor cells with a high rate drops below 5%, causing a complex cellular and sys- of glucose consumption and lactate production have temic adaptation mediated mainly by transcription of an isoenzyme PKM2 that is inactive. Cell proliferation factors inducible by hypoxia [21, 26]. The decrease in continues when metabolism is able to provide enough oxygen availability (hypoxia) stimulates cells to con- metabolic intermediates to ensure both energy regenera- sume glucose and produce lactate. In mammalian cells, tion and the synthesis of cell building blocks in sufficient this response is coordinated by the transcription factor quantities [21]. It is a fundamental role of PKM2 to [27, 28]. determine whether glucose is converted into lactate for Initially, a binary classifier is used to tell whether energy regeneration (active tetrameric form) according the cell’s status is epithelial or mesenchymal. A hybrid to the work in [10, 19, 22]. E/M phenotype is achieved with a metric developed by There is evidence of the critical role of PKM2 in George et al.inCancer Res. 77, 6415 (2017). tumorigenesis by promoting the Warburg effect. The use An additional central aspect to be considered in the of knockout showed that cancer cells reduced glucose plasticity of cancer is the complex network of inter- uptake, increased oxygen consumption, and decreased actions between epithelial and stromal cells. Such an lactate production, while its reintroduction reversed the approach can be seen in [24] to decode such plas- changes occurred. Regulation of the glycolytic activ- ticity. A hybrid metabolic state in cancer cells con- ity of PKM2 is probably crucial for aerobic glycolysis. tributes to metabolic plasticity, arguing that because The use of inhibitors of PKM2 molecules can reduce of this flexibility, they efficiently produce energy in the Warburg effect in a way that mimics the effects of both states. In intermittent hypoxia, a free oxygen the PKM2 knockout [10, 16, 19]. deficit is not sufficiently established to allow HIF-1α stabilization [29]. 3.2.1 Oncogene of c- protein: The c-Myc gene has a fundamental role in controlling growth, differenti- ation, and apoptosis, and its abnormal expression is 4. Interaction between species with different associated with many tumors. The processes of reg- metabolic pathway models ulation of cell growth and metabolism are closely interrelated. C-Myc is an oncogene ‘master regulator’, Heterotrophic organisms can produce ATP from two which controls several aspects of both processes. A different mechanisms: one inefficient due to its high mutated version of Myc is found in many , input consumption rate and another more efficient, which causes Myc to be constitutively (persistently) characterized by lower consumption of resources but expressed [17]. higher yield in ATP production. Organisms with a The first suggestion that c-Myc plays an important high consumption can reproduce more quickly, which role in regulating glycolysis was the observation that gives it a greater selective advantage compared to lactate dehydrogenase A converts pyruvate into lac- those of greater yield. However, both coexist in the tate as part of the glycolytic pathway. It was one of evolutionary process. Recently, Amado et al.[30] the 20 putative genes targeted by c-Myc, as noted in addressed this issue as the goal of understanding [23]. Subsequent work has shown that many other glu- the coexistence of these strains, despite the sup- cose metabolism genes are directly regulated by c-Myc, posed selective disadvantage of one of them. We will including the GLUT1 transporter, hexokinase 2 (HK2), briefly describe the model and comment on the main phosphofructokinase, enolase 1 (ENO1). results. Indian Academy of Sciences Conference Series (2020) 3:1 59

αATP C αATP D 4.1 Model description The parameters C Si and D Si describe the yield The purpose of the model is to study the invasion in the production of ATP. In the next section, we address of a C species in a population of another D species the equilibrium solution of the homogeneous model that is in balance. Species C is characterized by high version. The analysis will be done in terms of three yield in resource conversion in ATP, i.e., high utiliza- parameters as follows: •  = AD , which measures the ratio of resource tion resource (glucose) at the cost of being a slow AC process compared to that carried out by species D. consumption between types D and C. When On the other hand, species D consumes much more  > 1, strain D consumes more resources than resources but produces little ATP per mole of glucose strain C • Δ αATP /αATP ingested when compared to another. This is the case ATP = D C is the ratio of metabolic effi- when one produces ATP via fermentation, typical of ciency in energy production (ATP). When Δ < 1 species D, as opposed to aerobic respiration as is the type C is more efficient than type D AATP • Γ D case with species C. The breathing organisms share = ATP is the strength of energy conversion ratio AC the resources better while those that ferment further between types D and C. degrade the environment, depleting resources, and so they can be seen as cooperators (C) and defectors (D) 4.2 Homogeneous population respectively. Before proceeding to the simulation, we write the equa- Next, we will describe a structured or multilevel tions that describe the homogeneous population (where model for the problem of competition between species there is only one group). Let n (t) and n (t) be the C and D. The model consists of a population with D C population size of the species D and C, which evolve variable size that is subdivided into NG groups. Each C D as: i group has a population of size Pi = Pi + Pi that S varies over time. The total quantity S resource is equally −αATP JDS D JS n +JS n i / nD(t +1)=nD(t) 1+aD 1 − e D D C C − ν divided between groups receiving SG = Y NG, for every i =1,..., N(G). Each individual reproduces (9) when its internal energy exceeds the threshold Emax, S with each part having the half energy. Individuals com- −αATP JCS C JS n +JS n pete within each group and reproduce with the above nC(t +1)=nC(t) 1+aC 1 − e D D C C − ν rule. When the group reaches the value Pmax,itis divided into two groups with half of the individuals (10) each. where ν is the natural death rate and a = AATP /E . This approach is called the model multilevel, adopted C C max The equilibrium solution of the homogeneous model here as a mechanism to promote cooperation [1]. is easy to obtain as described in [30]. The stability of Resource consumption by species D and C is described the equilibrium solution is studied from the Jacobian by matrix. The invasive species C is introduced when the D population of species D is in balance. From this stability D JS (SG) Si (SG)= D D C C SG (5) analysis resulted the expression: JS (SG)Pi + JS (SG)Pi ν/ Γ ( ac) and ATP = Δ . (11) 1 − (1 − ν/ac) ATP C C JS (SG) This equation describes the line that separates the region Si (SG)= D D C C SG (6) JS (SG)Pi + JS (SG)Pi where species D will never be invaded to the region where species C can raiding D. S S where JC(S)=AC and JD(S)=AD. The conversion of resource into ATP that determines 4.3 Effect of toxins and hypoxia the reproduction of individuals of species C and D is a The fermenting species turns part of glucose into ATP function of the resources ingested by them. The energy and another part into some toxins (such as alcohol or Δ C C C variation is given respectively by Ei = JATP (Si ) other toxic substances). On the other hand, the respirator Δ D D D ATP or Ei = JATP (Si ), where the J functions are as converts glucose into carbon dioxide and water, not pro- follows: ducing toxins for the population. The toxin eventually ATP C ATP ATP C increases the death rate of both strains. Let us consider J (S )=A 1 − exp(−α S ) (7) C i C C i the homogeneous population. We start from the model ATP D ATP − −αATP D JD (Si )=AD 1 exp( D Si ) . (8) described by eq. (9) and introduce the η and β rates to 60 Indian Academy of Sciences Conference Series (2020) 3:1 measure the impact of toxins in type D and type C pop- ulations, respectively. Following [31], we adapted the model to include the hypoxia effect in the population competition, i.e.,  − θ − −α S[1 F1( (t))] nD(t +1)=nD(t) 1+aD 1 exp D  nC(t) + nD(t)

− [ν + F2(θ(t))] − η nD(t) S[1 − F (θ(t)] Figure 4. The size of the population as a function of Δ. Popu- n (t +1)=n (t) 1+a 1 − exp −α 1 C C C C n (t) + n (t) lation C is purple while population D is blue. The continuous C D line denotes stable equilibrium while the dashed line corre- sponds to the unstable balance. The parameters were set as − [ν + F2(θ(t))] − β nD(t) follows: Γ =4, = 10, S = 10, aC = 0.2, αC =1,β = 0.000,021, ρ η = 0.01 and γ = 100. θ(t +1)=θ(t) 1 − . (12) t +1 This result helped us to discuss and understand the

Here, ρ describes the oxygen decay and Fk(θ(t)) effect of increasing the pH in increasing cancer cells. = λ (1 − tanh(γθ(t)), k = 1 or 2 was proposed Our model is more general than [31], which is obtained k λ θ θ recently by Sabir et al.in[32], with λk, γ being pos- by setting 1 = 0 for both F1( (t)) and F2( (t)) in itive constants. Hypoxia is described by a dynamical eq. (12), where only the toxin’s effects are addressed. variable F2(θ(t)) which affects the death rate of both The result is summarized in figure 4 which shows pop- cell types. This term is different from natural death ν. ulations C and D as a function of Δ. Note that on the Also the metabolic efficiency is affected due to hypoxia. left of the dashed red line the population consists of So we introduce the term [1−F1(θ(t))], which multiply only strains C, while population D, on the right side of αC and αC in both equations above (eq. (12)). Oxy- the blue line, invades population C. In the intermediate gen varies over time according to eq. (12). Note that, region there is a regime of stable coexistence between when Fk(θ(t)) → 0, we recover the model that takes them. Thus, it is clear that toxins can explain species into account only the toxins effects [31]. We present coexistence. the equilibrium analysis and the Jacobian matrix in the Now we consider the model with hypoxia. Firstly, λ λ appendix. take 1 = 0.5 and 2 = 0 or (F2 = 0). We suppose that oxygen decay over time affecting the ATP production. 5. Result In figure 5 one can see that the same pattern observed in the previous figure is present. However, the coex- The structured population model by Amado et al. [30] istence region shrinks, i.e., now it is delimited by the shows the existence of a social dilemma. So, structuring may have been a key element in enabling the survival of species like species C in the evolutionary process. They found that the invasion region by strains C increases substantially, both in terms of Δ and Γ. However, tumor cells are favored by the acidic environment due to the presence of toxins as well as due to the scarcity of oxygen. First of all, we comment on the result of the effect of the toxins on the cell population observed in [30]. The model captured very well the effect of structuring on competition between species which produce energy over a wide spectrum of values that describe the trade- Figure 5. The size of the population as a function of Δ. Here, λ λ off between high yield (C) on one side and high pro- we study the impact of hypoxia when 1 = 0.5 and 2 =0. duction rate (D) of another in the production of ATP. Population C is purple while population D is blue. By compar- ing with figure 4 we see that the coexistence region decreased Later on, Fernández et al.[31] adapted the model to and population of type D invades population of type C for study other aspects observed in the same system, i.e., smaller values of Δ. The parameters were set as: Γ =4, fermentation produces toxins that can compromise cell  = 10, S = 10, aC = 0.2, αC =1,β = 0.000,021, η = 0.01 and survival. γ = 100. Indian Academy of Sciences Conference Series (2020) 3:1 61

The dashed blue line in figure 4 is around Δ˜ = 0.58. Comparing figure 4 with figures 5 and 6, for a given oxygen decaying rate (ρ), we can see that for a low value of Δ strains C suppress strains D and the popula- tion size remains the same. However, by increasing Δ, there is a special value of Δ < Δ˜ where cells C drop out (see green vertical line in figures 5 and 6). We can infer that type C cell survival also depends on oxygen concentration. Below a certain level (ρ), all individuals C die favoring type D. This is in agreement with what Figure 6. The size of the population as a function of Δ. one expects to observe in experiments, but we are just at λ λ Here, we study the impact of hypoxia when 1 = 0 and 2 the phenomenological level. In the model, it is possible = 0.05. Population C is purple while population D is blue. By to control the parameter ρ and it also can be calibrated comparing with figure 4 we see that the coexistence region with real data. Similar to what happens in the model decreases and population of type D invades population of type C for smaller values of Δ. The parameters were set as: Γ =4, [32], here the oxygen level decay to a constant value ρ  = 10, S = 10, aC = 0.2, αC =1,β = 0.000,021, η = 0.01 and which depends on parameter . γ = 100. The model has many parameters, which take into account different metabolic pathways, hypoxia, and tox- vertical dashed red line and vertical dashed green line. ins produced by tumor cells. But it is quite simple since it By decreasing λ1, when F1 → (0) the scenario depicted does not consider details of complex chemical reaction in figure 4 is recovered. chains taking place inside cells (see [24] for details). For In the model described by Sabir et al. [32], they sug- instance, at a molecular level Yu et al. [25] addressed a gest that hypoxia will affect the death rate. In this case, model in which they constructed a regulatory network we may set F1(θ(t)) = 0 and consider F2(θ(t)) = 0. of genes and metabolites to study cancer metabolic plas- Again, we observed in figure 6 that the coexistence ticity and design cancer therapies targeting metabolism. region also decreases. The impact of parameter λ1 is This is an important aspect to highlight some limi- different in F1 and F2. In other words, F2 is much more tations of our model which apply at a macro level sensible to the variation on λ compared to F1. (population). It is possible to introduce more realism in the model, 6. Discussions for instance, allowing αC and αD to be a function of Θ to study the ATP production. Alternatively, it is possible to Hypoxia has a significant role in the metabolic repro- interpret part of the toxins as lactate, which will be a fuel gramming of tumor cells. Mathematical modeling can for tumor cells in hypoxia conditions. There is a very be a means of assessing the effects of concentration, interesting feedback involving lactate production and duration, and frequency of oxygen in the period of tumor the metabolic pathways to be explored in the future. This progression. Also, hypoxia has been shown to increase work may evolve for developing more realistic models the aggressiveness and severity of tumor progression. In for tumor growth and maybe it can be used to predict [28] three levels of occurrence of hypoxia in the TME the response of tumor hypoxia in cancer treatment. are discussed: the acute form, the chronic form, and Data support the concepts that altered metabolism the intermittent form, and the role they play in tumor results from active reprogramming and that metabolic progression. In the case of cyclical hypoxia, several hall- adaptations can be clonally selected during tumorige- marks of cancer are triggered. It is postulated that the nesis. Altered metabolism should now be considered intermittent pattern has a great influence on EMT, inva- a core hallmark of cancer [33]. Quantitative omics- sion, and metastasis of tumor cells, strongly discussed methodologies have collected and supported molecular in [24]. level experimentally available information about can- Here we present two ways to introduce the hypoxia cer metabolism overall [34]. Bersanelli et al. [35] and effect in the model, by changing the efficiency of ATP Kristensen et al. [36] showed that advanced strategies production or increasing the death rate. Of course, one for integrative analysis of multi-omics datasets are non- can combine both ways, which enhance the tumor cell trivial problems of the dynamics of molecular systems, invasion. The idea here is just to show a model that but they pave the way for a more comprehensive and is able to consider different mechanisms in population deeper understanding of biological systems. The use of competition, where the target is a tumor cell population integrative approaches is mandatory in order to gain fur- invading normal cell population in the Warburg effect ther insights on oncological phenomena [37]. The next context. stage of our work is to investigate the problem from the 62 Indian Academy of Sciences Conference Series (2020) 3:1 integrative omics data perspective using mathematical Assuming absence of tumor cells, nD(t)=0,wecan modeling. determine the solution balance for the cooperators: α ˜CS nC(t)=− . (A.1) log 1 − ν˜ 7. Conclusion aC

The same study can be done when nC(t)=0.The So far we have presented an outline of the cancer equilibrium solution for nD(t)is dynamic modeling starting with the contextualization α˜ S ν˜ + η n (t) of the work from the perspective of complex systems. exp − D = 1 − D . (A.2) Then, we describe cancer in general and we chose the nD(t) aD Warburg effect as an object of research. The choice of We will see the general case in search for coexis- this effect is due to the fact that it is an apparently simple tence. The coexistence solution is obtained by solving mechanism that evokes changes in metabolic properties the eq. (12) system to nC(t) and nD(t) equilibrium, which of the cells. Leveraging recent results from the complex results in systems modeling involving competition from species Δ ν˜ + βn (t) ν˜ + ηn (t) with different metabolic pathways [30–32], we adapt the 1 − D = 1 − D (A.3) model for the study of cancer from the population point aC aD α of view by addressing the role of toxins and hypoxia ˜CS nC(t)=− −  nD(t). (A.4) effect on population competition and invasion. ν˜+βn (t) log 1 − D As discussed above, the origin of cancer refers to the aC molecular descriptive level closely related to the mech- Clearly, these equations are solved only numerically. anisms of gene transcription. We chose to study the Yet, there are two particular cases in which they have properties at a higher level of complexity and when we an analytical solution. The case where β = 0 simpli- have a better understanding, we can attack the problem fies expressions (A.4) and it allows to find the equilib- from a genetic point of view. Modeling the evolution rium populations for types D and C and determine the of cancer cells taking into account microenvironment stability of each one. The nD population size is parameters is important from the therapeutic point of ν Δ ν view. Mathematical insight shed light to elucidate many aD ˜ ˜ nD = 1 − 1 − − . (A.5) fundamental issues. η aC η

The population size nC is found by replacing nD in eq. (A.4). The general case β > 0 has to be studied Acknowledgements numerically. The second case arises when Δ = 1 where it is possible to find coexistence solution, except when The author UHPS would like to thank USP and IFSP for β = η. allowing this work to be carried out, Mauro Ongarelli Finally, we are not going to discuss stability here in this case. However, we write the Jacobian matrix: for the artistic illustration of the figures, Lucas Almeida and Enio Oishi Akira for reviewing the text, Gabriela  − ν − −α S Rodrigues for the professional proofreading, Ludmila J11 = 1 ˜ + aD 1 exp ˜D n (t) + n (t) Deute, Paulo Custodio, Paulo França and Rolf Simões C D n (t)2 for suggestions and Complex Systems Modeling Master × 1+α˜ S D − 2η n (t) D [n (t) + n (t)]2 D Program at USP. FFF thanks Victor Emmanuel Molina ⎧ C D ⎫ Camargo for useful discussions and São Paulo Research ⎨ S ⎬  exp −α˜D  Foundation, FAPESP for financial support. This study − α nC(t)+ nD(t) J12 = nD(t) aD ˜D S 2 was financed in part by the Coordenação de Aperfeiçoa- ⎩ [nC(t) + nD(t)] ⎭ mento de Pessoal de Nível Superior – Brasil (CAPES) ⎧ ⎫ ⎨ S ⎬ – Finance Code 001.  exp −α˜C  − α nC(t)+ nD(t) β J21 = nC(t) aC ˜CS 2 + ⎩ [nC(t) + nD(t)] ⎭ Appendix A: Equilibrium model analysis S J = 1 − ν˜ + a 1 − exp −α˜ In this section, we will show the equilibrium solu- 22 C C  nC(t) + nD(t) tion and present the Jacobian matrix for the model × α nC(t) − β described by eq. (12). To simplify the notation we define 1+˜CS 2 nD(t). [nC(t) + nD(t)] ν˜ = νF2(θ(t)) and α˜x = αx[1 − F1(θ(t))], for x =C,D. Indian Academy of Sciences Conference Series (2020) 3:1 63

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