Cancer is a complex disease of the genome that behaves as a clonal evolutionary process in populations of cells [1–4]. Although cancer is a diverse and multifactorial disorder with differing origins and degrees of malignancy, clonal expansion and the presence of Darwinian or natural selection are generally accepted as common features [4, 5]. Since Nowell first proposed the clonal evolution theory of neoplasia in 1976 [1], a broad range of studies have provided support for this model. In recent years, the use of next-generation sequencing (NGS) technologies for the investigation of tumor genomes has generated increasing evidence that most neoplasms grow as a clonally expanded cell population [3, 6–8]. The vast amounts of invaluable data generated by NGS have surpassed analysis and interpretation capacity. However, the intricate nature of clonal expansion and evolution in cancer makes it difficult to convert the experimental and clinical data into medical practices [9, 10]. Experimental data alone are not generally sufficient enough to address the complex problem of cancer. Consequently, focus has shifted toward devising mathematical/computational models for simplification and extraction of fundamental meaning from the complex biological processes of cancer.

Adult T-cell leukemia (ATL) is a life-threatening malignancy that manifests with very poor prognosis [11, 12]. ATL develops through a multistep leukemogenic process, the nature of which remains elusive [13]. Among the different types of cancer, ATL is a remarkably unique neoplasm in that it is directly caused by infection with human T-cell leukemia virus type-1 (HTLV-1), which is mainly transmitted via breastfeeding [14]. HTLV-1 infection and integration of provirus into the host genome are intrinsic and inevitable early events for ATL development [15]. HTLV-1 mainly survives in vivo by persistent clonal proliferation of infected cells [16]. Whereas the majority of HTLV-1–infected individuals remain asymptomatic carriers (ACs) throughout their lifetime, ~5% of them develop ATL after a long period of clinical latency [17]. Currently, there is no clear determinant to distinguish between individuals who will remain ACs and those who will develop ATL [18, 19]. Our Joint Study on Predisposing Factors of ATL Development (JSPFAD) group examined ATL risk factors and demonstrated that a proviral load (PVL; i.e., the percentage of infected peripheral blood mononuclear cells) of >4% is one of the risk factors for progression to ATL; however, PVL alone cannot predict development of the disease [19]. Similar to other types of cancer, clonal expansion of abnormal cells is a hallmark of ATL [20, 21]. Considering that the incidence of large clones increases with disease progression from the healthy AC state to the malignant states of smoldering (SM), chronic, or acute ATL [22–24], monitoring clonal expansion via an accurate method of detection is of great clinical importance [8, 23].

Generally, mutation patterns of cells can be used to define clones and monitor clonal expansion in different types of cancer [7]. ATL development has an advantage in that not only the mutation pattern but also the integration site of the provirus can be used to define clones and monitor clonal expansion [8, 23]. Individual infected cells can be uniquely characterized based on their integration site because, typically, a single integration of HTLV-1 occurs per host cell [25]. Detecting the clonality dynamics, including clonal status and alterations, requires an appropriate method for defining two main characteristics of clones, HTLV-1 integration site and clone size.

Research in this area would be greatly benefitted by an easier to understanding representation and description of how cancer develops in terms of clonal expansion, which is expected to be provided by appropriate models. A realistic model would provide a better understanding of cancer and would provide a comprehensive perspective on cancer processes by integrating clinical and biological data within a mathematical and computer science framework. As with other malignancies, suitable models for ATL would help to simplify the dynamics of cooperative and complex behaviors in cancer development [26, 27]. Quantitative NGS data have the potential for creating robust and reliable mathematical modeling approaches [28]. Increasingly complex mathematical models of cancerous growth (particularly of solid tumors) that are based primarily on mutation patterns have been developed [29, 30]. The prominent role of mathematical modeling in the detailed quantitative description of diseases, and the contribution of mathematical modeling to solving biological problems have been eloquently discussed by Tanaka and Ono [31]. Currently, there is a broad range of theoretical models available; however, empirical mathematical models are still limited [30].

Several mathematical modeling studies are available in the field of HTLV-1 research [32–38], although none of these studies have focused on modeling clonal expansion and its correlation with ATL development. The earliest mathematical model for HTLV-1 explored the correlation between the antiviral immune response, viral load and viral diversity [32]. Later Stilianakis et al. used a nonlinear differential equation and theoretical assumptions to describe HTLV-1 infection of CD4⁺ T-cells [34]. This model was further optimized to test different assumptions and/or alteration of the proposed differential equations [35–38]. Therefore, there is an obvious lack of a data-driven mathematical model that describes the role of clonal expansion of HTLV-1–infected cells in ATL development. Mathematical models that are data-driven and hypothesis-free are considered to be the most applicable in many situations and have the lowest risk of confirmation bias [31]. Moreover, there is currently no computational model available for ATL development. A model that not only reflects details of biological phenomena like mathematical models but also allows abstract visualization of the observed information like computational models would be most informative to biologists [39]. Establishing suitable expressive formalisms requires filling the gap between mathematics and computer science by using advantages of both approaches.

In this study, we used deterministic finite state automata (DFA), which are a concept in automata theory [40]. Automata are the main mathematical objects in computer science that are capable of applying sequential algorithms, formalism, to system description and specification [41, 42]. DFA can abstractly display evolutionary processes and other phenomena with a sequential order of events [40, 43]. DFA represent a framework to describe the behavior of clonal expansion as discrete-state systems. Our main goal was to illustrate clonality patterns and to design a conceptually clear framework based on real biological data on clonality obtained from individuals with different PVLs and progression states of ATL. We also categorized the observed clone sizes accurately based on our integration site–mediated clonality analysis approach. Moreover, we propose the first well-suited empirical model for intuitive description of clonal expansion in ATL.

Modern medicine has done much to eradicate and cure disease, but it has been less successful in some areas, such as cancer, which still remains one of the most common incurable diseases. Remarkable progress has been made recently in the genomics of cancer with the advent of NGS technologies. However, this explosion in rapidly generated, massive sets of loosely structured raw data has challenged our abilities to quantitatively analyze and draw knowledge from this information [52]. Formal modeling can address this problem by enabling appropriate simplification of real data and making sense of observed experimental data. Empirical modeling provides an accurate and complete picture of observed complex data, and it has many applications in the life science [53]. The merging of mathematics, computer science and biology in empirical models can reshape these fields by providing new ways of thinking about a problem. The virtue of mathematics in modeling is to confer clarity and precision to explanations, and to provide coherence and formalism to experimental observations [54]. The quantitative and objective power of mathematics allows understanding of otherwise hidden aspects of biological phenomena [28]. Computational models enable intuitive representation of the masses of biological data via their visualization capability, which in turn facilitates mechanistic understanding of disease [55].

The most effective and appropriate type of mathematical/computational modeling varies for each biological question. A practical model that is properly formulated to explain and interpret experimental and clinical data obtained from analyzing clonal expansion in ATL is greatly needed. We need a model that can imitate the components of our biological system (the clonality patterns and clone sizes) and reflect its properties intuitively.

In the current study, we aimed to organize and intuitively express data from NGS on clonal expansion of HTLV-1–infected individuals using finite automata theory. Finite automata theory can describe and analyze dynamic behaviors of systems, and it is capable of simply representing complicated processes [43, 52]. Finite automata theory, which is a well-developed formal system, is used in processing various strings and sequences, especially in DNA sequence processing [51]. DFA are a subtype of finite automata theory and are simple computational structures that can formally illustrate the size order and combination of observed clones (clonality patterns). Our model translates the observed data into formal mathematical language by formulating a precise relationship between a set of clones in terms of their sizes and presenting this relationship in an easily understood state diagram.

Conventionally, clonality has been described as polyclonal, oligoclonal and monoclonal [56, 57]. However, these pattern descriptions are not quantifiable. For instance, it is known that the monoclonal expansion that results in large clones is an intrinsic feature of ATL development [20]; however, absolute clone sizes to describe this phenomenon have not been determined. In recognition of this limitation, we categorized the observed clone sizes into defined groups by which we could intuitively assess the degree of clonal expansion. We defined four groups of clonality patterns and four groups of clone sizes. Thus we defined polyclonal as a pattern showing different combinations and large numbers of VS and/or S clones, oligoclonal as a pattern showing more than one B or VB clone in combination with large numbers of VS and/or S clones, and monoclonal as a pattern showing a single VB or B clone in combination with a background of VS and/or S clones. In this way, we could attribute a meaning to the observed clone sizes and assess their contribution to ATL progression. In other words, we quantified how large a clone must be to affect the clinical status of an infected individual.

Generally, it is known that competition between clones shapes their distribution [3], but we do not know how a clone wins this competition to undergo clonal expansion. Presumably, a clone needs to become large enough to gain a fitness advantage to out-compete other clones. Coexistence of large numbers of S and VS clones, as well as presence of limited numbers of B or VB clones together with large numbers of S or VS clones in each given sample was observed. Total number and type of isolated clones are provided in Table 1, and Table 2. Therefore, we suggest that small cell populations (VS and S) do not have a selective advantage and can coexist with other clones. ACs and patients with SM ATL with low PVL harbored only VS and S clones, whereas all dominant clones in aggressive ATL (acute) were VB. The observed clone sizes were sorted in ascending order, and then thresholds of 2⁷, 2⁹ and 2¹¹ cells were applied. Hence, observed clone sizes were categorized into four distinct groups. Over the threshold of 2¹¹ cells, the largest clones in the samples that had monoclonal patterns were categorized within the same group. Within the threshold range of 2⁹ to 2¹¹ cells, the two largest clones in the samples that had oligoclonal patterns were categorized in the same group. Within the threshold range of 2⁷ to 2⁹ cells, clones in the samples that had polyclonal patterns with PVLs > 4% were categorized in the same group. Below the threshold of 2⁷ cells, the clones in the samples that had polyclonal patterns with PVLs < 4% were categorized in the same group. This quantified categorization of clone size not only is more intuitive for biological interpretation but also facilitates the transferal of clone size information into our model.

To convert the complex nature of data on clonal expansion into a manageable level of simplicity, borrowed the aid of mathematics and computer science. Our proposed DFA describe the clonality status of infected individuals as the output of final states q1, q2 and q3. Transitions are described by the function δ, which specifies exactly one next state for each possible combination of state and input symbol. The rows in the transition table indicate the states Q, the columns the input symbols, and the table entries the transition function δ. We indicated the accepting state with an asterisk in the figures. F = q1, q2, q3 indicates the final states, which consist of a set of states Q.

The final state of q1 represents an early stage of clonal expansion in which clone sizes does not exceed the threshold of 512 infected cells. AC patients with PVL > 4% and the SM ATL patients with low PVL terminated in this state. The DFA of samples of clinically progressed patients with SM and chronic subtypes with maximum clone sizes of 2048 infected cells terminated at the final state of q2. The final state of q3 included samples of the SM, chronic and acute subtypes with clone size > 2048. Acute samples, which represent the final stage of ATL progression, were observed only in q3. VB clones were observed only in the samples whose DFA terminated at q3. In the current study, c4 (VB clone) was observed only once in each analyzed sample. Since, presence of more than one VB clone is theoretically possible, we put a loop on the q3 final state of our final DFA machine. In the case of observing such a sample, the clonality will be defined as a largely expanded oligoclonal pattern with q3 final state.

We conducted a cross-sectional analysis of HTLV-1–infected individuals with a broad range of PVLs, representing different progression states of disease. Although analyzing the same individuals over time is of great importance, obtaining these kinds of samples is difficult and needs to be addressed in future studies. However, having access to the JSPFAD biomaterials bank allowed us to obtain two longitudinal (2 years apart) samples from the same individual (F12 and F16). By analyzing these samples, we could directly examine the hypotheses that samples with a higher final state have a higher chance of disease progression. At the first time point (F12) the patient was diagnosed with chronic ATL and had a PVL of 83.81% and major clone size of 5377. At the second time point (F16), the patient had progressed to the acute stage and had a PVL of 51.42% and clone size of 4909. The PVL and size of the major clone at the second time point were presumably decreased because of therapy. However, the major clone with integration site of Chr9:123682855 (+) remained stable and dominant over the 2-year period. The DFA for both time points terminated in q3. Thus, it appears that reaching the final state of q3 is a factor that can be used as a risk indicator. Because the final state of this patient was already q3 at the first time point, progression to the acute stage was predicted by our DFA. As further validation of our DFA, the other patients with acute ATL also had DFA that terminated at q3, and thus we expected these patients to have a poor prognosis. Subsequently, we confirmed the poor prognosis of these patients (F13, F14, F15 and F16) by checking their clinical follow-up data, which showed that they had all died of the disease. However, AC patients (F1, F2 and F3) and patients with SM ATL with low PVL (F4) who showed the final state of q1 in the DFA remained clinically stable without disease progression in two years.

The data suggest that our final proposed machine (M) not only describes the clonality status of patients at single time points within a cross-sectional analysis but also opens the door for future analyses of longitudinal samples for predictive purposes. The predictive ability of this model with larger numbers of samples from the same individuals over time still needs to be examined.

We believe our model is an appropriate empirical model for this system because it uses real biological data without theoretical assumptions as well as the fewest number of variables and the simplest set of relationships to explain the clonality status of samples. This model has the potential to provide insight into clonal expansion of ATL, but we are still far from understanding exactly when, where and at which step clonal expansion and transformation occur and how they can be controlled. However, our multidisciplinary strategy for translating the data of clonal expansion into the computable language of mathematics via modeling opens new avenues to approach these relevant questions in future studies.

Currently, ATL patients are categorized into different subtypes of disease progression based on clinical manifestations [46]. These standard clinical criteria for diagnosis mainly include organ involvement, leukemic manifestation, and levels of lactate dehydrogenase and calcium [58]. However, molecular features that represent the disease status remain to be characterized. Considering that in clinical practice distinct therapeutic strategies are used for the treatment of different subtypes of ATL, accurate subtype classification is of great importance. Thus, there is demand for more robust classification of ATL subtypes mediated by a genomic feature, such as HTLV-1 clonal composition. This kind of analysis would be also helpful in clinical decision-making, such as monitoring the outcome of therapeutic interventions, based on analysis of the clonality status of patients before and after therapy [59]. In this respect, constructing an empirical model of clonal expansion would be one of the primary steps towards developing a powerful software tool for automated analysis and interpretation of individual clonality, which holds great promise for molecular diagnostics and personalized therapeutic interventions.

We used HTLV-1 integration sites as a stable fingerprint to identify infected cells and accurately monitor their clonal expansion. We isolated large numbers of integration sites and quantified the clone sizes of eighteen clinical samples by our high throughput and validated methodology. We defined a threshold system that categorizes the size of clones into discrete groups based on the number of infected cells in each clone. We could quantify polyclonal, oligoclonal and monoclonal patterns using this categorization. We found that harboring larger clones was strongly associated with the progression of a patient to the more aggressive type of ATL, whereas smaller clones were observed across all samples and had little impact on progression. All samples with low PVLs (<10%) had smaller clones, however those with higher PVL had both smaller clones and one or two dominant larger clones. For the first time, we suggested DFA as a formalism that can represent sequential order of clones. We found that our DFA accurately reflect the true patterns of clonal expansion for each sample. Analyzing a large cohort of clinical samples from the same patients over time with the appropriate formal models will provide new insights into the clonal expansion of ATL and will allow for possible clinical applications of clonality in molecular diagnostics and predictions of prognosis.