2.1 Longitudinal single-cell whole genome sequencing data

In their work [15], Salehi et al. passaged the TNBC cell line 184-hTERT to characterize subpopulation dynamics. They employed CRISPR-CAS9 to derive a p53 knock-out (KO) version of the 184-hTERT cell line from the parental p53 wild-type (WT) cell line. The parental (WT) line was cultured for 60 passages and sampled four times to infer copy number status via scWGS (1599 cells sequenced on average, ranging from 1135-2061, 6395 total). Similarly, the KO cell line was cultured for 60 passages, with a replicate KO dish established at passage 20, which resulted in two KO lineages which were sampled for scWGS a total of six times, establishing single-cell genomes for 1200 cells per passage (range of 728–1738 cells, 16795 cells total). None of the in-vitro experiments involved cisplatin treatment.

Expanding their study [15] to in-vivo models, the researchers established one HER2+ and three TNBC PDX tumor mouse models. From these four PDX tumors, a total of 27 lineages were established. Among these, six serially-passaged PDX tumors were never exposed to treatment, while 18 were exposed to cisplatin in at least one passage. The PDX tumors were passaged between four and seven times over a range of 353–927 days, with single-cell genomes established at each passage for an average of 1146 cells originating from TNBC PDX mouse models (range of 466–1845 cells, 95954 total), and 890 cells from the HER2 + PDX (ranging from 497-1831 for a total of 8009 cells). For all downstream analysis, we only considered those lineages which had no intermittent treatment (Fig 1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Overview of the serial-passaging experiments and example subpopulation dynamics in the SA1035-TNBC treatment arm.

(A) Sankey diagram showing data origin (cell line or PDX), replicate groups, and cisplatin treatment schedules. (B) Relative SP frequencies across origins and replicate groups, with colors indicating SP identities unique to each replicate group. (C) Schematic of serial-passaging in the TNBC-SA1035 PDX, illustrating tumor passaging under treatment and no treatment arms. Created in BioRender. Veith, T. (2026) https://BioRender.com/yq988eq. (D) Example SP dynamics from TNBC-SA1035 Replicate Group 2, highlighting SP frequency shifts over time. Shaded region indicates cisplatin administration.

https://doi.org/10.1371/journal.pcbi.1013897.g001

2.2 Defining subpopulations and replicate groups

We took the single-cell copy number calls generated by Salehi (2021), et. al., and aggregated them to chromosome arm–level profiles by annotating genomic bins and computing modal arm-level copy numbers (Sect 2.2.1). SPs were then identified by applying -means clustering () with Silhouette scores used to determine the optimal number of clusters (Sect 2.2.2). Finally, replicate groups were defined based on lineage origin and treatment exposure, and for each replicate group a subpopulation-frequency matrix was computed to capture temporal dynamics of SP composition across passages (Sect 2.2.3).

2.2.2 Subpopulation assignment via k-means clustering.

For each patient-derived xenograft or cell line, we aggregated the arm-level copy number matrices from all available single-cell datasets. Subpopulation (SP) identities were then determined by applying k-means clustering to the aggregated matrix. Specifically, for a given (with ), clustering was performed (with 25 random starts), and the average Silhouette score computed. The optimal number of SPs was selected as:

(1)

where is the average Silhouette score for the clustering with centroids. Each cell is thereby assigned to a subpopulation (denoted , ). We selected k-means clustering for its interpretability and suitability for discrete subpopulation (SP) identification in copy number space (S1 Text, Sect 5.7). K-means clustering was performed using the base package ‘stats’ in R (4.4.1) with Silhouette scores calculated using the ‘cluster’ (2.4.1) package.

2.2.3 Computing subpopulation-frequency matrices and replicate grouping.

Two serially-passaged lineages were considered members of the same biological replicate group if and only if they originate from the same PDX or cell line and have been exposed to cisplatin for a the same number passages. We implemented the replicate structure to explore the robustness of our inverse search, as finding one matrix that described the interactions of more than one dataset (containing the same karyotype-defined SPs) increases the confidence that interactions between those karyotypes are real.

Using available metadata, each cell was annotated with its passage (timepoint) and an experimental label that reflects treatment exposure. Although SPs were defined over all cells from an origin, cells were further grouped by timepoint. For each timepoint (or passage) within a replicate group, we computed a SP-frequency matrix , where each entry was defined as:

(2)

Here, represents the set of cells in replicate at timepoint . Thus, is the fraction of cells that belong to the subpopulation, .

2.3 Identifying candidate interactions

From the SP frequency matrices, we filtered for candidate interactions by identifying pairs where the growth rate of one subpopulation was significantly correlated with the frequency of another (see S1 Text, Sect 5.1).

2.4 Optimization routine

To capture the dynamics of SP interactions and selection pressures, we modeled frequency-dependent selection using the replicator equation. This equation described the change in frequency of each SP over time, incorporating the fitness of each SP and the population-average fitness. For SPs, the replicator equation is given by:

(3)

where is the frequency of the SP as defined in Eq (2), gives that SP’s fitness, and is the population-average fitness. The fitness of a given SP was defined as:

(4)

where is the payoff matrix where an entry is the payoff to an individual of SP when interacting with an individual of SP . Consequently, the population-average fitness is the weighted average of all individual fitness values:

(5)

Some interactions between SPs may have no fitness consequences. In that case the payoff matrix entry representing their interaction would be set to zero. Non-zero payoff matrix entries were inferred via the methods described in Sect 2.3.

The final payoff matrix was determined using a hybrid routine that combined model selection and parameter estimation. The overall structure of the matrix (i.e., the set of non-zero interaction coefficients) was estimated using a beam search algorithm. This search performs a guided, backward-elimination of parameters, starting from a full model. At each iteration, it evaluates simplified models created by setting a single interaction coefficient to zero. To prevent a greedy search, a beam width of 3 was used to maintain and expand upon the most promising model structures at each step.

The fitness of each candidate model structure was evaluated by its Bayesian Information Criterion (BIC) (see Sect 2.5). This inner optimization was performed using the sequential quadratic programming (SQP) algorithm, as implemented in MATLAB’s fmincon function (MATLAB 2025a). To ensure a robust estimation and avoid local minima, each SQP optimization was launched from 80 random starting points using the MultiStart framework. The final model selected was the one with the lowest BIC found across the entire beam search procedure.

Parameters were estimated by minimizing the negative log-likelihood assuming Gaussian residuals (see S1 Text, Sect 5.2).

The optimization routine was performed for multiple replicates simultaneously. Let } be the set of replicates within a replicate group, with entries as defined in Eq (2). To reduce the incidence of spurious subpopulations with noisy frequency dynamics we excluded from all downstream analysis a given subpopulation which never exceeds 10% frequency in any replicate. Suppose one such SP exists, then . Note that this leads to the payoff matrix having identical dimensions across all replicates of a given group during each iteration of the optimization routine.

2.5 Parsimonious parametrization based on iterative BIC calculation

We used the above optimization routine to select between models of different complexity as follows. We calculated the BIC and corrected Akaike Information Criterion (AICc) across all replicates within a given group. These were given by:

(6)

(7)

where is the negative log-likelihood calculated in Sect 2.4, is the total number of estimated parameters (the non-zero entries in the payoff matrix ) and is the total number of passages across all replicates.

During each iteration, the current set of non-zero payoff matrix entries was re-estimated (see Methods Sect 2.4), and the corresponding BIC and AICc were calculated. Each non-zero entry was then systematically considered for removal, one at a time, by setting it to zero and re-optimizing the model. For each candidate removal, the fit was re-evaluated, and the BIC and AICc recalculated. The removal that yielded the most favorable change was adopted, and the process repeated. To guard against local optima, entries that did not individually improve model fit were also tested in combination with other removals. This combinatorial search continued iteratively until only two non-zero entries remain. Lastly, the best combination of non-zero and zero entries (yielding the best BIC) across the entire search was retained. BIC was used as the primary model selection criterion because it applies a stronger penalty for additional parameters compared to AIC/AICc, which helps prevent overfitting, particularly in settings where the number of potential payoff matrix entries can be large relative to the number of sampling timepoints. For replicate groups with more than 2 SPs, allowing both ECO-K and FitClone to be applied, we computed AICc and BIC for each model using their respective likelihood functions. For model comparison, we required concordant improvement in both criteria: ECO-K was judged superior only if both AICc and BIC were lower than those of FitClone; otherwise FitClone was considered to provide the better fit (see Table 1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Comparison of ECO-K vs FitClone performance across various TNBC lineages.

The table summarizes the root mean squared error (RMSE), AICc, and BIC for both ECO-K and FitClone models. Each row represents an individual sample with column entries to denote replicate group (RG), the number of SPs in that replicate group, the number of non-zero interaction coefficients estimated by ECO-K, the number of those coefficients deemed significant during parametric bootstrapping, the mean absolute matrix entry value (payoff), the number of timepoints at which single-cell genomes were established, and at how many of those timepoints the sample had been exposed to cisplatin. Overall, ECO-K provided better fits (lower AICc and BIC) in two datasets, where FitClone was superior in two datasets. The remaining four datasets were unable to be analyzed by FitClone as it requires more than two SPs.

https://doi.org/10.1371/journal.pcbi.1013897.t001

2.6 Confidence interval estimation via parametric bootstrapping

To ensure the reliability of matrix entry estimates, we employed a subsampling-based method. For each , we generated subsampled data sets by randomly selecting unique timepoints (i.e., sampling without replacement) from the solution to Eq (3) using the payoff matrix found via ECO-K. Specifically, each subsample was formed by drawing a random subset of timepoints and the corresponding solution trajectories from the replicator equation. The number of timepoints used in the subsampled datasets was equal to the original number of timepoints available for that replicate, minus one.

For each subsampled data set , we performed the optimization routine as described in Sect 2.4, which resulted in subsampled payoff matrices for . After processing all subsampled data sets, we analyzed the resulting payoff matrix estimates and calculated the means, standard errors, p-values, and confidence intervals of these estimates to assess their variability and statistical significance.

In particular, the mean and standard error of the estimated payoff matrix entries were given by:

The 95% confidence intervals can be calculated using the percentiles of the subsampled distribution or using the standard error:

We tested the null hypothesis that a matrix entry was effectively zero (. This test was performed by calculating a -statistic(using the MatLab R2025 ttest2 function) for the mean of the bootstrapped estimates, , against a value of zero, using the parametric bootstrap standard error . Bootstrap parameters which returned a p-value < 0.05 were counted as significantly different from zero.

2.7 Analysis of resulting payoff matrices

2.8 Synthetic dataset generation

To test the robustness of our method, we generated 1000 artificial datasets with varying numbers of SPs and noise levels. For each dataset, we first randomly selected the number of SPs (, where ) and assigned initial SP frequencies drawn from a uniform distribution, normalized such that they sum to 1.

Next, we generated a random payoff matrix by drawing entries uniformly from the interval . To restrict interaction complexity while still allowing for a sufficient number of nonzero payoff matrix entries, we kept the top matrix entries by absolute value and set the rest to zero. We then solved the replicator dynamics (Eq 3) from to with these initial conditions. To assess how accurately we could infer the original payoff matrix, we used the following two performance metrics:

• Rank Correlation: , the Spearman’s rank correlation coefficient between the vectorized entries of the true matrix and the inferred matrix . This measures the model’s ability to recover the correct relative ordering of the payoff entries, independent of their absolute values (Figs 3A–3B, S2). The existence of an ESS and the qualitative features of the fitness landscape are determined by the relative ordering of payoffs. A high rank correlation indicates that the inferred payoff matrix preserves these strategic properties.
• Dynamic Range: , the difference between the maximum and minimum entries in the inferred matrix. The Spearman correlation coefficient between and the true range, , is used to evaluate how well the model captures the strength of selection, dictated by the magnitude of payoff differences (Figs 3C–3D, S3). An accurately inferred range indicates that the model correctly predicts the rate at which strategy frequencies will change under replicator dynamics.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Overview of the optimization routine used in our method.

The flowchart depicts the step-by-step process of optimizing the payoff matrix for SP interactions using ECO-K. The routine begins by initializing the interactions and setting up the optimization problem. An initial optimization was performed, and the BIC was calculated with all interactions. The method then evaluates the removal of matrix entries to determine if this improves the BIC score. All indices in the matrix were tested for removal, but only the one that gives the lowest BIC score was retained. This routine was designed to ensure the most parsimonious set of interactions was used to capture the subpopulation dynamics.

https://doi.org/10.1371/journal.pcbi.1013897.g002

3.1 Inverse game theory estimates karyotype interaction coefficients from interference patterns

We developed a comprehensive framework to model and quantify frequency-dependent interactions between SPs in genomically resolved time-series data (Fig 2). Briefly, we first identified candidate SP pairs for inclusion in the model by testing whether the growth rate of one SP correlated with the frequency of the other SP [13]. We then optimized a payoff matrix (where each entry reflects the interaction strength for the corresponding SP pair) by minimizing a negative log-likelihood function under a multi-start strategy (Eq (12)). After this optimization, we iteratively removed individual entries from the payoff matrix if their removal lowered the BIC, thereby pruning insignificant matrix entries (Methods 2.5). Finally, we performed a parametric bootstrap procedure on the reduced payoff matrix to determine which of the remaining interaction coefficients were significantly different from zero (Methods 2.6). This process produced a final payoff matrix of estimated frequency-dependent interaction coefficients, yielding a parsimonious representation of the dynamics.

To evaluate our framework’s performance, we generated 1,000 synthetic time-series datasets. The process began by creating a sparse random payoff matrix, , with entries , which was validated to ensure it produced an ESS (See Methods Sect 2.8). Using this ground-truth matrix and the replicator equation, we then generated a synthetic time series, , systematically varying both the number of SPs () and the level of observational noise (, see Methods Sect 2.8).

From each synthetic time series, we inferred a payoff matrix, , and assessed the framework’s ability to recover two key properties of the ground-truth matrix, : first, the rank order of its entries, and second, its range (the difference between the maximum and minimum values). The framework demonstrated high proficiency in recovering the rank order of interactions (Fig 3A–3B). For instance, in the low-noise, 2-SP condition, the Spearman rank correlation between true and inferred entries was strong and highly significant (, ; Fig 3A). This robust performance was consistent across all conditions; while the correlation strength decreased with added noise and dimensionality, it remained significant in all cases (Fig 3B). In sharp contrast, the framework was unable to reliably infer the range of the payoff matrix entries (Fig 3C–3D). For the same low-noise, 2-SP condition, the correlation between the true and inferred range was weak and not statistically significant (; Fig 3C). This poor performance was systematic, with the correlation failing to reach significance in most conditions (Fig 3D). Taken together, these results indicate that while our framework can robustly determine the relative importance and hierarchy of interactions, it does not capture their absolute magnitudes from the time-series data alone. Specifically, as illustrated in Fig 3C, the inferred range of payoff values—and their associated uncertainty intervals—can underestimate the true underlying range by approximately a factor of two, indicating that ECO-K estimates should be interpreted in relative rather than absolute terms.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Comparative analysis of framework performance on inferring payoff matrix rank and range.

(A) A density scatter plot correlation between true and inferred payoff matrix entry ranks for the 2-player, low-noise condition (, ). (B) Spearman correlations remain significant as noise and the number of SPs increase, though the effect size diminishes. (C) Scatter plot for the inference of payoff matrix range (max-min) for the same 2-player, low-noise condition (, ). (D) Performance in range inference is shown across all conditions. Spearman rank correlation: o * , ** , *** .

https://doi.org/10.1371/journal.pcbi.1013897.g003

3.2 Model-based inference of frequency-dependent interactions in TNBC

Copy number alterations play a critical role in cancer development and progression, impacting cellular phenotypes and patient outcomes. Shah and colleagues [15] recently performed serial-passaging (Fig 1) experiments followed by scWGS of TNBC cell lines and PDX tumor mouse models, allowing for longitudinal copy number inference. In-vitro and in-vivo lineages were cultured for 20–60 passages and sampled four to seven times for sequencing and copy number inference. A subset of lineages were exposed to cisplatin at various passages to evaluate whether the drug selects for certain copy number alterations.

We applied ECO-K to these TNBC cell lines and PDXs to estimate interaction coefficients under a frequency-dependent selection model. ECO-K assumes that subpopulation fitness depends on the frequencies of other subpopulations, and infers a payoff matrix that best explains the observed SP frequency trajectories. We grouped 10 lineages into seven replicate groups (Table 1). Two lineages and were considered replicates (i.e., members of the same replicate group), if and only if they originated from the same PDX or cell line and had been exposed to cisplatin for the same number of passages (Sect 2.2). We clustered single-cell genomes across all replicates within a given replicate group by their karyotype to define SPs (Sect 2.2), which resulted in an average of four SPs (between two and seven) per replicate. Thus if we assumed interactions between every pairwise set of SPs, ECO-K would find 55 non-zero entries across seven payoff matrices.

Applying ECO-K to these seven replicate groups, we assigned 39 interaction coefficients in total with an average of six interaction coefficients per group (Table 1). Parametric bootstrapping confirmed that 36 (92%) of these interactions estimated coefficients were consistently non-zero under the assumed model (Table 1). We considered a replicate group to be well-described by the frequency-dependent model if at least one replicate had a RMSE (i.e., difference between predicted and observed frequency trajectories) less than or equal to 0.05 and the matrix representing the inferred interactions between SPs in that replicate group had a mean absolute matrix entry (i.e., effect size) above 0.10. Out of seven replicate groups, six met these criteria. Interestingly, untreated cases had a 2.87 times higher proportion of positive matrix entries compared to cases exposed to Cisplatin (t-test: p-value = 0.0483; S4 Fig), suggesting that competitive dynamics are less intense under cisplatin exposure.

To clarify whether intrinsic growth differences alone between SPs could explain our observations, we compared the ECO-K framework introduced here to the FitClone method [15]. FitClone uses a Bayesian Wright-Fisher diffusion model, which estimates constant growth rates (referred to as selection coefficients) for each SP independently. Importantly, FitClone assumes that the growth rate of each SP is unaffected by the presence or frequency of other SPs within the tumor. In contrast, ECO-K explicitly captures frequency-dependent selection through a payoff matrix, where each entry quantifies how the presence of one SP affects the growth of another. Non-zero entries in the payoff matrix represent meaningful interactions between specific pairs of SPs, while zero entries imply interactions which do not have an effect on fitness. Here, each SP represents a distinct karyotypic configuration, analogous to a “strategy” in evolutionary game theory. Fits to all replicate groups are given in S5 Fig.

Among six replicate groups whose dynamics were well-explained by the frequency-dependent model, two could be better fit by the baseline frequency-independent FitClone model (TNBC-SA1035 Replicate Groups 1 and 2), and two were better fit by ECO-K (TNBC-SA609 Replicate Groups 1 and 2), while the remaining two (TNBC-SA535 Replicate Group 2 and 184-hTERT p53KO) had insufficient SPs (must be ) to be evaluated by FitClone (Table 1). We defined ‘better fit’ as having both a lower AICc and a lower BIC compared to the alternative model (Table 1). RMSE values are reported descriptively but were not used as the primary model selection criterion. Here we highlight one lineage where the frequency-dependent model provided a particularly good description of the observed dynamics (Fig 4): Replicate Group 2 in PDX TNBC-SA1035 (Fig 4).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Estimated frequency-dependent interaction coefficients between karyotype defined subpopulations in TNBC PDX mouse models.

(A) Observed (dashed lines) and replicator equation–predicted frequency dynamics (solid lines) for subpopulations (SPs) in TNBC-SA1035 Replicate Group 2. (B) Payoff matrix diagram illustrating interactions between SPs. Red bands represent a negative payoff matrix entry, green bands represent a positive entry. The width of the band represents the relative strength of the interaction, and the arrow gives the direction of the interaction (i.e., which SP receives the payoff). (C) Replicator phase diagram correlating to (B) (D) Ternary plots of replicator dynamics for selected SP combinations: A–F–H (left) and A–C–F (right). (E) Heatmap of copy number alterations (CNAs) across SPs, annotated by sampling timepoint and cluster. (F) Mean population fitness ( from Eq 3) trajectories under different SP compositions. Removing specific SPs alters overall fitness relative to the original composition (red).

https://doi.org/10.1371/journal.pcbi.1013897.g004

In TNBC-SA1035 Replicate Group 2, the best fit payoff matrix for the four karyotype-defined SPs (Fig 4E; Sect 2.2.2), had four non-zero entries (Fig 4B). The most notable changes occur between days 60 and 120 (Fig 4A), corresponding to the trajectory where the dominant subpopulation transitions from SP A to SP C. This is illustrated in a total velocity magnitude ternary plot (Fig 4C, Methods Sect 2.7). In Fig 4D, we focus-in on the A-H-F and A-C-F trilateral relationships to demonstrate how SP F (the only SP assigned non-zero coefficients relative to all other SPs) influences the evolutionary trajectory of the population towards dominance of SP C.

Potential mechanisms for interactions arise from analyzing the replicator dynamics and copy number profiles of the subpopulations (Fig 4E). Subpopulation F was the only subpopulation that interacted with all three other SPs (Fig 4B–4D) and was characterized by chromosome 1 loss and gain of chromosome 14p (Fig 4E). Chromosome 1 harbors multiple genes involved in metabolism (e.g., GLUL, involved in glutamine metabolism and NADPH-dependent enzymes) [17,18]. Its loss might impair SP F’s ability to synthesize certain metabolites, making it more dependent on metabolic exchange with other SPs, facilitating broad interactions.

Alternatively, if we consider specifically SP F’s interaction with SP C: SP F’s gain of 14p (Fig 4E) may promote SP C due to upregulation of TGFB3 [19], a signaling factor that supports proliferation and immune evasion [20,21]. This compensates for SP C’s loss of 8q, which may have impaired its autonomous survival metabolic flexibility [22].

Collectively, these results support the hypothesis that karyotypic alterations can drive frequency-dependent interactions in specific contexts (Table 1).