Clonal Evolution

Neutral Evolution

17 min read

Neutral Evolution

Definition

Neutral evolution in cancer occurs when clonal diversity arises from mutation and genetic-drift without differential selection — all lineages have equal probability of producing surviving offspring. It is the evolutionary state that dominates the periods between selection events (Turajlic et al., 2019).

flowchart TD
    subgraph Neutral["Neutral Evolution (Default State)"]
        N1["Somatic mutation arises<br>in a cell lineage"] --> N2["No fitness effect<br>(passenger / neutral)"]
        N2 --> N3["Frequency determined by<br>timing of occurrence + drift"]
        N3 --> N4["Older mutations: high VAF<br>(arose in small population)"]
        N3 --> N5["Younger mutations: low VAF<br>(arose in large population)"]
        N4 --> N6["Expected: 1/f distribution<br>(delta=0 special case)"]
        N5 --> N6
    end
    
    subgraph Selection["Positive Selection (Episodic)"]
        S1["Driver mutation arises"] --> S2["Fitness advantage<br>s ≈ 0.4% per driver"]
        S2 --> S3["Lineage expands faster<br>than neutral expectation"]
        S3 --> S4["Excess high-VAF mutations<br>above calibrated null"]
    end
    
    Neutral -->|"Between sweeps"| Selection
    Selection -->|"After sweep completes"| Neutral
    
    N6 --> Test["Selection Inference"]
    S4 --> Test
    Test --> T1["Estimate delta independently"]
    T1 --> T2["Compare observed VAF distribution<br>to delta-calibrated neutral null"]
    T2 --> T3["Deviations = selection signal"]
    T3 --> T4["No deviation = neutral dynamics<br>dominate observable history"]

Figure: Neutral evolution as the default state and its relationship to positive selection. Under neutrality, mutation frequency is determined by timing and drift alone — the expected distribution is 1/f when delta=0. Selection produces excess high-frequency mutations that deviate from the neutral expectation. Selection is episodic; neutral evolution dominates between selection events. Selection inference requires independently estimating delta before testing for deviations from the neutral null. Synthesized from Turajlic et al. (2019), Graham & Sottoriva (2017), Bozic et al. (2016), and Greaves & Maley (2012).

Theoretical Basis

Neutral theory in molecular evolution, formalized by Kimura (1968, 1983), posits that most genetic variation within and between species is due to random drift of selectively neutral mutations rather than adaptive selection. This framework was applied to cancer by Williams et al. (2016), who demonstrated that in some cases intratumor-heterogeneity is explainable by neutral evolution alone — the observed distribution of mutation frequencies follows the 1/f^2 distribution predicted by neutral theory in a growing population (Turajlic et al., 2019).

The Frequency Spectrum Under Neutral Evolution

The 1/f^2 Distribution (delta = 0 Special Case)

Under neutral evolution in a well-mixed, exponentially growing population with no cell death (a pure birth process), the number of mutations m(f) as a function of allele frequency f follows a 1/f^2 distribution. This is the null model against which selection is tested: deviations from 1/f^2 — specifically, an excess of mutations at high frequency — suggest positive-selection (Turajlic et al., 2019).

This distribution arises from the Luria-Delbruck model of mutation in growing populations (Luria & Delbruck, 1943). Each new mutation arises at frequency 1/N (where N is the population size at that moment), and as the population doubles, the frequency halves. Older mutations are therefore more abundant simply because they arose in a smaller population — not because of selection.

Graham & Sottoriva (2017) formalized the 1/f test as an operational method: under neutrality in a growing population, the cumulative number of mutations at frequency f follows a 1/f distribution — the number of mutations doubles each time the frequency halves. Applying this test across 14 solid cancer types, approximately 30% of cancers showed no evidence of subclonal selection, suggesting that neutral evolution is common in established tumors.

The Death-Birth Ratio (delta): Generalizing 1/f

Bozic et al. (2016) showed that the 1/f distribution is a special case — it holds only when the death rate d = 0 (pure birth process). In real tumors, significant cell death occurs, and the death-birth ratio delta = d/b becomes the critical parameter determining the passenger mutation frequency spectrum. When delta is close to 1 (slow net growth but high cell turnover), the frequency spectrum shifts dramatically upward: far more mutations appear at high frequencies than 1/f^2 would predict.

flowchart TD
    Title["Passenger Mutation Frequency Spectrum as a Function of delta = d/b"]
    
    subgraph D0["delta = 0 (Pure Birth)"]
        A1["1/f^2 distribution"]
        A2["~1 passenger above 1% VAF"]
        A3["0 clonal passengers (m_c = 0)"]
        A4["rho_1 = 0 (no fixation possible)"]
        A5["Graham & Sottoriva (2017) null"]
    end
    
    subgraph D072["delta = 0.72 (Fast Growth)"]
        B1["Near-1/f^2, slight elevation"]
        B2["~few passengers above 1%"]
        B3["rho_1 approx 0.04"]
        B4["Star-like phylogeny"]
        B5["Metastasis-like parameters"]
    end
    
    subgraph D099["delta = 0.99 (Slow Growth)"]
        C1["Dramatically shifted upward"]
        C2["m_s approx 150 mutations >1% VAF"]
        C3["m_c approx 1.5 clonal passengers"]
        C4["rho_1 approx 0.60, rho_2 approx 0.36"]
        C5["Linear phylogeny"]
    end
    
    subgraph D0997["delta = 0.997 (MSS CRC fitted)"]
        E1["Amplified approx 333-fold vs delta = 0"]
        E2["m_s proportional to 1/(1-delta)"]
        E3["2-3 clonal passengers expected"]
        E4["Near-critical: high turnover,<br>slow net growth"]
        E5["TCGA colorectal (Bozic et al., 2016)"]
    end
    
    D0 -->|"Increasing delta →<br>more cell death + turnover"| D072
    D072 --> D099
    D099 --> D0997

Figure: How delta shifts the expected passenger mutation frequency spectrum. At delta = 0 (pure birth), the 1/f^2 distribution applies and essentially no mutations reach high frequency. As delta increases toward 1, cell turnover amplifies the number of observable mutations at all frequencies — the m_s formula contains 1/(1-delta), producing orders-of-magnitude amplification. Clonal passengers (mutations reaching 100% frequency without being truncal) become common when delta is close to 1. Phylogenetic tree shape transitions from star-like (low delta) to linear (high delta). Synthesized from Bozic et al. (2016) and Graham & Sottoriva (2017).

Fixation of neutral mutations. When d > 0, neutral passenger mutations can reach fixation (100% of cells) because the founding lineage can die out, leaving a later-appearing mutation as the sole ancestor. The probability that the k-th surviving passenger mutation fixates is rho_k approx (u/(u - log delta))^k, where u is the exome-wide passenger mutation rate (~0.015 per cell division). For fast-growing tumors (delta = 0.72), rho_1 approx 0.04 — fixation is unlikely. For slow-growing tumors (delta = 0.99), rho_1 approx 0.60, rho_2 approx 0.36, and even rho_5 approx 0.08 — multiple clonal passenger fixations are expected.

Frequency spectrum. The expected number of subclonal mutations above frequency alpha is m_s = u(1 - alpha)/((1 - delta)alpha). The expected number of clonal passenger mutations is m_c = delta*u/(1 - delta). For delta = 0.99, u = 0.015, there are ~150 mutations at >1% frequency, ~15 at >50%, and ~1.5 clonal. For delta = 0.999, these numbers are tenfold higher. In contrast, for delta = 0 (pure birth), there is on average only a single passenger above 1% frequency.

Clonal does not equal truncal. When delta is close to 1, multiple passenger mutations reach fixation during clonal expansion — they become clonal (present in all cells) but were NOT present in the founding cell. This means that not all clonal mutations are truncal in the phylogenetic sense. The number of clonal passengers collected during expansion can be substantial. Each such fixation event records a bottleneck during which competing intermediate-clones were lost — only the winner’s passengers survive to be sampled.

Tree shape. The death-birth ratio determines whether phylogenetic trees of passenger mutations are star-like (delta = 0.72, fast growth) or linear (delta = 0.99, slow growth). For intermediate delta = 0.97, mixed tree shapes are most likely. This provides a theoretical null for interpreting observed phylogenetic-tree shapes.

Empirical fit. Fitting the model to 42 TCGA colorectal cancer samples yielded delta approx 0.997 for microsatellite-stable (MSS) cancers, between premalignant (delta approx 0.999) and invasive (delta approx 0.99) estimates. This confirms that colorectal cancers operate with delta extremely close to 1 — high cell turnover with slow net growth.

Formal Resolution: The 1/f Test and delta-Generalized Spectrum Are Nested

The central theoretical relationship between the 1/f test (Graham & Sottoriva, 2017) and the delta-generalized spectrum (Bozic et al., 2016) is that they are nested models, not competing ones. The 1/f distribution is the delta = 0 special case of the Bozic 2016 frequency spectrum:

F_k(alpha) = 1 - [u/(u - log(1 - alpha(1 - delta)))]^k

When delta = 0, this reduces to 1 - [u/(u - log(1 - alpha))]^k, which approximates the 1/f form. When delta > 0, the spectrum shifts upward by a factor proportional to 1/(1 - delta).

This nesting has three practical implications:

1. The 1/f test is a joint hypothesis. The standard 1/f test tests “neutrality AND pure birth (delta = 0)” simultaneously. A rejection of the 1/f null could mean either (a) subclonal selection is present, or (b) delta > 0 with no selection at all. This means the 1/f test, applied without calibrating for delta, has unknown specificity — it may misclassify neutral tumors with high delta as showing selection.

2. delta must be estimated independently before selection can be inferred. The practical resolution is a two-step procedure: (i) estimate delta from the VAF distribution (or from orthogonal growth-rate data, or by fitting the Bozic 2016 model to the low-frequency tail where selection signatures are weakest), then (ii) test observed mutation frequencies against the delta-calibrated neutral null. Only deviations beyond what delta predicts constitute evidence of selection.

3. The ~30% neutrality estimate is a lower bound. Graham & Sottoriva (2017) found that ~30% of cancers across 14 types showed no evidence of subclonal selection by the 1/f test. Given that delta varies across tumor types — and for MSS colorectal cancer delta approx 0.997, amplifying mutation counts ~333-fold — the 1/f test has variable sensitivity. Tumors with high delta are more likely to be classified as “showing selection” because their neutral spectrum already deviates from 1/f. The true prevalence of neutral subclonal dynamics may be higher than 30%. Graham & Sottoriva (2017) acknowledge this implicitly by noting that “a specific combination of selected subclones could masquerade as 1/f” — the Bozic 2016 framework converts this qualitative caveat into a quantitative correction.

This resolution does not invalidate the 1/f test. It clarifies its domain of applicability: the test is reliable when delta is known to be small (fast-growing tumors, delta ~ 0.72), and it requires delta calibration when cell turnover is unknown or expected to be high. See branching-process-model for the unified mathematical architecture spanning both frameworks.

Growth Model Caveat

The 1/f neutral null is derived under the assumption of exponential growth (Graham & Sottoriva, 2017). Under gompertzian-growth or other decelerating growth models, the expected null distribution would differ: fewer cell divisions occur late in growth, depleting the low-frequency mutation tail. A Gompertzian-growing tumor without selection could therefore deviate from 1/f and be misclassified as showing selection. The magnitude of this effect remains unquantified — it is an open question whether the 1/f null requires Gompertzian correction (see gompertzian-growth).

The Gompertzian caveat compounds the delta caveat: both non-exponential growth (Gompertzian deceleration) and non-zero death rate (delta > 0) can shift the expected neutral spectrum away from 1/f. Disentangling these two effects from genuine selection requires independent estimates of both the growth curve shape and delta.

The Big Bang Model

Sottoriva et al. (2015) demonstrated a mechanism for neutral-dominated tumor growth: after initial transformation, colorectal tumors grow predominantly as a single clonal expansion in which numerous intermixed subclones co-exist without stringent selection. In this “Big Bang” model, the timing of a mutation — not selection for it — determines its prevalence. Early-arising private mutations become pervasive throughout the final tumor because they had more cell divisions to expand; late-arising mutations remain localized regardless of fitness advantage.

The study’s empirical design is unusually granular for cancer genomics: 349 individual tumor glands were laser-capture microdissected from 15 colorectal tumors (11 carcinomas, 4 adenomas), with each gland representing a spatially localized clonal population. Mutations were classified into six spatial categories: public (present in all glands), side-specific (one side of the tumor), side-variegated (patchy on one side), variegated (scattered throughout), regional (localized to one region), and unique (single gland).

All 15 tumors exhibited Big Bang dynamics: public alterations were uniformly present, most private alterations showed pervasive but variegated spatial patterns, and — critically — no tumor showed evidence of a selective sweep. The absence of sweeps was universal across both adenomas and carcinomas. This finding demonstrates that extensive intratumor-heterogeneity can arise from neutral dynamics during a single expansion, without ongoing selection.

A key finding restricted to carcinomas was the “born to be bad” signature: variegated alterations (mutations present in scattered glands across the tumor) appeared exclusively in carcinomas, not in adenomas. This suggests that early subclone intermixing and pervasive private mutation spread may reflect the emergence of an invasive phenotype — the spatial dynamics of mutation spread differ between benign and malignant lesions.

Because late sweeps are rare under Big Bang dynamics, the genomic profile of the early tumor (when it contained only ~10,000-100,000 cells) can be reconstructed from the final, macroscopic tumor. This “primordial tumor profile” is recoverable precisely because selection has not overwritten the early mutation record — a property that would be lost if successive selective sweeps had repeatedly purged the population.

The Big Bang mechanism complements the delta-driven fixation described by Bozic et al. (2016): both frameworks show that neutral processes can generate high-frequency and clonal mutations without selection, but they operate through different mechanisms. Sottoriva’s Big Bang emphasizes spatial intermixing during a rapid single expansion; Bozic’s delta framework emphasizes cell turnover and stochastic fixation. A real tumor could exhibit both: rapid spatial expansion (Big Bang) with high cell turnover (delta close to 1), producing a mutation landscape shaped by both timing and death-birth dynamics.

When Neutral Evolution Dominates

Neutral evolution is the default state. Before an adaptive mutation occurs, the population evolves neutrally. After a clonal-sweep is complete and the entire population carries the adaptive mutation, the dynamics revert to neutral (Turajlic et al., 2019). Thus, neutral evolution can both precede and follow selective events.

In some tumors, neutral dynamics may dominate throughout most of the observable evolutionary history. In multiple myeloma, detection of neutral evolution dynamics correlated with progression-free and overall survival and was associated with the presence of a strong clonal (truncal) oncogenic driver, which might explain the lack of ongoing selection (Johnson et al., 2017, cited in Turajlic et al., 2019). This suggests a clinically significant pattern: tumors driven by a single potent truncal driver may experience a prolonged period of neutral subclonal dynamics, while tumors requiring multiple weaker drivers may experience repeated selective sweeps.

Turajlic et al. (2019) situate neutral evolution within a four-mode taxonomy of cancer evolution: linear (sequential sweep), branching (divergent subclones from a common ancestor), neutral (drift-dominated), and punctuated (rapid bursts of genomic change). Neutral evolution is both a mode in its own right and the inter-sweep default that operates within the other three modes.

Neutral Evolution and the Molecular Clock

Under neutral evolution, passenger mutations accumulate at a rate governed by the background mutation rate, with no effect on cell fitness. This makes them a temporal record: the number of passenger mutations unique to a given cell lineage is proportional to the time elapsed since that lineage diverged from its ancestor (Greaves & Maley, 2012). This principle — the molecular-clock — depends critically on the assumption of neutrality for the clock mutations.

The molecular clock enables four classes of evolutionary inference from neutral passenger accumulation (Graham & Sottoriva, 2017; Gerstung et al., 2020):

  1. Relative timing of clonal branching: comparing passenger loads on different branches of a phylogenetic-tree reveals the relative order and temporal separation of branching events.
  2. Timing of driver acquisition: the ratio of clonal to subclonal mutations times when driver mutations occurred — Gerstung et al. (2020) showed many drivers precede diagnosis by years to decades.
  3. Timing of copy-number alterations: mutations present on both copies of a gained segment occurred before the gain; single-copy mutations occurred after.
  4. Timing of mutational process shifts: comparing mutational-signature composition in early (clonal) vs. late (subclonal) mutations reveals which mutational processes were active at different evolutionary epochs.

The clock’s precision depends on the constant-rate assumption: that the neutral mutation rate is stable over evolutionary time. Gerstung et al. (2020) found this assumption holds in ~60% of samples; in ~40%, mutational spectrum shifts indicate rate changes that reduce clock precision.

The molecular clock and neutral evolution are thus mutually reinforcing concepts. Neutral evolution provides the theoretical justification for treating passenger mutations as a clock; the clock, in turn, provides empirical evidence that neutral dynamics dominate long stretches of tumor evolutionary history — because if selection were pervasive, the clock signal would be continuously disrupted by clonal expansions that reset the passenger count.

Significance for Subclonal Inference

The 1/f^2 neutral tail of low-frequency mutations is an inevitable consequence of tumor growth and complicates clonal inference from variant-allele-fraction data. Mutations in this tail are abundant but uninformative about clonal structure — they represent the ongoing background of neutral mutation in the most recent cell generations (Turajlic et al., 2019).

The clonal not-equal-truncal insight from Bozic et al. (2016) adds a second complication: mutations that appear clonal (CCF = 1.0) may be passengers that fixed during expansion, not truncal mutations present in the founding cell. Standard subclonal reconstruction methods that equate clonal with truncal will overestimate the number of truncal events and misdate the timing of branching events. Distinguishing truncal from clonal-but-not-truncal requires independent evidence — timing relative to copy-number changes, or phylogenetic analysis across multiple samples.

The driver fitness effect s approx 0.4% estimated by Bozic et al. (2010) and Greaves & Maley (2012) means that selection, when it does operate, is extraordinarily weak by organismal standards. This blurs the boundary between neutral evolution and weak selection: a driver with s = 0.4% requires many cell generations to produce a detectable frequency shift, and during that time, drift and neutral dynamics continue to operate. The observable signature of selection may therefore be subtle and gradual rather than abrupt — a shift in the frequency spectrum rather than a discrete sweep.

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