Heterochrony as temporal redistribution of developmental burden

Abstract

Heterochrony is central to evolutionary developmental biology because shifts in developmental onset, rate, or termination can generate morphological divergence. Yet it remains largely descriptive, identifying when developmental trajectories change without specifying how commitments are reorganized across interacting subsystems. Here, I propose an explicitly phenomenological framework that interprets heterochronic change as a temporal redistribution of developmental burden. Development is represented as a structured state of subsystem-specific masses and energetic expenditure rates, while developmental burden is introduced as a latent comparative index linking developmental organization to the energetic expenditure associated with organized growth. A heterochronic trajectory is decomposed into two operations: transformation of subsystem contributions and remapping of developmental time. In this framework, redistribution denotes an organizational reconfiguration of developmental burden through changes in subsystem weighting, coupling, priority, onset, rate, or termination. It does not necessarily presuppose a conserved resource, nor does it necessarily entail the transfer of energetic or material resources. Progenesis illustrates this distinction: reproductive maturity is attained earlier relative to somatic development and is represented as an earlier developmental endpoint, thereby redistributing burden in time without necessarily involving resource transfer. This separates changes in developmental organization from the stronger hypothesis of resource-linked allocation or trade-offs, which requires independent evidence. It also distinguishes instantaneous from cumulative burden and somatic development from reproductive maturation, preserving the biological differences among canonical heterochronic modes. The formalism generates testable expectations: prolonged development may reduce instantaneous burden while extending cumulative commitment, early termination may truncate somatic burden, and subsystem prioritization should produce temporally asymmetric growth or maturation profiles. Applications to amphibian paedomorphosis, human neural development, hominin evolution, dinosaur ontogeny, and bat limb development demonstrate its scope. The equations serve as formal organizing relations for comparative hypothesis development rather than as a parameterized predictive model. By connecting classical heterochronic categories with measurable growth, metabolic, and maturation data, the framework provides a common language for investigating how developmental timing, subsystem organization, and energetic feasibility jointly shape morphological evolution.

1 Introduction

Heterochrony has long provided evolutionary developmental biology with a framework for relating ontogenetic change to phylogenetic transformation (Gould, 1977; Alberch et al., 1979; McKinney and McNamara, 1991). By focusing on changes in developmental onset, rate, duration, and sequence, it shows how substantial morphological divergence may arise through reorganization of developmental schedules without requiring wholly novel structures.

Gould established the conceptual importance of heterochrony in evolutionary theory, Alberch and colleagues made it analytically tractable by representing development as ontogenetic trajectories in size and shape space, and McNamara clarified its nomenclature. Together, these contributions provided a rigorous descriptive foundation. A remaining challenge, however, is to express the canonical modes as comparable transformations of a shared developmental baseline while preserving the distinctions among changes in rate, onset, termination, and reproductive timing.

Classical heterochronic terminology remains primarily descriptive: it classifies changes in developmental timing but does not directly represent how those changes alter the relative contributions of interacting subsystems through ontogeny. This matters because development is not a single scalar trajectory. Subsystems differ in growth rate and energetic cost, and they interact under shared physiological and developmental constraints. Similar adult phenotypes may therefore arise from different schedules of subsystem development and burden (Smith, 2001; McNamara, 2012).

The present study addresses this problem with an explicitly phenomenological framework informed by thermodynamic considerations. Its central claim is that heterochronic change can be interpreted as a temporal redistribution of developmental burden. In this framework, redistribution denotes an organizational reconfiguration of developmental burden across time and subsystems through changes in subsystem weighting, coupling, priority, onset, rate, or termination. It does not necessarily presuppose a conserved resource, nor does it necessarily entail the transfer of energetic or material resources. Development is represented as a vector of subsystem-specific masses and energetic expenditure rates, and a heterochronic trajectory is decomposed into subsystem transformation and temporal remapping. Resource limitation, allocation, and trade-offs are additional system-specific hypotheses that should be imposed only when supported by independent biological evidence.

The framework is deliberately limited in scope. It neither proposes a microscopic thermodynamic law of development nor replaces the classical descriptive vocabulary of heterochrony. Instead, it provides a comparative structure in which heterochronic modes can be expressed as transformations of a shared developmental baseline. The equations therefore provide formal organizing relations for comparative hypothesis development rather than predictive laws. Prediction would require system-specific definitions, parameter estimates, and independent validation of the state variables, weights, efficiencies, and temporal maps.

This definition establishes two levels of interpretation. At the organizational level, the framework describes changes in developmental scheduling, duration, coupling, and subsystem priority. At the resource-linked level, it asks whether those changes covary with metabolic allocation or reciprocal trade-offs; this stronger interpretation requires independent evidence. Developmental burden schedules can therefore be compared without assuming that every heterochronic shift redistributes a conserved resource.

2 Heterochrony beyond description

Heterochrony is defined here as evolutionary change produced by shifts in the onset, rate, or termination of a developmental process relative to a comparative baseline. That baseline need not be a directly observed ancestor: it may be reconstructed phylogenetically, approximated by a closely related extant taxon, or estimated from a reference population. For a theoretical framework, “comparative developmental baseline” is therefore more precise than “ancestral ontogeny” (McKinney and McNamara, 1991; McNamara, 2012).

This definition also clarifies why heterochrony cannot be reduced to a single temporal metric. Development comprises overlapping programs of growth, differentiation, and maturation across neural, appendicular, axial/trunk, metabolic, and reproductive systems. Because these programs interact, a change in subsystem priority may alter their ordering, duration, or relative emphasis. Such an organizational change does not, by itself, demonstrate transfer of a conserved resource.

Within classical heterochrony, developmental change may generate either paedomorphosis or peramorphosis, depending on whether descendant development yields less or more developed adult morphology relative to the comparative baseline. This distinction becomes especially important once somatic development and sexual maturation are considered separately. Adult phenotype depends not only on how rapidly or how long body form develops, but also on when reproductive maturity is reached relative to somatic completion. A retained juvenile morphology does not therefore identify a single underlying developmental mechanism (Gould, 1977; McNamara, 1986, 2012).

Neoteny denotes a reduction in the rate of somatic development. Juvenile ancestral features are retained because body form changes more slowly through ontogeny. By contrast, progenesis denotes precocious sexual maturation relative to somatic completion. Here reproduction is reached earlier, and development is truncated before the ancestral somatic trajectory is fully completed. Thus, both neoteny and progenesis may produce paedomorphic adults, but they do so through different developmental relations: neoteny by slowing somatic development, progenesis by advancing reproductive maturity. Postdisplacement refers to delayed onset of development of a trait or subsystem relative to the baseline, whereas predisplacement refers to earlier onset. Acceleration denotes an increased rate of somatic development, and hypermorphosis denotes delayed termination or extension of development (McNamara, 1986, 2012).

A major step toward formal analysis was provided by Alberch and colleagues, who recast heterochrony in terms of ontogenetic trajectories in size–shape space. Their contribution was not simply to rename the classical categories, but to show that developmental change could be represented as a trajectory and that phyletic morphological differences could be analyzed as transformations of that trajectory. In this sense, heterochrony became more analytically tractable because developmental rate, timing, duration, and scaling could be discussed within a comparative framework rather than as isolated descriptive labels (Alberch et al., 1979).

To move beyond description, development can be treated as the maintenance and expansion of organized mass under continuing energetic expenditure. Heterochrony then becomes a problem of how developmental burden is scheduled. Under otherwise comparable energetic scaling, slower development may reduce instantaneous burden while extending its duration, whereas faster development may compress burden into a shorter interval. Delayed onset shifts commitment to a later phase. In progenesis, reproductive maturity is attained earlier relative to somatic development; the present model represents this relation as an earlier developmental endpoint. These are conditional comparative expectations, not automatic consequences of the formalism.

3 A phenomenological thermodynamic framework for developmental burden

To formalize heterochrony as temporal redistribution, development must first be represented as a structured state rather than as a single scalar process. Here, the term “vector” is used in the sense of a developmental state vector, not as a spatial vector. Its components do not represent physical directions, but collect subsystem-specific developmental masses at a given time. Likewise, the matrix notation is not intended to imply an abstract mathematical operation detached from biology. It is used as a compact bookkeeping device for specifying how developmental components are reweighted or coupled. In this sense, records the state of multiple developmental subsystems, whereas R specifies how those subsystem components are redistributed in a heterochronic transformation.

Throughout this section, the vector-valued developmental state and energetic expenditure-rate vector are written in bold, matrices such as and R are written in bold Roman capitals, and scalar quantities or scalar-valued functions, such as , , , , and , are written in italic type. Standard mathematical operators such as diag, exp, and max are written in upright Roman type. A dot denotes a rate with respect to chronological time . Subscripts are kept distinct throughout: H for hindlimb-related, F for forelimb-related, Ax for axial/trunk-related, B for brain-related, S for somatic, G for gonadal or reproductive, and MR for metamorphic remodeling. In this notation, bold uppercase R denotes a redistribution matrix, whereas lowercase r denotes a dimensionless scalar subsystem-weighting coefficient. For example, is the general heterochronic redistribution matrix, and later case-specific matrices such as use scalar weights such as , , and to indicate relative subsystem priority. Here, abbreviates the ordered subsystem set hindlimb, forelimb, axial/trunk, and brain-related; the axial/trunk component is written as Ax in equations to avoid ambiguity.

where denotes hindlimb-related developmental mass, forelimb-related developmental mass, axial and trunk-related developmental mass, and brain-related developmental mass. The axial/trunk term includes the vertebral column, pelvis–trunk support, thoracic configuration, and postural stabilization structures relevant to locomotor and body-orientation development. Locomotor function is therefore represented as an emergent relation among hindlimb, forelimb, and axial/trunk components, rather than as a separate variable overlapping appendicular anatomy. This decomposition is illustrative rather than exhaustive, but it is sufficient to represent the structured character of ontogeny. The subsystem-specific comparative baseline trajectories are illustrated in Figure 1A. In biological terms, Equation 1 is a bookkeeping list: at any time t, it records how much organized developmental mass is assigned to each of the four subsystems.

Figure 1

Let the corresponding vector of energetic expenditure rates be

In Equation 2, the four entries on the right-hand side are the energetic expenditure rates of the hindlimb, forelimb, axial/trunk, and brain-related subsystems, respectively. Biologically, records how rapidly energy is being used by each subsystem at time t. Subsystem-specific conversion efficiencies are represented by the diagonal matrix

In Equation 3, is the diagonal efficiency matrix and each is the positive conversion-efficiency coefficient of subsystem . Biologically, a larger coefficient means that more organized developmental mass is supported per unit energetic expenditure. The diagonal form treats subsystem efficiencies separately and does not itself introduce cross-subsystem coupling. With these definitions, a mass-specific developmental-burden index can be written as

The row vector simply sums subsystem components, so . The numerator is the corresponding efficiency-weighted energetic expenditure rate. Equation 4 therefore defines as an instantaneous developmental-burden index: it increases when energetic expenditure rates are high or conversion efficiencies are low, and decreases when the same organized developmental mass can be maintained more efficiently. Equation 4 is not intended as a complete microscopic law of development, but as a phenomenological comparative index.

Equation 4 is defined only over biologically meaningful developmental intervals for which . In numerical or empirical applications, time points at which organized developmental mass is absent, or at which the denominator approaches zero, should be excluded from the model domain or treated with an explicitly stated regularization convention.

The formulation makes two points explicit. First, development entails not only the accumulation of form but also the maintenance and expansion of organized mass under energetic expenditure. Second, a common burden index permits distinct developmental trajectories to be compared in terms of when and where developmental commitments are concentrated.

Let denote a comparative baseline developmental trajectory. A heterochronic trajectory may then be written as

In Equation 5, is the transformed heterochronic developmental-state vector and is the comparative baseline state. Here, is a redistribution (state-reweighting) matrix acting across subsystems and is a temporal remapping function that shifts, stretches, compresses, delays, or advances developmental time. Equation 5 therefore states that a heterochronic trajectory differs from the comparative baseline in two separable ways: changes the relative weighting of subsystems, whereas changes the schedule on which the baseline trajectory is traversed. In the present phenomenological treatment, must map a non-negative baseline state to a biologically admissible non-negative transformed state. Elementwise non-negativity is a sufficient special case, and diagonal self-weighting is the simplest illustrative form; inhibitory off-diagonal coefficients are admissible when this state-level condition remains satisfied. The subsystem-redistribution and temporal-remapping components are shown schematically in Figures 1B, C, respectively.

In this framework, redistribution denotes an organizational reconfiguration of developmental burden across time and subsystems through changes in subsystem weighting, coupling, priority, onset, rate, or termination. It does not necessarily presuppose a conserved resource, nor does it necessarily entail the transfer of energetic or material resources. Formally, the redistribution matrix acts on the comparative subsystem state vector, not on the scalar burden index or on a conserved pool. It changes the pattern and magnitude of subsystem contributions to the developmental trajectory and, through the associated expenditure-rate vector, to the derived burden schedule. The matrix need only map admissible baseline states to admissible transformed states; it is not assumed to conserve total mass, to be stochastic, or to have weights that sum to a fixed total. Conservation or bounded-budget constraints may be added in system-specific applications when justified by independent evidence. Consequently, Equation 5 alone does not determine the direction or magnitude of a burden change; that determination requires the transformed expenditure-rate vector and efficiencies specified in Equation 6.

In this context, “diagonal” refers to self-weighting entries of the redistribution matrix, not to the straight arrows in Figure 1B. In a diagonal form, each subsystem is mapped only to its own weighted output: hindlimb-related mass to , forelimb-related mass to , axial/trunk-related mass to , and brain-related mass to . By contrast, an off-diagonal entry represents cross-subsystem coupling, in which development in one subsystem modifies the burden, growth, or feasibility of another. For example, increased hindlimb emphasis may require coordinated pelvic and axial/trunk stabilization. Thus, Figure 1B should be read as a schematic representation of self-weighting terms with optional biologically admissible cross-subsystem coupling.

The illustrative applications below use diagonal redistribution matrices, in which each subsystem is reweighted independently. However, the general framework also allows one subsystem to influence another. For a coupled system, the redistribution matrix can be written as

Here, n is the number of modeled subsystems, and index those subsystems. This expression separates two types of developmental effects. The diagonal matrix contains the subsystem-specific self-weighting terms , which describe changes in the contribution of each subsystem to its own transformed state. The matrix C contains the coupling terms between different subsystems. In , the first index i identifies the subsystem receiving the effect and j identifies the subsystem exerting the effect. The condition means that the diagonal elements of C are zero because self-weighting is already represented in . The plus sign in Equation 5a denotes an algebraic combination of self-weighting and cross-subsystem effects; it does not imply that all effects increase development. A positive represents a facilitating association, whereas a negative represents an inhibitory association. Negative coupling is admissible only when the resulting transformed developmental state remains biologically meaningful and non-negative. The coupling coefficients may phenomenologically summarize coordinated effects arising from shared developmental precursors, endocrine signaling, biomechanical dependence, or shared pleiotropic regulation. However, they should not be interpreted as a mechanistic genetic model of pleiotropy, which would require explicit genetic or regulatory variables and empirically defined causal relationships.

Cross-subsystem coupling is conceptually distinct from direct competition for a limited developmental resource. Coupling describes how one subsystem influences another, whereas a resource trade-off arises when multiple subsystems draw from a shared and bounded developmental budget. When independent biological evidence supports such a constraint, the model can additionally impose

Here, is the transformed developmental amount of subsystem i at time t, is its non-negative cost per unit of development, and is the total developmental budget or capacity available at that time. It is an available upper bound, rather than an instantaneous developmental-burden measure. Equation 5b states that the combined demands of all subsystems cannot exceed the available budget. Consequently, an increase in the demand of one subsystem leaves less budget available to the others, thereby representing a direct developmental trade-off. The budget constraint is optional and does not follow automatically from the redistribution matrix in Equation 5a. Both the coupling coefficients and the developmental-budget parameters require system-specific empirical estimation.

Let denote the subsystem-specific energetic expenditure-rate vector associated with the heterochronic trajectory . The heterochronic developmental burden is therefore

In Equation 6, is the heterochronic energetic expenditure-rate vector, is the transformed developmental state, and is the efficiency matrix defined in Equation 3. Equation 6 applies the burden calculation in Equation 4 to the heterochronic trajectory. Its numerator is the efficiency-adjusted energetic expenditure rate and its denominator is the total transformed developmental mass. Biologically, it estimates the instantaneous burden required to support the reorganized developmental state at time t. The corresponding burden accumulated across the full developmental interval is

Equations 6 and 7 distinguish between instantaneous burden, , and cumulative burden, . A heterochronic mode may reduce the former while increasing the latter, or vice versa. This distinction is one of the main theoretical advantages of the formalism. Here denotes the initial time of the developmental interval and its endpoint; when the heterochronic mode changes the developmental endpoint, the upper limit is replaced by the transformed endpoint introduced below in Equation 14. The distinction is visualized in Figure 1D, where the hatched area represents the cumulative heterochronic burden rather than the baseline burden.

The panels of Figure 1 summarize this sequence: subsystem-specific baseline trajectories (Figure 1A) are transformed by subsystem redistribution (Figure 1B) and temporal remapping (Figure 1C), yielding instantaneous and cumulative burden indices (Figure 1D).

Although developmental burden is introduced as a phenomenological index, it can be operationalized empirically. Equation 4 defines the general comparative quantity, whereas Equations 8–10 present possible energetic and thermodynamic approximations. With body-mass or subsystem-mass trajectories and appropriate metabolic proxies, a first-order operational form is

In Equation 8, is the instantaneous developmental burden, is body mass or subsystem-specific developmental mass, and is the total energetic expenditure rate associated with development. The proportionality sign indicates a scaling relationship rather than an exact equality. Biologically, burden increases when more energy is expended per unit developmental mass. The corresponding cumulative burden is

In Equation 9, is the total burden accumulated between and ; the integral adds the instantaneous burden over time. Thus, even a moderate instantaneous burden can produce a large cumulative burden if it persists for a long developmental interval. In practice, direct energetic reconstruction will rarely be available across taxa. Empirical application would therefore rely on comparative proxies such as growth curves, metabolic rate, ossification timing, reproductive maturation, organ-specific growth, histological maturation, or developmental transcriptomic timing. Earlier thermodynamic work has also suggested that development may be interpreted in terms of specific entropy production or entropy production per unit mass (Zotin and Zotina, 1967; Zotin and Zotin, 1997; Lamprecht, 2003). In that stronger thermodynamic sense, burden may be approximated by mass-specific entropy production rate,

In Equation 10, denotes the developmental entropy-production rate and denotes mass, so their ratio is entropy production per unit mass. Biologically, this ratio can serve as a thermodynamic proxy for developmental burden when direct energetic measurements are unavailable. The entropy-production form is one possible biologically motivated operationalization, not a requirement of the framework. The model does not demand exact lifetime energetic reconstruction for every organism; rather, it provides a comparative language connecting developmental schedules with mass trajectories, timing data, and metabolic proxies. Recent work in developmental energetics and metabolism has renewed interest in how energetic costs, metabolic state, and developmental progression are coordinated during embryogenesis and early development (Ghosh et al., 2023; Garfinkel et al., 2024).

4 Canonical heterochronic modes

The canonical heterochronic modes can be represented as special cases of the general transformation in Equation 5. They differ in which component of the developmental schedule changes and, in some cases, in the relation between somatic development and reproductive maturation. Table 1 compares their formal representations, burden interpretations, and illustrative examples.

If development is slowed, as in neoteny,

In Equation 11, is the neotenic temporal-remapping function and is its developmental-pace coefficient. Because , mapped developmental time advances more slowly than chronological time. Acceleration is represented by

In Equation 12, is the acceleration remapping function and is its developmental-pace coefficient. The condition means that mapped development advances faster than the baseline schedule. Neoteny therefore stretches developmental progression across a longer interval, whereas acceleration compresses it into a shorter interval. These remappings specify changes in developmental pace, not energetic consequences by themselves. Lower instantaneous burden under neoteny and higher instantaneous burden under acceleration are conditional expectations that follow only when the relevant state scaling, expenditure relationships, and conversion efficiencies are otherwise comparable. They must therefore be evaluated empirically rather than treated as mathematical consequences of Equations 11, 12.

Postdisplacement and predisplacement can be represented as shifts in developmental onset:

In Equation 13a, is the delayed-onset remapping function and is the positive amount of delay. The condition prevents mapped developmental time from becoming negative. Predisplacement, which advances developmental onset, is written as

In Equation 13b, is the advanced-onset remapping function, and the same positive measures how far onset is shifted earlier. Biologically, Equations 13a and 13b change when developmental burden begins. Delayed onset shifts developmental commitment to a later phase and may leave greater early capacity for other processes, whereas advanced onset imposes commitment earlier and may increase early competition among subsystems. These timing shifts do not by themselves demonstrate transfer of a conserved resource. Equivalently, a non-negative developmental-time domain may be enforced by the zero-bounded form .

Changes in developmental endpoint may be represented separately as

where denotes the developmental endpoint of the baseline trajectory and denotes the developmental endpoint of the transformed trajectory. Unlike postdisplacement and predisplacement, which modify when development begins, hypermorphosis and progenesis modify when development ends. Hypermorphosis extends the developmental interval, whereas progenesis truncates it. In classical biological usage, progenesis is especially associated with earlier reproductive maturity and consequent truncation of somatic development, whereas hypermorphosis prolongs the developmental interval.

The formalism also permits explicit subsystem prioritization. A hindlimb/axial-trunk-priority developmental pattern may be represented as a special case of Equation 5, combining subsystem redistribution with an explicit temporal remapping:

Here, is the transformed developmental-state vector and is the comparative baseline state. In Equation 15, denotes a case-specific hindlimb/axial-trunk-priority matrix. The subscript stands for Hindlimb, Forelimb, Axial/trunk, and Brain-related subsystems. The four lowercase coefficients , , , and are dimensionless scalar weights assigned to these subsystems, respectively. The identity temporal-remapping term isolates subsystem redistribution from changes in developmental traversal rate, while the inequalities and specify stronger developmental emphasis on hindlimb and axial/trunk support relative to forelimb-related development. This formulation does not require brain-related developmental burden to be reduced. Such a case represents a possible prioritization schedule rather than a direct empirical measurement of developmental burden. Here, prioritization means altered relative weighting or developmental emphasis; it does not, without additional evidence, establish transfer of a conserved energetic resource or a trade-off between subsystems. Thus, the same formal structure may describe scheduling alone in one system and resource-linked burden redistribution in another.

This unified representation treats neoteny, acceleration, displacement, hypermorphosis, and progenesis as distinct organizational redistributions involving developmental pace, onset, endpoint, subsystem weighting, or reproductive timing. Progenesis is especially informative: reproductive maturity is attained earlier relative to somatic development and is represented in the model as an earlier developmental endpoint. This temporal reorganization can redistribute developmental burden without necessarily involving the transfer of energetic or material resources. The energetic consequences of any heterochronic mode remain conditional and must be evaluated empirically.

5 Illustrative biological applications

The examples in Table 1 are illustrative rather than exhaustive or confirmatory. They were selected to show how paedomorphosis, peramorphosis, onset shifts, and subsystem-specific prioritization can be represented within a common comparative framework while retaining their biological differences.

5.1 Alpine newt: neoteny versus progenesis

A particularly useful class of examples comes from amphibian paedomorphosis, where neoteny and progenesis have been explicitly distinguished as alternative developmental routes to similar adult outcomes (Denoël and Joly, 2000). In the Alpine newt (Triturus alpestris, also referred to as Ichthyosaura alpestris), these two processes were compared directly, showing that paedomorphic morphology may arise either through slowed somatic and metamorphic development or through earlier sexual maturation with truncation of somatic ontogeny. In the present framework, this distinction is especially important because similar adult morphology need not correspond to the same developmental-burden schedule.

For this system, a schematic developmental state vector may be written as

Here, denotes the Alpine-newt developmental-state vector. Its components are for somatic developmental mass, for gonadal or reproductive developmental mass, and for metamorphic-remodeling mass. Biologically, Equation 16 is a three-part inventory of somatic, reproductive, and metamorphic development. A neotenic transformation of this state can be written as

In Equation 17, is the neotenic trajectory, is its diagonal redistribution matrix, and is the comparative baseline trajectory. The two matrix entries equal to 1 preserve the relative somatic and reproductive weights, whereas reduces the relative contribution of metamorphic remodeling. The condition slows traversal of the baseline developmental schedule. Biologically, neoteny therefore combines slower progression with reduced relative remodeling. Although metamorphic transformation burden may decrease, cumulative burden need not decrease because prolonged ontogeny continues to require somatic and reproductive maintenance.

By contrast, a progenetic trajectory may be written as

Here, and denote the baseline and progenetic times of reproductive maturity, respectively, so that . These quantities are distinct from and , which denote the endpoints of the full modeled developmental interval. In Equation 18, is the progenetic trajectory, is its diagonal redistribution matrix, and is the comparative baseline. The condition increases relative reproductive weighting, whereas reduces relative metamorphic-remodeling weighting. The identity map leaves the overall traversal rate unchanged, while truncates the developmental interval. Biologically, progenesis represents reproductive maturity attained earlier relative to somatic development; this is expressed by and the earlier endpoint. It therefore differs from neoteny, which slows somatic and metamorphic progression. The two modes may produce similar paedomorphic adults while generating different developmental-burden schedules. The axolotl remains a familiar comparative example of neotenic paedomorphosis (Wakahara, 1996).

5.2 Hominin bipedalism as an illustrative hindlimb/axial-trunk priority case

The hominin case is included as a hypothesis-generating example, not as a confirmatory test. The fossil record is compatible with the possibility that hindlimb, pelvic, and axial/postural reorganization preceded the strongest phase of encephalization, and independent analysis supports later large-scale increases in hominin brain size (Püschel et al., 2024). Evidence for bipedal adaptations in Sahelanthropus tchadensis remains contested (Macchiarelli et al., 2020; Cazenave et al., 2025; Williams et al., 2026). Accordingly, Sahelanthropus is treated only as a debated early data point. The staged pattern modeled below is a formal interpretation of comparative ordering, not a claim that fossil measurements directly quantify developmental burden.

The specimen-level fossil measurements used to illustrate this case are summarized in Supplementary Data Sheet 1. Measurement codes and operational variables are defined in Supplementary Tables S1, S2, published values and derived ratios are given in Supplementary Tables S3, S4, and detailed Burtele foot measurements are provided in Supplementary Table S5. The corresponding source-value and calculation audit trail is provided in Supplementary Data Sheet 2. These materials provide a published-measurement basis for comparison while keeping the main text focused on the theoretical framework; they are not treated as an independent test of Equation 20 or as direct proof of heterochrony.

For this case, the developmental state is represented as

where denotes the developmental state of the hominin case at time , hindlimb-related developmental mass, forelimb-related developmental mass, axial/trunk-related developmental mass, and brain-related developmental mass. Here is restricted to axial and trunk-support structures relevant to posture, body orientation, and pelvic–trunk stabilization, rather than to undifferentiated body mass.

To represent this hypothesis, namely that hindlimb and axial/trunk support may have been relatively emphasized earlier whereas neural development became more strongly emphasized later, the heterochronic trajectory is written as

where denotes the comparative baseline developmental trajectory, the time-dependent redistribution matrix, and the temporal remapping function.

Equation 20 decomposes the illustrative hominin trajectory into two components: a change in relative developmental priority across subsystems and a change in the schedule on which the comparative baseline is traversed. It does not, by itself, establish the proposed evolutionary sequence or its energetic mechanism.

The redistribution matrix is defined piecewise as

where denotes the switch time separating an earlier and a later developmental phase, the relative weighting assigned to hindlimb-related development during the early phase, the relative weighting assigned to forelimb-related development, the relative weighting assigned to axial/trunk support development, the early-phase weighting of brain-related development, and the later-phase weighting of brain-related development.

Thus, in the early phase , hindlimb-related and axial/trunk-related development are up-weighted relative to forelimb-related development through and . The early brain-related coefficient is left flexible rather than forced below , so the formulation does not assume that brain-related developmental burden is reduced. In the later phase , brain-related development becomes relatively more emphasized through . The distinction between and is intentional: the former denotes the early-phase weighting of brain-related development, whereas the latter denotes the increased weighting assigned afterward.

The entries equal to 1 in the later-phase matrix are a normalization convention, not arbitrary empirical constants. All subsystem weights in this illustrative formulation are expressed relative to the comparative baseline. Thus, after the switch, the hindlimb-, forelimb-, and axial/trunk-related components are returned to baseline/reference weighting, meaning that they are no longer given additional relative up-weighting beyond the baseline schedule. This does not imply that locomotor or postural development ceases. The later-phase matrix is written in this normalized form to isolate the relative increase in brain-related weighting; a different common baseline constant would be mathematically equivalent after rescaling, so only the relative contrasts are interpreted biologically.

The temporal remapping function is written as

In Equation 22, maps chronological time t to developmental time, is the switch time, and is the post-switch developmental-pace coefficient. The condition means that developmental time advances more slowly after the switch. Biologically, the mapping leaves the earlier schedule unchanged and prolongs the later developmental phase without creating a discontinuous jump in developmental time.

Accordingly, before the switch time, development follows the baseline schedule t. After the switch time, development continues at a slower pace. The form chosen in Equation 22 preserves continuity at , so that the schedule changes smoothly rather than abruptly.

Equations 19-22 therefore define a schematic two-phase hypothesis rather than an empirical reconstruction. The early phase assigns greater relative weight to hindlimb, pelvic, and axial/trunk support; the later phase increases brain-related weighting and permits a more prolonged schedule. In concise terms, the model represents earlier postural-locomotor organization followed by later and more prolonged neural expansion. Supplementary Note S3 states the corresponding fossil-sequence expectations.

Although Equations 21, 22 use one switch time for simplicity, the model is not restricted to a single developmental change. Development can instead be divided into phases. Let and denote the beginning and end of the modeled developmental interval, and let denote the internal transition times separating successive phases, so that . Each phase can be assigned its own redistribution matrix:

Here, specifies the relative weighting of the developmental subsystems during phase . Equation 22a therefore allows subsystem priorities to change more than once during development. The weighting is held constant within each phase but may differ between successive phases. The parenthesized superscript in is a phase label, not an exponent. Developmental tempo may also differ among phases:

Here, describes developmental tempo during phase . Values below 1 indicate slower progression relative to the baseline, a value of 1 indicates the baseline rate, and values above 1 indicate faster progression. Because each phase begins from the value reached at the preceding boundary, developmental time remains continuous and does not jump at a transition. The original single-switch model is recovered when , which gives two phases separated by one internal transition time.

Equations 22a, 22b describe distinct phases separated by discrete boundaries. If abrupt changes are biologically unrealistic, the redistribution matrices can instead change gradually:

where

In Equation 22c, is the initial redistribution matrix, and is the change in subsystem weighting associated with transition k. The logistic transition function gradually activates that change around . In Equation 22d, controls how rapidly the transition occurs: smaller values produce a gradual change, whereas larger values approach a discrete switch. The multiple-transition formulation therefore allows developmental reorganization to occur in several stages or through gradual changes. The single-switch model remains a minimal illustrative case, not a claim that hominin developmental reorganization occurred at one precisely defined or deterministic moment.

Any empirical calibration must also distinguish ontogenetic developmental time from evolutionary or phylogenetic time. The single-switch parameter in Equations 21, 22, and the multiple-transition parameters in Equations 22a–22d, are phenomenological breakpoints in comparative developmental schedules, not fossil ages. Fossil morphology and stratigraphic information can constrain the ordering and plausible ranges of postural-locomotor and neural changes, but they cannot directly estimate ontogenetic transition times without sufficiently dense juvenile growth series. Accordingly, the fossil measurements in the Supplementary Material are used as comparative constraints rather than for direct calibration of these transition parameters. Future estimation would require integration of fossil morphology and chronology with phylogenetic reconstruction and ontogenetic data from extant comparative taxa.

Evidence for prolonged human neural development, including transcriptional neoteny and extended neocortical myelination (Somel et al., 2009; Miller et al., 2012), is compatible with the later, prolonged neural phase of this hypothesis but does not demonstrate the preceding postural-locomotor redistribution. Species-specific neuronal tempo may also be linked to mitochondrial metabolism (Iwata et al., 2023). In burden terms, delayed neural maturation may reduce instantaneous burden while extending cumulative commitment, illustrating why the two quantities must be distinguished.

5.3 Hindlimb-priority redistribution in Psittacosaurus and limb timing in bats

Hindlimb-priority redistribution may also be illustrated comparatively outside hominins. In Psittacosaurus, hindlimb-dominant relative growth has been associated with an ontogenetic shift toward bipedality (Zhao et al., 2013), and such a case may be represented formally by a redistribution matrix in which one subsystem receives stronger developmental emphasis than others. A schematic state vector may be written as

Here, denotes the Psittacosaurus developmental-state vector. Its components are for hindlimb developmental mass, for forelimb developmental mass, and for axial or trunk-support developmental mass. Biologically, Equation 23 is a three-component inventory used to compare how strongly these subsystems are emphasized. The simplest transformed form is

In Equation 24, is the transformed Psittacosaurus trajectory, is its diagonal redistribution matrix, and is the comparative baseline trajectory. The coefficients , , and are the relative weights of hindlimb, forelimb, and axial/trunk development. The condition expresses stronger hindlimb emphasis relative to the forelimb. Biologically, Equation 24 represents a hindlimb-priority growth pattern while keeping axial/trunk support explicit; the direction of the axial/trunk effect should be estimated from data rather than assumed. In burden terms, Equation 6 would then contain a larger early hindlimb contribution than the baseline case.

Bat limb development provides a living example of mosaic, subsystem-specific heterochrony rather than a single global change in developmental rate. Nojiri et al. (2023) found that limb-ridge onset was advanced, whereas later aspects of forelimb development were prolonged or delayed: the period of forelimb organogenesis was extended, and the appearance of the fingers and claws was delayed. Hindlimb development was especially advanced relative to forelimb development, a pattern associated with the need for neonates to cling soon after birth. Thus, the bat example combines onset shifts and changes in developmental duration and should not be assigned to a single organism-wide heterochronic mode (Nojiri et al., 2023; Zhu and Tabin, 2023). This example directly illustrates why the present framework treats heterochrony as subsystem-specific rather than as a single organism-wide clock. The corresponding placement is shown explicitly in Table 1.

A natural extension would assign distinct temporal-remapping functions to different subsystems, for example separate clocks for forelimb and hindlimb development. Even without that extension, the present model captures the possibility that heterochrony changes relative subsystem priority rather than a single organism-wide rate. Such priority shifts should first be interpreted as changes in scheduling; a resource-linked interpretation requires independent evidence of altered allocation, metabolic covariation, or reciprocal trade-offs.

5.4 Onset shifts and limiting peramorphic cases

Predisplacement and postdisplacement are limiting cases of temporal relocation: they alter when a developmental program begins without necessarily changing the magnitude of its burden. Reviews of tetrapod limb evolution show that onset shifts recur in limb development (Smith, 2001; Zhu and Tabin, 2023). Within the present framework, advanced or delayed onset changes the phase at which a subsystem begins to incur developmental commitment. Whether that change generates resource competition or transfer is an empirical question, not a consequence of the timing shift itself.

Classical peramorphic examples remain useful as complementary limiting cases. In McNamara’s nomenclatural treatment, the Irish elk, Megaloceros, is cited as a classic example of hypermorphosis, illustrating how extension of the developmental endpoint may generate exaggerated adult morphology (McNamara, 1986). Spatangoid echinoids provide a second comparative example of prolonged developmental elaboration in which heterochrony has been treated explicitly in relation to morphological evolution (McNamara, 1989). Within the present framework, such cases are not invoked as detailed energetic reconstructions, but as conceptually useful illustrations of schedules in which cumulative burden may increase if positive burden is sustained across the extended interval. Conversely, examples of accelerated developmental subprograms, such as accelerated spinal cord development in the direct-developing frog Eleutherodactylus coqui and transcriptional heterochrony in blowflies (Schlosser, 2003; Skaer et al., 2002), represent the opposite temporal limit: developmental commitment is compressed into a shorter interval, while any increase in instantaneous burden remains a conditional empirical expectation rather than a necessary consequence of acceleration. Bolitoglossa occidentalis remains a classical salamander example of progenetic paedomorphosis in which heterochronic mechanisms of morphological diversification were analyzed in detail (Alberch and Alberch, 1981).

Taken together, these examples demonstrate the scope of the formalism rather than validate it. The framework provides a common language for comparing paedomorphosis, peramorphosis, onset shifts, subsystem-specific limb timing, and staged postural-locomotor and neural development. Confirmation in any particular system requires operational variables and evidence appropriate to that system.

6 Discussion

6.1 Conceptual contribution of the framework

The principal contribution of this study is to supplement the classical descriptive vocabulary of heterochrony with a compact comparative formalism. Neoteny, acceleration, displacement, hypermorphosis, and progenesis retain their established biological meanings but can also be represented as transformations of a shared developmental baseline. The framework thereby makes their common structure explicit without treating them as mechanistically identical (Alberch et al., 1979; McNamara, 1986, 2012).

The framework offers three analytical advantages. First, it represents development as a structured state of interacting subsystems rather than as a single trajectory. Second, it separates changes in subsystem weighting from changes in developmental time. Third, it distinguishes instantaneous from cumulative burden. This last distinction allows prolonged, compressed, and truncated schedules to be compared without assuming that their energetic consequences follow directly from timing alone.

6.2 Neoteny, progenesis, and developmental burden scheduling

The contrast between neoteny and progenesis illustrates the organizational meaning of redistribution. Both may produce paedomorphic adults, but neoteny slows somatic progression, whereas in progenesis reproductive maturity is attained earlier relative to somatic development. The present model represents the progenetic relation as an earlier developmental endpoint, which reorganizes the temporal distribution of developmental burden without necessarily implying resource transfer. Similar adult morphologies therefore need not reflect the same developmental transformation or burden schedule (Denoël and Joly, 2000; McNamara, 1986, 2012).

Heterochrony also need not operate as an organism-wide change in developmental tempo. By allowing subsystem-specific weighting, the framework accommodates cases in which neural, appendicular, axial/trunk, metabolic, or reproductive programs change in relative emphasis. Timing can therefore encode developmental priority even when direct evidence of energetic reallocation is unavailable.

Organizational redistribution and a resource-linked interpretation of redistribution are related but evidentially distinct hypotheses. The former requires evidence for altered ordering, weighting, coupling, onset, rate, duration, or termination of developmental programs. The latter additionally requires metabolic covariation, a bounded developmental budget, reciprocal delay, or another independent indicator of allocation or trade-off. Resource-linked redistribution is therefore a stronger, system-specific claim rather than the default meaning of redistribution. Conversely, the absence of resource transfer does not exclude redistribution in the organizational sense defined here.

6.3 Human bipedalism, developmental priority, and evo-devo interpretation

The hominin application remains suggestive rather than demonstrative. Comparative evidence is compatible with postural-locomotor reorganization preceding the strongest phase of encephalization, and analysis of hominin brain-size tempo supports later large-scale encephalization (Püschel et al., 2024). However, fossils do not directly recover developmental schedules, and the interpretation of Sahelanthropus tchadensis remains disputed (Macchiarelli et al., 2020; Cazenave et al., 2025; Williams et al., 2026). The model therefore treats earlier hindlimb, pelvic, and axial/trunk priority followed by later neural expansion as a testable scenario, not a reconstructed fact. Evidence for prolonged human neural development supports the plausibility of the later phase but does not establish the earlier one (Somel et al., 2009; Miller et al., 2012).

The framework also permits a causally explicit and non-deterministic formulation of bipedal evolution. In evolutionary biology, functional or teleological language can be legitimate shorthand for traits maintained by natural selection, but it does not imply intentional or future-directed causation (Ayala, 1970). Moreover, the current utility of a trait does not by itself establish the historical process responsible for its origin (Gould and Vrba, 1982). Accordingly, proposed advantages such as improved vigilance, freeing of the hands, and reduced solar heat load are treated here as competing hypotheses concerning possible selective benefits or later consequences of upright posture, rather than as predetermined developmental goals or established single causes of bipedal origins.

This caution is consistent with evidence specific to bipedal evolution. The conventional claim that terrestrial bipedalism immediately “freed the hands” may exaggerate the behavioral transition (Richards, 1986), whereas heat-balance simulations suggest that the thermoregulatory advantage of upright posture was too limited to be regarded as a sufficient first cause (Chaplin et al., 1994). Evidence that arboreal bipedalism enables access to flexible supports (Thorpe et al., 2007), together with field evidence that a savanna-mosaic habitat did not increase terrestrial bipedalism in wild chimpanzees (Drummond-Clarke et al., 2022), also challenges simple open-country explanations based on surveillance or reduced solar exposure. These proposed benefits may therefore have reinforced, modified, or been co-opted during the later evolution of bipedalism, but their causal roles must be evaluated independently rather than inferred solely from their present-day usefulness.

Within the present framework, an arboreal-orthograde scenario is therefore represented as a testable hypothesis of changing functional and developmental weighting: forelimbs become increasingly specialized for grasping and positional control, while hindlimb-pelvic and axial/trunk systems assume greater roles in support, propulsion, and body orientation. This formulation specifies a comparative developmental scenario; it does not establish the ancestral locomotor state, imply resource transfer, or assume that any anticipated benefit acted as a predetermined developmental goal.

This scenario is broadly consistent with biomechanical reviews that place the emergence of upright walking in an arboreal context, and with pelvic and femoral evidence from Ardipithecus ramidus indicating a combination of bipedal and arboreal locomotion (Lovejoy et al., 2009; Crompton et al., 2023). Nevertheless, the locomotor repertoire of the Pan-Homo last common ancestor remains unresolved because no universally accepted fossil representative exists and the earliest hominin record is fragmentary (Tocheri et al., 2008; Stamos and Alemseged, 2023). The scarcity and uneven preservation of associated pre-adult remains further limit the reconstruction of developmental schedules in fossil hominins (Bolter et al., 2025). Preservation bias, particularly in tropical forest settings, further limits inference from absence (Kidwell and Flessa, 1995; Scerri et al., 2022). The framework can therefore compare alternative scenarios, but it cannot adjudicate the ancestral state from the fossil record alone.

For that reason, the value of a mathematical treatment of heterochrony lies not in directly proving an arboreal common ancestor, but in providing a coherent comparative framework for evaluating alternative evolutionary scenarios when direct ancestral evidence is fragmentary. By integrating extant developmental data, fossil morphology, and phylogenetic reconstruction, the present formalism makes it possible to formulate and compare the hypothesis that the human lineage involved hindlimb- and axial/trunk-prioritized allometric heterochrony emerging from an arboreal-orthograde background, rather than a de novo departure from an already terrestrial quadrupedal precursor. In this sense, the formalism is justified by conceptual clarity and by methodological necessity, but its paleoanthropological application remains hypothesis-generating. Recent developmental work on the human ilium is especially relevant here, because it links pelvic form to heterochronic and heterotopic reorganization associated with early bipedal adaptation, thereby reinforcing the usefulness of a developmental reorganization framework (Senevirathne et al., 2025). Importantly, the two-step model of Senevirathne et al. (2025) is not assumed here to correspond directly to the phenomenological switch time in Equation 21. Rather, it is cited as independent developmental evidence that bipedal pelvic form involved heterochronic and heterotopic reorganization, whereas is a schematic modeling device for representing a change in subsystem priority.

The quantitative basis for this illustrative human-bipedality case is provided in Supplementary Data Sheet 1. That supplement does not introduce new fossil measurements; rather, it assembles published measurements and derived ratios for Ardipithecus ramidus (ARA-VP-6/500), the Burtele foot (BRT-VP-2/73), Australopithecus afarensis specimens including AL 288-1, DIK-1-1f, and KSD-VP-1/1, Little Foot (StW 573), and the Nariokotome Homo erectus skeleton (KNM-WT 15000). In the terminology of the present model, these published values provide an illustrative fossil-measurement basis that is compatible with a hindlimb/axial-trunk priority interpretation. They do not independently prove the proposed hindlimb/axial-trunk prioritization, directly demonstrate heterochrony in extinct hominins, or show that heterochrony alone caused bipedalism. Data provenance is described in Supplementary Note S1, specimen-by-specimen interpretation in Supplementary Note S2, and fossil-specific limitations in Supplementary Note S4.

6.4 Limitations, empirical testability, and concluding remarks

The framework is intentionally phenomenological rather than mechanistically predictive. It does not derive subsystem masses or energetic expenditure rates from first principles, and the illustrative decomposition into hindlimb, forelimb, axial/trunk, and brain-related components would need to be redefined for other systems. The efficiency matrix and burden index are comparative constructs, and Equation 4 is meaningful only where organized developmental mass is positive. The formalism can organize and compare hypotheses, but prediction requires system-specific parameterization, independent data, and prospective validation. The hominin application is correspondingly hypothesis-generating because the available fossil measurements do not reconstruct complete ontogenetic burden trajectories.

Formal model comparison becomes appropriate only after empirical parameterization. A natural null model, denoted , sets and , representing neither subsystem redistribution nor temporal remapping. Biologically, this null model leaves both subsystem organization and developmental timing unchanged; therefore, the modeled trajectory is identical to the comparative baseline. Here, I is the identity matrix, which leaves all subsystem weights unchanged. Alternative models may allow temporal remapping alone, subsystem redistribution alone, or both processes together, and may subsequently incorporate one or more transition times or non-diagonal coupling terms. Because the present study specifies neither a likelihood function nor parameters estimated from observations, the Akaike information criterion (AIC) cannot be calculated for the current illustrative equations.

Once candidate models have been fitted to the same response data under a common likelihood, AIC, or the small-sample correction AICc, could compare them by balancing goodness of fit against the number of estimated parameters. Overfitting should additionally be controlled by limiting or regularizing switch points and off-diagonal coefficients, evaluating predictive performance through cross-validation or independent data, and reporting uncertainty in estimated transitions. These procedures are requirements for future empirical applications rather than analyses claimed in the present theoretical study.

These limitations do not preclude empirical evaluation. Once the state variables are operationalized, the framework yields comparative expectations testable with growth curves, organ-size measurements, ossification sequences, histological maturation indices, metabolic proxies, or transcriptomic timing. Under otherwise comparable energetic scaling, neoteny may reduce instantaneous burden while prolonging cumulative commitment; hypermorphosis may sustain burden over an extended interval; and in progenetic cases, reproductive maturity should be attained earlier relative to somatic development and be represented in the model by an earlier developmental endpoint. Subsystem-prioritized cases should first be tested as asymmetric scheduling patterns. An energetic or resource-linked interpretation requires additional metabolic or trade-off evidence. These are comparative hypotheses, not parameter-free numerical predictions.

More broadly, the framework treats developmental timing as an organizational variable whose feasible trajectories may be constrained by energetic conditions. Metabolic evidence can therefore be used to evaluate resource-linked interpretations, but timing alone does not establish them. This layered interpretation is consistent with recent work linking developmental tempo to metabolic state and biochemical rate regulation across species (Diaz-Cuadros et al., 2023; Diaz-Cuadros and Pourquié, 2023).

In conclusion, the framework preserves the classical categories of heterochrony while representing them as structured changes in subsystem organization and developmental time. What is redistributed is developmental burden across time and subsystems. The diagonal and single-switch formulations are minimal illustrative cases; empirical applications may require biologically admissible cross-subsystem coupling and multiple or smoothly varying developmental transitions. Such redistribution does not necessarily presuppose a conserved resource, nor does it necessarily entail the transfer of energetic or material resources. Progenesis demonstrates this distinction: reproductive maturity is attained earlier relative to somatic development and is represented in the model as an earlier developmental endpoint. The empirical task is therefore sequential: first identify the organizational transformation, and then test whether it is associated with energetic allocation or trade-offs in the biological system under study.

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