How can phenotypic variation be reconstructed from multiscale behavioral dynamics?

Averaging behavior hides the scales where individuals differ

Behavioral variation is often treated as noise to be averaged away, or summarized with time-averaged quantities such as mean speed or the frequency of discrete behaviors. The paper argues that this discards biological signal: two individuals can share an overall exploration-versus-exploitation tendency yet differ in the finer movements they use throughout. Because behavior evolves nonlinearly across multiple timescales, comparing dynamical phenotypes requires a representation that resolves differences across scales without prescribing in advance which scales matter.

The authors' goal is therefore to reconstruct the structure of behavioral variation from finite observations, identify the dynamical scales at which individuals actually differ, and relate that variation to the internal and environmental variables shaping it. The constraint that makes this hard is finite sampling: at sufficiently fine resolutions, observed differences become indistinguishable from the fluctuations expected between identical dynamics.

Reduced transfer operators turn trajectories into comparable dynamics

The method starts from a fine-scale Markov model among N short-time movements, estimated from an individual's trajectory. It then identifies q metastable states and builds reduced q-state models, with increasing q capturing faster dynamics. Each individual trajectory is projected onto this family of Markov models, so that at every resolution q the pair of individuals can be compared through their coarse-grained transition structure.

At each resolution the dynamical difference between two individuals, D_q, is measured and compared with a finite-sampling uncertainty scale, epsilon_q. Only the excess of D_q over epsilon_q is summed across all statistically resolvable scales, producing a single multiscale dissimilarity D. This is the key design choice: the data, not the analyst, determine which resolutions contribute. As q grows, sampling uncertainty eventually makes finer differences unresolvable, and those scales drop out automatically.

The resulting pairwise dissimilarities define a nonlinear geometry of dynamical phenotypes, which diffusion maps reconstruct as a low-dimensional embedding. The authors show that the recovered coordinates locally align with the dominant modes of the metric tensor mapping parameter variation onto emergent dynamical differences. In other words, the directions along which the embedding separates individuals most strongly are the parameter directions to which the dynamics are most sensitive, so the geometry is a representation of how parameter variation is expressed in behavior rather than an arbitrary visualization.

A three-well model recovers a known two-dimensional parameter plane

The authors test the approach on overdamped Langevin dynamics in a triple-well potential, varying the depths of the three minima to generate four dynamical phenotypes. Parameter variation is constrained to a two-dimensional plane, so the geometry to be reconstructed is known in advance. This lets the study check three things: whether the dissimilarity identifies the resolution at which differences emerge, whether it suppresses structure that finite data cannot resolve, and whether it recovers the known low-dimensional organization.

The scale dependence appears directly in the pairwise distances. Two phenotypes are already well separated at q = 2, whereas distinguishing another pair requires the finer q = 3 description that resolves transitions between the two shallower wells. Beyond q = 3, no additional metastable structure separates the phenotypes, and finer subdivisions contribute little. Meanwhile epsilon_q increases with q and approaches D_q at sufficiently fine resolution, so the multiscale dissimilarity retains the informative resolutions and suppresses the rest. Shortening the observation time causes fine-scale differences to lose significance earlier.

Diffusion maps then recover two dominant non-trivial coordinates, matching the imposed two-dimensional parameter variation and separating all four phenotypic groups. This is a validation result, not a biological finding: it shows the construction behaves as intended when the answer is known.

Run speed is a major bacterial axis and tracks CheB abundance

For E. coli, the authors analyze a dataset combining motility trajectories from wild-type cells with strains in which the expression of the adaptation proteins CheR and CheB was controlled and quantified. CheR and CheB regulate receptor methylation, and the chemotaxis network is well characterized, yet substantial phenotypic variation persists even within isogenic populations.

In the reconstructed phenotypic space, variation in run speed emerges as a major axis distinguishing individuals, consistent with earlier work identifying run speed as an important component of non-genetic behavioral diversity. Unexpectedly, run speed itself is strongly associated with CheB abundance, a relationship the authors note is not directly predicted by the canonical chemotaxis network and whose mechanistic origin remains unclear. Phenotypic variation aligns much more strongly with CheB abundance than with CheR, consistent with the greater sensitivity to CheB reported previously.

The authors are explicit that these associations, particularly between CheB abundance and run speed, remain correlative; establishing a causal basis would require targeted perturbations. They also observe that wild-type and mutant cells occupy different regions of the space, with wild-type cells enriched where fast runs are more frequent and search performance is comparatively high across a broad range of scales. The suggestion that native CheR and CheB regulation constrains variation away from narrow specializations is offered as a possibility, not a demonstrated mechanism.

Paramecia experience concentrates fish in a capture region

In larval zebrafish, the authors analyze bout-sequence data from fish raised under different prey conditions. Prior exposure to paramecia reshapes behavioral dynamics and reduces inter-individual differences, with a decrease in phenotypic space entropy of H_paramecia minus H_naive equal to -1 (-1.5, -0.5) bits at p < 0.0005. Naive fish occupy a broader range of behavioral phenotypes, whereas paramecia-experienced fish converge toward a more stereotyped repertoire.

Significant differences persist up to roughly q = 20 states. Comparing state usage at that resolution, naive fish more frequently visit states enriched in routine turns, high-angle turns, and shadow-avoidance turns. Fish raised with paramecia instead increase time in a state enriched in Approach Swims and J-turns, the canonical prey-capture sequence, and they also increase sequence length in that state. The authors note that these fine effects are recovered without relying on eye-convergence measurements, and that at coarser scales paramecia-raised fish spend less time on exploratory behaviors.

The entropy result is a population-level statement about spread in phenotypic space, not a claim that every individual shifted in the same direction. The authors also report that for fish hunting rotifers (n = 46 across the two conditions), the entropy difference is not consistently significant across bandwidths, possibly reflecting the small sample size.

Rotifers reverse the exploratory pattern seen with paramecia

The effects are prey-specific. Fish raised with rotifers show enhanced time performing exploratory behaviors, sequences of slow forward movements and routine turns, and reduced visits to the prey-capture state that paramecia rearing enhances. They also spend more time without converged eyes, in contrast to the paramecia pattern, and the duration of sequences in that capture state is reduced.

The authors offer possible explanations rather than conclusions: differences in prey motility may let rotifers be captured more rapidly, leaving more time for exploration, and differences in nutritional value could also contribute. They state that establishing whether larvae acquire genuinely prey-specific repertoires, and identifying the mechanisms, will require additional replicates and controlled manipulations of prey exposure. This is the paper's clearest internal contrast: the same experience manipulation moves phenotypic spread and state usage in different directions depending on prey type.

Positions in the space are not yet proven individual traits

The significance testing ensures that retained differences exceed what finite sampling of identical dynamics would produce, which rules out the trivial case. It does not establish that an individual would return to the same position if recorded again. Demonstrating consistent individual differences would require repeated measurements of the same individual, the standard route in behavioral ecology.

The authors frame this as a data requirement rather than a flaw in the construction: repeated recordings would place each individual at several points in the space, allowing within-individual spread to be compared directly against between-individual spread. Following the same animals through development would show whether they occupy fixed positions, converge with experience, or move along the manifold in a stereotyped way. The reconstructed spaces are presented as a starting point that can be aligned with independent measurements, such as single-cell expression in bacteria or whole-brain imaging in zebrafish, so that an individual's position could be mapped onto internal state rather than inferred.

Let the data choose the scale, then read the sensitive directions

The paper's distinctive priority is the ordering of operations: first determine which dynamical resolutions survive finite-sampling uncertainty, then interpret the resulting embedding as directions of dynamical sensitivity. That ordering is what separates this from analyses restricted to a single coarse-graining scale, and it is what allowed the zebrafish data to resolve additional effects of prey experience within one construction.

The most decision-relevant comparison is between the two prey conditions. Paramecia rearing reduces phenotypic spread and concentrates fish in a capture-associated region, with an entropy decrease of -1 (-1.5, -0.5) bits, while rotifer rearing shifts fish toward exploration and does not show a consistently significant entropy difference at n = 46. For bacteria, the axis to watch is run speed and its association with CheB, which the authors flag as correlative. Readers weighing whether this framework applies to their own system should ask whether their recordings are long enough for fine-scale differences to remain resolvable, since shorter observation times push the resolvable scale coarser.

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