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Unraveling phenotypic variation in multiscale behavioral dynamics

Gautam SridhariD, Claire WyartiD, Antonio Carlos CostaiD

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Abstract

Behavioral variability reflects differences in internal and external parameters and provides a substrate for natural selection. However, uncovering its structure is challenging because behavior evolves nonlinearly across multiple timescales. Our goal is to reconstruct phenotypic variation from finite behavioral observations without prescribing the scales at which individuals should be compared. To achieve this, we introduce a multiscale dissimilarity that compares dynamics across all statistically resolvable resolutions by constructing reduced transfer operators at progressively finer scales and weighting their differences by finite-sampling uncertainty. This dissimilarity defines a geometry of dynamical phenotypes, which we reconstruct using diffusion maps. We show that the resulting phenotypic coordinates locally follow the parameter directions to which dynamics are most sensitive. We validate our approach in a stochastic model and apply it to bacterial and larval zebrafish behavior. In bacteria, we identify run speed as a major component of inter-individual variability, associated with the chemotaxis protein CheB. In zebrafish, we find that prior prey experience reshapes behavioral dynamics and phenotypic variability, with paramecia experience concentrating fish in a prey-capture-associated region of phenotypic space. Together, these results provide a framework for identifying internal and environmental variables that structure dynamical phenotypes across scales.

In-depth summary

Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (11 Oct 2026) · CC BY 4.0

Comparing coarse-grained dynamics across resolutions, weighted by sampling uncertainty, yields phenotypic spaces linking E. coli run speed to CheB and showing paramecia experience narrows zebrafish variability.

Behavior and other dynamic phenotypes play out over many timescales, so time-averaged statistics or one fixed coarse-graining (such as the q = 7 representation used before for zebrafish) can miss where individuals actually differ. Here the data decide which resolutions carry differences that can be resolved, and the resulting diffusion-map axes are tied to the parameter directions that the dynamics are most sensitive to.

Key results

  1. ShownIn E. coli, the largest dynamical differences between cells appear at the scale that separates fast runs from slow runs. Run speed, and not only tumble bias, is therefore a major axis of phenotypic variation.

    Dissimilarity peaks near q = 3 and can no longer be resolved by q ≈20. Regressing v1, v2 on speed and tumble bias gives R^2 = 0.555 (0.546, 0.567; bootstrapped 95% CI). Run speed predicts a cell's position better than tumble bias does. Data: 5399 RP437 tracks and 1374 fluorescent-cell tracks. · From gene expression to behavioral variation in…; Bacterial (Escherichia coli) navigation dataset; Figure legends

  2. ShownRaising larval zebrafish with paramecia narrows how widely individuals spread in phenotypic space. Experienced fish cluster in the region linked to efficient frontal prey capture.

    Entropy difference (raised-with minus naive) is −1 (−1.5, −0.5) bits, p-value < 0.0005, comparing n = 65 naive and n = 83 raised-with fish. The result holds for kernel bandwidths from 0.05 to 0.3 (p-value < 0.001). The rotifer comparison (n = 46 fish) was not consistently significant across bandwidths. · Larval zebrafish navigation: multiscale…; Assessing variability through phenotypic space…; Figure legends

  3. ShownThe method resolves zebrafish differences up to q ≈20, well past the earlier fixed q = 7 representation. At that fine scale, paramecia-experienced fish spend more time in a state made of Approach Swims and J-turns, typical of prey capture.

    At q = 20, raised-with fish use state qi = 10 more, and its geometric mean dwell rises from 1.5 (1.45, 1.55) to 1.64 (1.58, 1.69) bouts (p-value < 0.0005). Naive fish favor qi = 8 (routine, high-angle and shadow-avoidance turns). Rotifer-raised fish instead visit qi = 10 less. · Larval zebrafish navigation: multiscale…; Figure legends

  4. SuggestedAmong mutant cells, phenotypic position tracks CheB more than CheR, and run speed also varies with CheB. In simulated search, two mutant-rich corners of the simplex-like space specialize in short or long range, while the group richest in wild-type cells performs well across length scales.

    v1 and v2 both vary with CheB; CheR effects appear mainly at high CheB (no coefficients are given). G1 (75.9% wild-type) is near average at short range, best near the arena scale and still near optimal at large scales. G3 (19.3% wild-type, low CheB) is best at short range. G2 (21.4% wild-type) excels only beyond the ≈ 10^3µm arena scale. · From gene expression to behavioral variation in…; Symbolic sequence and trajectory simulations for…; Resource detection estimation from simulated…; Figure legends

  5. ShownIn a simulated triple-well system with four phenotypes, the dissimilarity finds the resolution at which groups diverge and discounts fine-scale structure that finite data cannot resolve. Diffusion maps then recover the imposed two-dimensional parameter plane.

    Blue and green separate at q = 2, while blue and orange need q = 3. Beyond q = 3, εq rises toward Dq. The diffusion map shows two dominant non-trivial modes that separate all four groups (25, 29, 24 and 28 realizations; Texpt = 10^4s). · Multiscale total variation dissimilarity: a…; Overdamped Langevin dynamics from a 3 well…; Figure legends

Methods, data and reproducibility

Real-data trajectories are time-delay embedded (the toy model uses its full state space) and k-means partitioned into ensemble transition matrices, then coarse-grained into q-state Markov models via reversibilized eigenvectors. Per-q excesses of total-variation distance over a resampled null uncertainty are summed and embedded with self-tuning, density-normalized diffusion maps.

  • Simulated overdamped Langevin dynamics in a triple-well potential; four phenotypes vary the well depths within a 2D parameter plane · n = 25, 29, 24 and 28 realizations per phenotype
  • Motility tracks of isogenic wild-type RP437 cells from the Dufour et al. dataset, recorded at 10Hz in quasi-2D confinement · Escherichia coli · n = 5399 tracks · Kaggle 10.34740/kaggle/dsv/20089164
  • Fluorescent strains (CheB-mYFP, mCherry-CheR, inducible, native cheR/cheB deleted) with single-cell protein abundance measurements · Escherichia coli · n = 1374 tracks · Kaggle 10.34740/kaggle/dsv/20089164
  • Larval zebrafish bout sequences (wild-type Tubingen, 6-7 days post fertilization), tail tracked at 700Hz across 14 sensory contexts · zebrafish (Danio rerio) · n = 463 fish · Kaggle 10.34740/kaggle/dsv/20089164
  • Zebrafish hunting paramecia: naive fish vs fish raised with paramecia · zebrafish (Danio rerio) · n = 65 naive, 83 raised with (148 fish)
  • Zebrafish hunting rotifers: naive fish vs fish raised with rotifers · zebrafish (Danio rerio) · n = 46 fish across two conditions
Methods & tools
Maximally predictive Markov models (time-delay embedding, k-means microstates), Perron-Frobenius transfer operator approximation, Multiscale total-variation dissimilarity with permutation-based uncertainty, Self-tuning, density-normalized diffusion maps, Weighted fuzzy c-means clustering, Principal component analysis, PCHIP spline potentials for overdamped Langevin simulation, 2D Gaussian kernel density estimate with bootstrap and permutation tests, Single-cell fluorescence microscopy, Custom MATLAB tracking code, Custom online tail-tracking software
Reproducibility gaps
No software versions, environment files or compute details are reported. The raw E. coli and zebrafish recordings come from earlier studies, and tracks were extracted with custom MATLAB code and custom tracking software.
Limitations (6)
  • Estimating transfer operators assumes the dynamics are roughly stationary within each recording (an adiabatic approximation). The authors note this often does not hold, and longer recordings might be better treated as trajectories through phenotypic space. · stated by the authors
  • Each cell or fish contributes a single finite-time recording, so the space captures variation among realizations. Showing that positions reflect stable individual differences would require repeated recordings of the same individuals. · stated by the authors
  • The links between CheB abundance and run speed, and other protein-behavior associations, are correlative. The authors state that targeted perturbations are needed to establish causality. · stated by the authors
  • Prey-specific effects in zebrafish need more replicates and controlled manipulation of prey exposure. For rotifer-hunting fish (n = 46), the entropy difference was not consistently significant across bandwidths, possibly because of the small sample. · stated by the authors
  • The method assumes individuals share a common state space, so cross-species comparisons may need dynamical similarity analysis instead. Non-reversible, cyclic dynamics may suit Generalized Perron Cluster-Cluster Analysis better than the simple eigenvector clustering used here. · stated by the authors
  • Zebrafish prey and non-prey assays used different pixel sizes (27 vs 58 µm/pixel) and arena sizes. The prey-exposure axis along v1 may therefore partly reflect acquisition differences, and no control for this is described. · noted in this summary

Read this if you compare behavioral or other stochastic dynamics across individuals and want the data to choose the timescales of comparison, giving interpretable, low-dimensional phenotypic spaces.

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