Index of Dispersion (IOD) in Nonequilibrium Systems

• IOD distributions are dimensionless ratio metrics that quantify fluorescence variance relative to the mean, clearly distinguishing equilibrium from nonequilibrium dynamics.
• They are computed using sliding-window statistics on NIR fluorescence data from SWCNTs in actomyosin networks, effectively capturing motor-driven irreversibility.
• Systematic concentration-dependent shifts in IOD, assessed via KL divergence, offer a robust indicator for the emergence of nonequilibrium behavior.

The index of dispersion (IOD) is a dimensionless statistical measure defined as the ratio of the variance to the mean of a time series, frequently applied in the analysis of fluorescence intensity fluctuations to quantify dynamical changes and deviations from equilibrium in nonequilibrium systems. In the context of near-infrared (NIR) fluorescence from single-walled carbon nanotubes (SWCNTs) embedded in actomyosin networks, IOD distributions provide a sensitive, non-invasive optical metric for probing molecular motor-driven irreversibility and nonequilibrium dynamics, as demonstrated in studies utilizing ATP-fueled myosin contractility (Hendler-Neumark et al., 5 Dec 2025).

1. Mathematical Definition and Computation

The IOD in a time window $\left[t_0, t_0+w\right]$ of width $w$ frames is given by: $$\mathrm{IOD}(t_0, w) = \frac{\mathrm{Var}\bigl[I(t_0),\dots,I(t_0 + w)\bigr]}{\langle I(t_0),\dots,I(t_0 + w)\rangle}$$ where: $$\langle I(t_0),\dots,I(t_0 + w)\rangle = \frac{1}{w+1} \sum_{k=0}^w I(t_0 + k)$$

$$\mathrm{Var} = \frac{1}{w} \sum_{k=0}^w \bigl[I(t_0 + k) - \langle I\rangle\bigr]^2$$

IOD is computed per region of interest (ROI) without detrending or high-pass filtering, using sliding-window statistics. For each ROI $i$, NIR intensity $I_i(1:150)$ is segmented, then statistics are calculated in non-overlapping windows of $w$ frames. Each window, indexed by $m$ with start $t_0 = 1 + (m-1)w$, yields:

• Mean: $$\mu_{i,m} = \frac{1}{w}\sum_{k=0}^{w-1} I_i(t_0 + k)$$
• Variance: $$\sigma^2_{i,m} = \frac{1}{w-1} \sum_{k=0}^{w-1} [I_i(t_0 + k) - \mu_{i,m}]^2$$
• IOD: $\mathrm{IOD}_{i,m} = \sigma^2_{i,m} / \mu_{i,m}$

These values are pooled over ROIs and time windows for construction of IOD distributions.

2. Experimental Context and Acquisition Protocol

SWCNT NIR emission intensities are acquired under $\Delta t \approx 300$ ms (exposure 50–200 ms), yielding precisely 150 consecutive frames per ROI (50 frames pre-intervention, 100 post-myosin addition). Segmentation of SWCNT positions is performed using MSER in Fiji, with subsequent fluorescence extraction in MATLAB. IOD values are estimated using window lengths $w=50$, $25$, or $10$ frames, depending on time-resolution requirements.

No data preprocessing (detrending or filtering) is applied, as dynamic changes are intended to be detected via windowed statistics alone. This ensures that the IOD directly reflects the underlying fluctuation structure relevant to the system's nonequilibrium properties.

3. Characteristics of IOD Distributions: Equilibrium vs. Nonequilibrium

Under equilibrium (pre-myosin addition, 25-frame window), IOD distributions feature:

• Median $\approx$ 9–10, IQR $\approx$ 6–12
• Skewness $\approx$ 1.1, excess kurtosis $\approx$ 0.5
• Unimodal, sharply peaked form (per KDE)

Following myosin addition and ATP-induced activation:

• Median IOD rises to $\approx$ 50, IQR $\approx$ 30–80
• Strong increase in skewness ($>2.5$) and excess kurtosis ($>6$)
• Emergence of a heavy right tail in the distribution; KDE exhibits pronounced shift and broadening, departing from pre-myosin profile

The degree of this transition exhibits clear concentration dependence:

• 50% myosin: median $\approx$ 30, IQR $\approx$ 18–45, moderate broadening (skewness $\approx$ 1.8)
• 1% myosin: median $\approx$ 15, IQR $\approx$ 9–22, subtler effects in later time windows

No parametric distribution (gamma, log-normal, etc.) is fitted to IOD; density estimation employs nonparametric KDE methods for all conditions.

4. Statistical Divergence and Quantification of Nonequilibrium Transition

The Kullback–Leibler (KL) divergence $D_{\mathrm{KL}}$ quantifies deviation between the equilibrium (pre-myosin) IOD distribution $q(x)$ and various post-myosin distributions $p(x)$: $$D_{\mathrm{KL}}(p \Vert q) = \int p(x) \ln\!\frac{p(x)}{q(x)}\,dx$$ For $q(x)\equiv\mathrm{IOD}_{26\text{–}50}$ (baseline), and $p(x)$ for subsequent 25-frame windows, $D_{\mathrm{KL}}$ values show:

• Buffer and 1% myosin: negligible divergence ($\approx 0$)
• 50% and 100% myosin: $D_{\mathrm{KL}}$ increases (e.g., 0.20–0.70 in windows 51–150), systematically correlating with motor concentration

Empirical window-by-window results:

Time Window	Buffer	1% Myosin	50% Myosin	100% Myosin
51–75	≈ 0	0	0.20	0.50
76–100	≈ 0	0	0.30	0.70
101–125	≈ 0	0.10	0.25	0.60
126–150	≈ 0	0.07	0.20	0.50

This systematic concentration-dependent rise in divergence serves as an effective order parameter for the emergence of nonequilibrium irreversibility.

5. Interpretation and Dynamical Significance

The post-addition increase in IOD and the broadening/heavy-tailing of IOD distributions reflect enhanced NIR fluorescence fluctuations resulting from nonequilibrium, ATP-driven myosin activity. The increasing $D_{\mathrm{KL}}$ with myosin concentration quantifies the statistical departure from equilibrium and marks the degree of irreversibility in the actomyosin network.

Stationarity analyses (KPSS, DF-GLS) show a dose-dependent escalation in the fraction of nonstationary fluorescence traces, rising from $\approx$ 18% in baseline to $\approx$ 77% at 100% myosin. This is consistent with time-evolving, directed network reorganization and breaking of time-reversal symmetry.

A plausible implication is that IOD distribution shifts, alongside KL divergence, can serve as robust, minimally invasive optical indicators for assessing the dynamical regime (equilibrium vs. irreversible nonequilibrium) in active biopolymer systems.

6. Methodological Constraints and Considerations

Key methodological points:

• No parametric modeling of the IOD distribution was performed; reliance is solely on KDE for density estimation
• Short window sizes may bias skewness and kurtosis estimates, potentially underscoring higher-order moment interpretation
• Stationarity tests (KPSS, DF-GLS) assume linear trends or unit roots and may misclassify complex nonlinear fluctuation dynamics, though this is mitigated by buffer controls and block-bootstrap confidence intervals

A plausible implication is that while IOD statistics provide sensitive probes for nonequilibrium transitions, careful selection of time-window parameters and statistical controls is essential to ensure interpretational reliability.

7. Application Scope and Broader Impact

The application of IOD distributions as optical metrics, using single-emitter NIR SWCNT traces, establishes a minimally invasive strategy for interrogating nonequilibrium behavior and irreversible dynamics in actomyosin networks. The approach is generalizable to other active biopolymer systems, providing a quantitative, statistical method for mapping the onset and concentration dependence of dynamical irreversibility without perturbing network assembly or requiring photobleaching-prone probes (Hendler-Neumark et al., 5 Dec 2025).