How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time

We provide a tool for data-driven modeling of motility, data being time-lapse recorded trajectories. Several mathematical properties of a model to be found can be gleaned from appropriate model-independent experimental statistics, if one understands how such statistics are distorted by the finite sampling frequency of time-lapse recording, by experimental errors on recorded positions, and by conditional averaging. We give exact analytical expressions for these effects in the simplest possible model for persistent random motion, the Ornstein-Uhlenbeck process. Then we describe those aspects of these effects that are valid for any reasonable model for persistent random motion. Our findings are illustrated with experimental data and Monte Carlo simulations.

Figure 1

First of seven steps comparing theory with experimental motility data. (a) Trajectory starting at the origin and sampled with time lapses $\mathrm{\Delta }t=4\min$ [alternating black and red (light gray) dots]. The number of time lapses is $N+1=257$. (b) Scatter plot of secant-approximated velocities $(u_x,u_y)$ [see Eq. (1)] from (a) for sample time $\mathrm{\Delta }t=8\min$ [red (light gray) dots in (a)]. (c) Distribution of the squared secant-approximated velocities $u^2=u_x^2+u_y^2$ for sample time $\mathrm{\Delta }t=8\min$. The inset shows the same histogram with lin-log axes. The red (light gray) straight line is the graph of the exponential distribution given in Eq. (63), with parameters resulting from the fit to the power spectrum shown in Fig. 2. Pearson's ${\chi }^2$ goodness-of-fit test gives a $p$ value of 0.20 for this exponential distribution. So this aspect of these trajectory data is consistent with an exponential distribution and hence with the OU theory.

Figure 2

Comparison of experimental data and three model simulations done with parameter values determined from experimental data. $N+1=129$ consecutive positions were recorded with time lapse $\mathrm{\Delta }t=8\min$. (a), (c), and (e) Experimental data compared to theory. Theoretical curves have parameter values obtained from a fit to the power spectrum in (e). The fitted parameters are persistence time $P=17.8\min$, diffusion constant $D=1.1{\left(\mu \mathrm{m}\right)}^2/\min$, and localization error ${\sigma }_{\mathrm{pos}}=1.1\mu \mathrm{m}$. (b), (d), and (f) Data from three Monte Carlo simulations of the OU model. The model parameters used in simulations were those estimated from the experimental data in (e). Data of same color are generated from the same simulated trajectory. Blue (solid) curves are the theoretical curves, while dashed lines connect simulated points. (a) and (b) Mean-square displacements calculated with Eq. (A1) and the theoretical curves from Fürth's formula [Eq. (81)]. (c) and (d) Autocovariance of the secant-approximated velocities calculated with Eq. (A2) and with theoretical curves from Eqs. (78, 79, 80). The simulated data in (b) and (d) are correlated between panels for pairs of the same color (gray scale), because they were obtained from the same trajectory and because the mean-square displacement essentially is a double integral of the velocity autocovariance. (e) and (f) Power spectrum $P_u\left(f_k\right)$ of the secant-approximated velocities and the fit using Eq. (73) and maximum-likelihood estimation (see Sec. 8c). The inset in (e) shows that the power spectral data are uncorrelated by construction for the OU process. That makes a goodness-of-fit test straightforward when the distribution of the power spectral values around the expected value is known [Eqs. (74) and (76)]. Red (light gray) points indicate the expected number of counts in each bin. A ${\chi }^2$-goodness-of-fit test gives a $p$ value equal to 0.91, which, by being larger than 5%, shows that theory and data are consistent with each other according to the highest level of significance in common use.

Figure 3

Test of the OU model using the statistics of the acceleration of secant-approximated velocities. (a) and (b) Black dots show the acceleration of secant-approximated velocities in the directions (a) parallel $a^{\parallel }$ and (b) orthogonal $a^{\perp }$ to the velocity for the experimental trajectory shown in Fig. 1 sampled with $\mathrm{\Delta }t=8\min$. [Red (light gray) dots in Fig. 1 show the $N+1=129$ sampled positions.] The blue (solid) lines show the theoretical expected values [Eq. (66)] for parameter values obtained by fitting to the power spectrum in Fig. 2, resulting in persistence time $P=17.8\min$, diffusion constant $D=1.1{\left(\mu \mathrm{m}\right)}^2/\min$, and localization error ${\sigma }_{\mathrm{pos}}=1.1\mu \mathrm{m}$. (c)–(e) Elements of the variance-covariance matrix with the expected values obtained from Eq. (70). Red (light gray) data points with error bars were obtained by binning data shown as black data points in bins on the $u$ axis of secant speeds.

Figure 4

Goodness-of-fit and fitted values for model parameters for experimental trajectory in Fig. 1 recorded with three different values of the time lapse $\mathrm{\Delta }t$. (a) Resulting $p$ values from ${\chi }^2$-goodness-of-fit tests of the ratio of experimental and fitted power spectral values $2|{\widehat{\vec{u}}}_k{|}^2/P_u\left(f_k\right)t_{\mathrm{msr}}$ (see Sec. 8d). Also shown are the mean and standard deviation of the fitted parameters for (b) the diffusion coefficient $D$, (c) the persistence time $P$, and (d) the localization error ${\sigma }_{\mathrm{pos}}$ versus time lapse $\mathrm{\Delta }t$. The values shown for the standard deviations were obtained from fits to power spectra generated from trajectories simulated with Eq. (32) plus a localization error with standard deviation ${\sigma }_{\mathrm{pos}}$. The numbers of point in the trajectories are $N+1=257$, 129, and 65, respectively. Notice how the fitted parameter values do depend significantly on the duration $\mathrm{\Delta }t$ of the time lapse for $\mathrm{\Delta }t\ge 8\min$, but also that the values obtained for different values of $N$ are not independent, as they stem from the same experimental trajectory sampled with different values for $\mathrm{\Delta }t$.

Figure 5

Velocity autocovariance function with and without discretization effects for the OU process: open red (light gray) circles, ${\varphi }_j^{\left(\mathrm{true}\right)}=\langle {\vec{u}}_j^{\left(\mathrm{true}\right)}·{\vec{u}}_0^{\left(\mathrm{true}\right)}\rangle$ [Eqs. (61) and (62)]; solid line, $\varphi \left(t\right)$ for the OU process [Eq. (20)]; closed black circles, $\varphi \left(t_j\right)$ for velocities ${\vec{v}}_j=\vec{v}\left(t_j\right)$ recorded instantaneously with time lapse $\mathrm{\Delta }t/P=0.5$. The closed black circles fall exactly on the solid line by definition of what they represent.

Figure 6

Fürth's formula for the mean-square displacement of persistent random motion described by the OU process: solid line, mean-square displacement in Eq. (22); dotted line, asymptotic behavior of Fürth's formula, the function $t\mapsto 4D(t-P)$. The dotted line intersects the time axis at $t=P$.

Figure 7

Statistics of the two components of the secant acceleration defined in Eq. (40), parallel and orthogonal to ${\vec{u}}_j$, respectively. (a) Simulated trajectory, generated by iterating Eqs. (28) and (32) using Eq. (38) with $\mathrm{\Delta }t/P=0.5$ and the number of points $N=10000$. The inset shows the first 129 points of the trajectory, corresponding to the length of the experimental trajectory in Fig. 1. Red (light gray) dots mark the sampled positions. Also shown is the acceleration of secant-approximated velocities for the (b) parallel and (c) orthogonal directions relative to the velocity, respectively. (d)–(f) Elements of the variance-covariance matrix of the accelerations in (b) and (c). The solid lines in (b)–(f) are the exact expressions for the expected values for the two directions given in Eqs. (44) and (48). Dashed lines in (b), (d), and (e) are the results for infinitesimal sampling time $\mathrm{\Delta }t/P\to 0$ from Eqs. (45) and (49) and dotted lines are the expected value for the continuous model [Eqs. (2) and (3)]. The error bars are standard errors on the mean calculated as if all values falling in a given bin on the first axis are uncorrelated. Thus the error bars shown underestimate the true error bars, not by much, however, judging from the scatter around the fitted curves.

Figure 8

Comparison of power spectra of secant-approximated-velocity $P_u^{\left(\mathrm{true}\right)}\left(f_k\right)$ (solid line) and tangent velocity $P_v^{\left(\mathrm{aliased}\right)}\left(f_k\right)$ (dashed line) for the the OU process. Data points are power spectral values from a trajectory consisting of $N=10000$ positions generated by iterating Eqs. (28) and (32) using Eq. (38). The sampling time is $\mathrm{\Delta }t/P=0.5$. The frequency axis is discrete because the measurement time is finite, $\mathrm{\Delta }f=1/t_{\mathrm{msr}}$ with $t_{\mathrm{msr}}=N\mathrm{\Delta }t$, making $\mathrm{\Delta }f/f_{\mathrm{sample}}=1/10000$. Note that the time averaging done in Eq. (6) makes ${\vec{u}}^{\left(\mathrm{true}\right)}$ a low-pass-filtered version of $\vec{v}$, as borne out by this figure's comparison of their power spectra.

Figure 9

The two components of the measured secant acceleration, i.e., parallel and orthogonal to the measured secant velocity ${\vec{u}}_j$, including localization error. (a) Same trajectory as in Fig. 7, except a Gaussian distributed localization error with standard deviation ${\sigma }_{\mathrm{pos}}=0.2\sigma P^{3/2}$ was added to each position. The time increment is $\mathrm{\Delta }t/P=0.5$ and the number of data points is $10000$. Also shown is the acceleration of secant-approximated velocities for the (b) parallel and (c) orthogonal directions relative to the velocity. (d)–(f) Elements of the variance-covariance matrix for the accelerations in (b) and (c). The solid lines are the exact expressions for the expected values for the two directions found in Eq. (66). The dashed line in (b) is the expression in Eq. (44), valid in the absence of localization errors, while the dotted line is the limiting case in which localization errors dominate and $\varepsilon$ in Eqs. (66) and (67) tends to $-\frac{1}{2}$. Error bars are standard errors on the mean.

Figure 10

Comparison of $P_u\left(f_k\right)$ (solid line) with $P_u^{\left(\mathrm{true}\right)}\left(f_k\right)$ (dashed line) for the OU process. Data points in (a) are power spectral values from a trajectory consisting of $N=10000$ positions generated by iterating Eqs. (28) and (32) using Eq. (38) and adding a Gaussian distributed localization error with standard deviation ${\sigma }_{\mathrm{pos}}=0.2\sigma P^3/2$ to each position. The frequency axis is discrete because the measurement time is finite, $\mathrm{\Delta }f=1/t_{\mathrm{msr}}$ with $t_{\mathrm{msr}}=N\mathrm{\Delta }t$, making $\mathrm{\Delta }f/f_{\mathrm{sample}}=1/10000$. Here $P_u\left(f_k\right)$ is shifted up compared to $P_u^{\left(\mathrm{true}\right)}\left(f_k\right)$ by an amount growing from 0 at $f=0$ to its maximum at $f=f_{\mathrm{Nyq}}$; see Eq. (73). (b) Histogram of the ratios $2|{\widehat{\vec{u}}}_k{|}^2/P_u\left(f_k\right)t_{\mathrm{msr}}$, which is supposed to be $\mathrm{\Gamma }$ distributed with shape parameter 2 and scale parameter 1 [Eq. (76)]. The red (light gray) dots show the expected number of counts in each bin. Error bars are the square roots of the expected number of counts.

Figure 11

Velocity autocovariance function with discretization effects and effects of localization errors. The solid line, closed circles, and open circles are the same as in Fig. 5. In particular, with the present section's notation, open circles show ${\varphi }_j^{\left(\mathrm{true}\right)}=\langle {\vec{u}}_j^{\left(\mathrm{true}\right)}·{\vec{u}}_0^{\left(\mathrm{true}\right)}\rangle$ for $\mathrm{\Delta }t=P/2$. Triangles show ${\varphi }_j=\langle {\vec{u}}_j·{\vec{u}}_0\rangle$, which equals ${\varphi }_j^{\left(\mathrm{true}\right)}$ except for $j=0,\pm 1$, where ${\varphi }_j$ is shifted up by $4{\sigma }_{\mathrm{pos}}^2$ and down by $2{\sigma }_{\mathrm{pos}}^2$, respectively, relatively to the values of ${\varphi }^{\left(\mathrm{true}\right)}$. The magnitude of the localization error is ${\sigma }_{\mathrm{pos}}=0.2\sigma P^{3/2}$.

Figure 12

Mean-square displacement according to Fürth's formula (solid line), with the effect of localization error included. Here ${\sigma }_{\mathrm{pos}}=0.2\sigma P^{3/2}$. The localization error shifts the graph shown in Fig. 6 up by a constant value $4{\sigma }_{\mathrm{pos}}^2$.

Figure 13

Power spectra including the effects of finite measurement time. Power spectra for the tangent velocity, the secant-approximated velocity, and the secant-approximated velocity with localization error, respectively. Closed circles are the results including the ends of the sum in the Fourier transforms [see Eqs. (88) and (89)], while open circles are without these end points [see Eqs. (25), (60), and (73)]. The parameters are the sample time $\mathrm{\Delta }t=P/2$, the localization error ${\sigma }_{\mathrm{pos}}=0.2\sigma P^{3/2}$, and the length of the trajectory $N=32$.