Information Theory as an Experimental Tool for Integrating Disparate Biophysical Signaling Modules

Here, we introduce and apply a novel software package called CAIM (calcium imaging) designed to calculate AIS, MI, and TE using the Inform library [55,56]. CAIM was designed as a collaborative effort between biologists and physicists to allow researchers to easily and rigorously apply information theory to real biological datasets. CAIM takes time series datasets (Figure 8A) and plots intensity vs. time for ROIs that can be drawn within the CAIM program (Figure 8B). We generated mosaic time series in FIJI with nine ROIs in four quadrants: real actin data in the top left, real calcium data in the bottom left, randomized actin data in the top right, and randomized calcium data in the bottom right. This setup allows for an easy analysis between different conditions.

CAIM enables binarization of time series datasets via a variety of thresholding techniques and displays the binarized time series (Figure 8C). This function enables researchers to easily assess how accurately the applied binarization represents their data. For our analysis, we have elected to binarize our time series using the mean threshold as a cutoff between signal and noise. Such binarization is not strictly necessary for information theory time series analysis, however, another tool called the Java Information Dynamics Toolkit (JIDT) can be used for continuous datasets [34] but is beyond the scope of this study.

Once the signal is binarized, CAIM can be used to calculate AIS, MI, and TE between ROIs. The results are displayed in tables with individual ROIs being identified by the color used to demarcate them on the time series window. Because AIS does not involve comparisons between conditions, a single value is calculated per ROI. Pairwise comparisons are made between each ROI for MI and TE. Because MI is not directional, only a single value per pair is displayed. Measurement of an ROI’s MI with itself provides the information content of the timeseries as defined by the binarization conditions. As TE is directional, CAIM produces a separate value for each direction of each pairwise comparison. Source ROIs determine the row of the table and target ROI determines the column. Each data point is statistically compared against a randomized group of 1000 random time series and values with p < 0.05 colored blue. This provides an easy way to visually detect trends.

Due in part to its model-free nature, an information theoretic analysis does not readily distinguish between biologically interesting signal fluctuations and technical artifacts. Strong photobleaching, for example, causes all regions to decrease signal in parallel, and such time series have high mutual information that does not reflect biological behavior. Much of the jGCaMP8S data that we present here rely on comparatively weak subthreshold fluctuations that may be substantially affected by imaging noise which may be registered as signal due to our mean threshold binarization approach. Further, these images were collected using single-plane confocal microscopy, which provides relatively fast clear images, but only visualizes a single surface of the cell, and thus, may miss important factors outside the imaging plane. These potential confounding factors are limitations to be kept in mind and addressed in future work.

We measured the active information storage (AIS) of actin and calcium dynamics in each of our ROIs as compared with randomized control time series (Figure 9A–D, Supplemental Movies S2–S4). We imaged nine ROIs on each of 10 explants which were collected from three separate rounds of injection. For both signal types, AIS is significantly higher than the randomized control, suggesting that both actin and calcium dynamics are informed by their previous behavior. This conclusion is consistent with our qualitative observations. Actin pulses seem to be somewhat consistent in duration with a refractory period between pulses. The calcium dynamics, in contrast, seem to change gradually over time, but in both cases the future behavior of the signal is well predicted by its past.

Interestingly, actin dynamics have significantly higher AIS than calcium signaling. This finding suggests that actin dynamics are better correlated with their own history than are calcium dynamics.

To quantify correlative relationships within actin and calcium dynamics in nearby cells, we measured pairwise MI between all combinations of nearby cells in each of our explants for both channels imaged (Figure 8A and Figure 9E). Then, we compared these values with random datasets generated by measuring MI between these ROIs and randomized ROIs generated from the same explant. We further measured MI between the randomized ROIs. The calcium and actin dynamics both showed significantly increased MI as compared with the randomized datasets generated from their respective channels. Moreover, calcium dynamics showed significantly increased MI as compared with actin dynamics. These data demonstrate that information is shared in both the actin and calcium dynamics of nearby cells, and suggest that more information is shared between cells via the channel of calcium signaling than via the actin dynamic signaling.

To quantify causal relationships within actin and calcium dynamics, we measured pairwise TE between all combinations of nearby cells in each of our explants for both channels imaged (Figure 9F). Because TE, unlike MI, is directional, we measured TE in both directions for each cell pair. We also generated three random datasets: one dataset in which the sending cell time series was randomized, one dataset in which the receiving cell time series was randomized, and one dataset in which both were randomized.

We did not detect any significant difference in TE between the real actin dynamics dataset and either the ”randomized sender” or the ”randomized receiver” group, and the real data contained significantly less TE (p = 0.0012, Mann–Whitney test) than the ”random sender and random receiver” group. Thus, we conclude that there is no detectable transfer of information from one cell to another in the actin signal.

In contrast, we do detect significantly higher TE in the real calcium dynamics dataset than in either of the three randomized controls. This finding suggests that information may be passed from one cell to another cell via the calcium channel but not the actin channel.

Having measured MI and TE between cells for both channels, next, we sought to measure the information dynamics between channels. We began using the same approach that we used for the individual channel analysis, i.e., we measured pairwise MI between the two channels for each pair of ROIs (10 A). We found significantly higher MI between real actin and calcium dynamics than randomized versions of these datasets, although the MI between channels was significantly lower than the MI within the same channel between ROIs. Taken together, these data suggest that actin and calcium dynamics are not independent in animal cap explants.

We followed our inter-channel MI analysis by measuring TE between pairs of ROIs. We found that actin dynamics had significant TE to calcium dynamics, but that this TE was not bidirectional (Figure 10B). This was somewhat surprising, as we had previously found that actin did not have significant TE between cells while calcium did. This data suggest a complex, directional relationship between actin and calcium in animal cap explants.

To more highly resolve the information dynamics between actin and calcium, we measured MI within each individual ROI in order to disentangle intracellular and intercellular effects. The real dataset has significantly greater MI than the randomized datasets, indicating that these two signals share information within a single cell (Figure 10C).

Finally, we measured TE between the actin and calcium channels for each ROI (Figure 10C). We found that, consistent with our previous pairwise ROI data, on an individual cell basis, actin dynamics demonstrate significantly higher TE to calcium dynamics than calcium dynamics due to actin dynamics. Likewise, TE from actin to calcium dynamics is significantly higher than that of the randomized controls, while TE from calcium to actin is insignificantly different from random controls. These data further support our conclusion of a directional information flow from actin dynamics to calcium dynamics and suggest that this interaction happens at the individual cell level.

Many aspects of information flow are consistent between actin and calcium dynamics. The modalities both display significant AIS, and both display significant MI with neighboring cells. These data suggest that both modalities are largely affected by their own histories but, at the same time, share information with their neighbors.

However, information flow patterns in actin and calcium dynamics share some key differences, indicating that they are not simply manifestations of a single signaling regime. Actin dynamics display significantly greater AIS than calcium dynamics do, while calcium dynamics display greater MI between cells than actin dynamics do. Taken together, these data point to a greater role for actin dynamics in the maintenance of a cell’s state, and calcium plays a greater role in mediating coordination between cells. This trend is further supported by our TE analysis, which shows significant information transfer in the calcium modality but not in the actin modality. Information analysis of crosstalk between the actin and calcium reveals that actin dynamics affect calcium dynamics, but calcium dynamics do not detectably affect actin dynamics (Figure 10A).

We propose (Figure 10D) that actin establishes relatively stable (high AIS) cell collectives (high MI) but does not directly mediate information flow between cells (low TE). This interpretation is consistent with the role of supracellular actin cables in tissue compartmentalization [57], and suggests that this type of compartmentalization may manifest at some level very early even in seemingly poorly organized tissue. Calcium, in contrast, does mediate information flow between cells (high MI and TE), but is less persistent than actin architecture (relatively low AIS). Finally, because actin does affects calcium dynamics (high TE, Figure 10A) we propose that actin dynamics establish compartments through which calcium mediated information flows.

Ultimately, the goal of this analysis is to inform experimental interventions to allow us to understand and control tissue behavior at the systems level. The effective information framework discussed in the introduction provides us with a quantitative way to translate our findings from simple observation to inferences of causality via future interventional experiments. Because calcium dynamics but not actin dynamics display significant intercellular transfer entropy, we predict that perturbations in one cell are likely to affect the calcium dynamics of distant cells, but not the actin dynamics. However, we predict that intervening upon actin dynamics will affect calcium dynamics, even in nearby cells, but that affecting actin dynamics should not perturb calcium signaling. Thus, the analysis that we provide here is a key step in establishing a robust, quantitative, and experimentally supported systems-level understanding of collectivity in embryonic stem cells. This case study demonstrates how the application of information theory can be used to identify new experimental interventions. Iterating the application of information theory and experimental intervention, thereby, can allow the potential for unraveling features of biological behavior across space and time that are otherwise hard to identify due to the multiscale causal interactions in complex living systems.

Fertilized embryos were raised at 14 °C in 0.1X Marc’s modified ringer solution until the onset of gastrulation. Frogs were maintained in accordance with the IACUC protocol M2020-35.

Embryos were injected into 2–4 cell embryos, with jGCaMP8S [58] being injected at 600 ng/µL and lifeAct-mCherry being injected at 100 ng/µL. mCherry-Lifeact-7 was a gift from Michael Davidson (Addgene plasmid # 54491, http://n2t.net/addgene:54491, Accessed on 12 August 2022, RRID:Addgene_54491)

Eight-chambered Nunc Lab-Tek II Chambered coverglasses were coated with 0.1 mg/mL human plasma Fibronectin (F2006, Sigma Aldrich, Burlington, MA, USA) for 60 min at 37 °C in 1X PBS [59]. Then, the dishes were washed once with PBS, once with Danilchick’s For Amy (DFA) buffer that was replaced with fresh DFA buffer [59]. The superficial ectoderm of the animal pole of Nieukoop and Faber Stage 10 embryos was surgically removed in 0.1X Marc’s modified ringer solution [60]. The deep ectoderm was then isolated and transferred to the Fibronectin coated coverglass chambers. Explants were allowed to adhere and spread overnight at 18 °C and imaged the following day.

Explants were imaged on a Leica Stellaris Sp8 confocal microscope for 1 h each with a capture rate of one frame per minute. jGCaMP8S and LifeAct-mCherry were imaged in parallel with separate excitation wavelengths and separate detectors. Specific imaging conditions are available in the representative metadata file (Supplementary Methods S1).

Nine 100 by 100 pixel ROIs were selected per explant time lapse. ROIs were chosen that would remain in a single cell despite cell motion during the time lapse, and were in a domain of the cell that appeared to display actin dynamics. Each explant contained a mixture of ROIs selected at the explant periphery and more internal. An effort was made to choose ROIs from juxtaposed cells where possible with cells exhibiting prohibitive cell motion being discarded.

Eighteen arrays of numbers 1–60 were generated in Microsoft Excel. A different array was used for each ROI, and separate arrays were used for the two channels. The 18 arrays and randomization macros are attached in Supplementary Method S2.

Statistics were performed using GraphPad Prism. Pairwise significance was assayed using Mann–Whitney tests because we could not assume Gaussian distributions of our datasets. To compare real datasets to multiple randomized datasets, pairwise comparisons were used instead of ANOVA tests to prevent false positives resulting from differences between the randomized data sets, and the null hypothesis was only rejected if the real dataset was significantly different from each randomized dataset.

Active information, mutual information and transfer entropy were measured using the novel CAIM software package [55]. CAIM facilitates application of the Inform software package [56] to time series datasets.

We thank Douglas Moore for his work developing CAIM. We thank Vaibhav Pai, Santosh Manicka, Samantha Payne, Kelsie Miller, and Sophia Jannetty for helpful comments on the manuscript.

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/ijms23179580/s1.

Conceptualization, P.M., S.I.W. and M.L.; formal analysis, P.M.; funding acquisition, P.M., S.I.W. and M.L.; investigation, P.M.; methodology, P.M. and S.I.W.; project administration, M.L.; resources, M.L.; software, S.I.W.; supervision, M.L.; writing—original draft, P.M.; writing—review and editing, P.M., S.I.W. and M.L. All authors have read and agreed to the published version of the manuscript.

S.I.W. gratefully acknowledges the support of the Defense Advanced Research Projects Agency (DARPA) under Cooperative Agreement No. HR0011-180200022. M.L. gratefully acknowledges support via grant 62212 from the John Templeton Foundation and grant TWCF0606 of the Templeton World Charity Foundation. Research reported in this publication was also supported by National Institute of Dental & Craniofacial Research (NIDCR) of the National Institutes of Health under the award number 1F32DE027606-01A1.