A thermodynamic perspective on mammalian neural crest ingression

Significance

Although many biophysical cues for cell ingression at the neural crest have been identified, a cohesive framework to unify their influences has been elusive. Moreover, cell ingression at the mammalian neural crest has not been thoroughly described. This study characterizes dynamic neural crest cell ingression in the murine embryo and offers a higher-order perspective of the process. It suggests that when the sum of three free energy parameters is sufficiently favorable, cell ingression can proceed spontaneously. A thermodynamic perspective can potentially unify various inputs to morphogenesis.

Abstract

The ingression of neural crest cells from an ectodermal to a mesodermal layer is regulated by instructive, directional cues and potentially stochastic, biophysical parameters such as differential cell adhesion and tension heterogeneity. However, a cohesive framework in which to consider how various influences contribute to ingression remains elusive. Here, we observe the cell behaviors of the murine neural crest in three dimensions over time and apply a free energy framework to more wholly understand why cells ingress. Guided by work on granular matter that provides a path by which to define the roles of stochastic mechanisms in nonequilibrium systems, we measured and manipulated biophysical parameters in vivo. The data suggest that an energy barrier to cell ingression is overcome by a combination of relatively favorable cell adhesion energies, high cell shape fluctuations, and entropic cell packing configurations. Under those conditions, cell ingression may proceed spontaneously. Recognized biophysical cues likely tilt these parameters to make the process more robust. The results imply that dissipative mechanisms which transiently disorder tissue may underlie some morphogenetic events. Variations of a thermodynamic framework can potentially be applied to integrate various inputs that drive morphogenesis in different contexts.

Cell ingression, or delamination, defines cell movement from one tissue layer to another as in epithelial to mesenchymal transition (1–3). This process occurs at multiple sites in the embryo to deliver specialized cells to organ primordia (4–7). A well-studied example is the ingression of cells from the neural crest into cranial mesoderm from where they subsequently migrate to craniofacial regions and differentiate into various cell types (8–10). The specification (11), differentiation potential (12, 13), and postingression migration of neural crest cells (9, 14) are increasingly understood. Compelling advances have also been made to explain the ingression event, although a cohesive mechanism remains elusive.

There is evidence for inductive and permissive drivers of cell ingression. Directional cues that promote cell movement from the epithelium into mesoderm include biochemical (15–18), electrical (19–21), and stiffness (22) gradients that may, in part, promote and orient cell protrusive activity (23–26). An increase in mesodermal stiffness that triggers neural crest cell ingression in Xenopus is downstream of planar cell polarity (PCP) signaling (22). Although PCP signaling is required for neural crest cell ingression in Xenopus and zebrafish (27, 28), VANGL-dependent PCP is not required in the mouse (29, 30).

Other drivers do not necessarily require a directional cue. For example, transition of the repertoire of cadherins alters cell adhesion affinities, potentially favoring neural crest admixing with mesoderm (27, 31–37). Although the neural crest must be properly specified for cadherin expression to change, the result would be akin to cell sorting. Discontinuity of the basement membrane (BM) due to incomplete maturation or dissolution (23, 36, 38–45) may be necessary but is not sufficient for cell movement through it (41, 42). In the zebrafish myocardium, epithelial crowding underlies tensile heterogeneity among cells (46), a state that can be associated with disruption of the epithelial-mesodermal boundary (47). In conceptually related work, downregulation of cellular tension in Xenopus promotes ingression (48). Crowding stress can also lead to live cell extrusion (49). Finally, enhanced aerobic glycolysis is required at the onset of neural crest cell ingression in the chick, and the expression of genes encoding rate-limiting glycolytic enzymes diminishes among postingression migratory cells (50). We infer from these advances that mechanical factors drive and constrain ingression, and cells likely overcome an energy barrier to delaminate. They suggest that combinations of instructive and potentially stochastic parameters promote cell ingression in different contexts.

Interestingly, as shown by the measurement of structural entropy at different stages of the mouse embryo, progression toward greater structural order during development is punctuated by periods of disorder, such as during gastrulation (51). That observation supports the idea that stochastic processes may contribute to some morphogenetic processes. The potential value of considering development from an alternative perspective such as thermodynamics is that we may be able to place multiple parameters into a framework and better understand how they interact. Here, we apply a free energy framework to assess whether neural crest cell ingression could be a spontaneous process. Since embryos function far from thermodynamic equilibrium (52–55), we combine a nonequilibrium framework (56) with biophysical measurements in the mouse embryo.

Results

Cell Ingression in the Murine Neural Crest.

In various systems, ingressing cells exhibit stereotypical bottle shape morphology through apical constriction that is mediated by contractions of nonmuscle myosin II, cell rounding, and blebbing (57–64). However, these processes are not consistent between zebrafish and chick embryos (65, 66) and are emerging for the mouse neural crest (49). To visualize cell ingression in the mouse cranial neural crest in 3D, we performed time-lapse lightsheet imaging of live, intact Wnt1:Cre2;mT/mG embryos at embryonic day E8.5. This transgenic marker labels a wide region of neural epithelium that includes the cranial neural crest. As such, it allows analysis of every neural crest cell but prevents visualization of cell protrusive activity within the epithelium because protrusions cannot be resolved from adjacent cell membranes. However, protrusions can be visualized at the boundary of labeled neural crest as they ingress into unlabeled mesoderm.

For orientation, the 12 somite embryo demonstrates the neural crest and ingressed cells transiting to facial primordia (Movie S1). We first observed evidence of cell delamination within the midbrain of 4 somite embryos among rosette-like cell clusters that were separated by 1 to 10 cell diameters (Fig. 1A and Movie S2). The central, leading cell of a cluster descended basally and became rounded as neighboring cells moved in to fill the gap, akin to T2 transitions in vertex models of epithelia (67) (Fig. 1 B and C and Movies S3 and S4). Ingressing cell clusters entered mesoderm in separate, stalactite-like chains (Fig. 1D and Movie S5) that gradually coalesced (Fig. 1E and Movie S6) into a coherent subepithelial collection (Fig. 1F). Subsequently, ingressed cells converged and extended into a cone-like configuration as they migrated deeper (Movie S7). A summary image and schematic are given to outline the progression of ingressing “stalactites” (Fig. 1 G and H). We focused further on mechanisms underlying the initial ingression rather than the later migration of neural crest cells.

Fig. 1.

Neural crest cells ingress in small groups with the lead cell undergoing a T2 transition. (A) Cell ingression at the cranial neural crest of a 4 somite Wnt1:Cre2;mTmG embryo. Individual lead cells are marked by solid colors while neighboring, preingressed cells are highlighted in light green in the Lower panel. Inset: Intact mouse embryo, 4 somite stage. The dashed red line indicates the axial level shown in the Top panel. The white line indicates the neural crest (NC). Bottom panel: basal/mesodermal, or deep to superficial, view of ingressing leader cells. Images are representative of 3 embryos at 4 somite stage. FB indicates forebrain. (Scale bars represent 30 µm.) (B) Apical and whole cell rendering of a T2 transition within the neural crest. Left panels: epithelial planar sections showing the apical area of the central cell (white) and immediate neighbors in the same plane. Right panels: three-dimensional renderings of the central white cell and its neighbors (green). Images are representative of 3 embryos at 5 to 6 somite stage. (Scale bars represent 10 µm.) (C) Schematic of neural crest ingression. As a neural crest cell ingresses (white) into the mesoderm (red), its apical surface diminishes, and neighboring cells (green) move in to occupy the vacated space. (D) 3D rendering of actual neural crest cells within an ingressing “stalactite.” Lead cell (yellow) is closely followed by a chain of neural crest cells moving into the mesoderm. (E) Still images from Movie S2 showing lead cells (blue and pink) of adjacent ingressing stalactites coalescing after ingression. (F) Neural crest cells (green in the axial plane of the midbrain of a 7 somite Pax3:Cre;mTmG embryo showing preingressed cells in the epithelium (red) and a coalesced group of ingressed cells (blue). (G and H) Summary of imaging findings. (G) Axial plane image of a lead ingressing cell (white arrowhead). Neural epithelium and neural crest in green, non-neural ectoderm in purple. (H) Schematic summarizing progression toward ingressing stalactites that will coalesce.

Individual cells that were tracked within the preingressed neural crest exhibited a broad distribution of trajectories, especially along the apicobasal axis (SI Appendix, Fig. S1 A and B). Moreover, cell shape fluctuations that have been shown previously to be associated with cell rearrangements and sorting in other contexts (68, 69) did not correlate with any particular direction of displacement (SI Appendix, Fig. S1B). These observations raise the possibility that a subset of neural crest cells move toward the mesoderm without the need for a directional cue in a stochastic manner.

A Mesodermal Stiffness Gradient Deep to the BM.

A durotactic mechanism due to stiffening mesoderm contributes to the initial migration of neural crest cells in Xenopus (22). However, the surface indentation used in association with epithelial dissection in Xenopus (22, 70) are arguably not optimal methods for measuring subsurface stiffness gradients. To measure stiffness in the mouse embryo, we used a 3D magnetic tweezer system that is suitable for bulk mesoderm (71–73). Magnetic beads were injected into the neural folds of mouse embryos and imaged live using a spinning-disk confocal microscope. Beads were actuated with a uniform magnetic field gradient such that the degree of displacement of individual beads reflected the stiffness of their locations (Methods). Stiffness of the neural crest was comparable to that of the underlying mesoderm immediately deep to the BM but increased progressively beginning at a depth of 60 to 80 µm (SI Appendix, Fig. S1C). In Xenopus, a stiffness gradient also contributes to deeper migration of ingressed neural crest cells (70). The stiffness gradient in the mouse suggests that a durotactic mechanism is less likely to precipitate ingression than it is to drive subsequent migration. Although other directional cues may be present in the mouse, we wondered whether a stochastic process might suffice to explain cell ingression.

A Thermodynamic Perspective of Cell Ingression.

Observations and conjecture have been applied to describe evolution and embryonic development as dissipative processes (52, 53, 55, 74). However, it is not clear whether periods of disorder contribute to specific developmental events. Guided by our observations of cell shape fluctuations among ingressing cells (more on this below), we postulated that fluctuation-driven transition could be a driver of cell ingression. This concept is akin to increased effective temperature facilitating barrier crossing in equilibrium systems (Fig. 2A).

Fig. 2.

Simulated effect of cell fluctuations. (A) Hypothetical energy landscape (blue curve) that a neural crest cell (black circle) would follow to reach another metastable state within mesoderm (gray circle). In principle according to Helmholtz free energy (F), a cell can overcome an internal energy barrier (ΔU) if the entropic term (effective temperature T_eff x entropy S) is sufficiently high. (B) The initial and final states of vertex model simulations in which neural crest cells (green) move spontaneously from the epithelium (blue) toward mesoderm (red) as a function of fluctuation amplitude. (C) T_eff, given by the ratio of particle diffusivity (D) to susceptibility (χ), was positively correlated with amplitude of cortical fluctuations in simulations (n = 48 simulations per amplitude). (D) Plot of system energy versus configurations of trajectories for single cell ingressions. The blue curve represents a deterministic path wherein the closest edge separating an ingressing cell from the mesenchyme is contracted to a point manually before the system is allowed to relax in the absence of fluctuations. That curve shows there is an energy barrier that must be overcome for cells to ingress into mesoderm. The red curve is the average of 96 simulations starting from the same initial configuration but with stochastic cortical fluctuations added and without manual interference (SD shaded). The curves were shifted to normalize the time step of ingression. Two representative curves are shown as orange and purple dotted curves. (E) Rates of ingression as a function of adhesion affinity. Intercellular line tensions were unchanged (0%) in cells that maintained their epithelial character. To mimic the acquisition of mesodermal adhesion character as cells undergo epithelial-to-mesenchymal transition, line tensions were altered to partially resemble that of mesoderm from (e.g., the blue curve represents line tension of 20% mesodermal and 80% epithelial). Cortical fluctuations were maintained at 12% (as in Fig. 2E). (F) Rates of ingression as a function of cortical fluctuation amplitudes. Amplitudes are represented as the SD (% of the mean) of cell shape fluctuations. Adhesion preference for mesoderm was set at 60% (as in Fig. 2D). For (D and E), solid color curves represent the mean (of means of cell positions) and shading represents SE [sigma/square root(n)] of 48 simulations for each condition. (G) Schematic of free energy parameters we considered in the neural crest. The internal energy barrier (U) that needs to be overcome for ingression to proceed includes cell adhesion affinities and the BM. Cell shape fluctuations represent effective temperature, and cell packing configurations represent entropy.

To identify the level of stochastic fluctuations for cells to ingress, we created a two-dimensional vertex model of cell ingression that employed active cell shape fluctuations which are essential for cell intercalation (59, 75, 76) and passive, dissipative relaxation that is central to any nonequilibrium setting (Fig. 2B and Methods). To relate this model to work on nonequilibrium granular systems (77, 78), we showed that increased cell shape fluctuations in the model lead to an increase in the effective temperature (56) (Fig. 2C, SI Appendix, Fig. S2 A and B, and Methods). Compared to a deterministic path in which junctions were rearranged to force cells beyond an energy barrier, fluctuations resulted in flattening of the energy landscape. “Hot” cells moved freely into mesoderm with effectively no barrier (Fig. 2D).

One of the most extensively documented biophysical parameters in model organisms including the mouse is expression of mesodermal cadherins by neural crest cells prior to ingression, thereby increasing their sorting preferences for mesoderm (13, 35, 79–87). As expected, when simulated neural crest cell adhesion affinities were increasingly changed from epithelial to mesodermal [represented by changing intercellular line tensions (47) in the model], ingression of hot cells into mesoderm became correspondingly more efficient (Fig. 2E).

When simulated neural crest cell fluctuation amplitudes were low, cell movements toward mesoderm were relatively slow and incomplete with some cells becoming trapped within the epithelial layer despite a high adhesion preference for mesoderm (60% as in Fig. 2E). Conversely, when fluctuation amplitudes were high, ingression occurred more rapidly and completely (Fig. 2 B and F and Movies S8 and S9). These simulations suggest that adhesion preferences alone are insufficient and that relatively high amplitude cell shape fluctuations could drive stochastic cell ingression.

Biophysical Parameters in the Murine Neural Crest.

To test our model predictions in vivo, we considered that the internal energy of the system must either diminish and/or be overcome by the combined effect of fluctuations (effective temperature) and entropy for ingression to be spontaneously favorable (conceptually reflected by a reduction in free energy—Fig. 2 A and G). We recognized that internal energy (U) of the neural crest cannot reliably be measured in vivo because it includes numerous, immeasurable entities. However, we assumed that 1) intercellular adhesion energies represent an important, if not dominant, component of internal energy that we can empirically infer as a relative parameter, and that 2) the BM represents a barrier to ingression that we can manipulate (below).

Interfacial tension reflects the combined effect of cortical tension that raises it and adhesion energy that lowers it if adhesion affinities are favorable as they would be between like cells. We applied established methods to segment cells in 3D [using Ilastik (88), SI Appendix, Fig. S3A] and to generate meshes of cellular geometries [using foambryo (89), Fig. 3A]. Interfacial tensions were then inferred from interface morphologies (Fig. 3B). Relatively high interfacial tensions distinguished the neural crest from the adjacent neural epithelium. Although tensions were heterogenous throughout the crest, patches of relatively high tensions were observed within it, especially among cells that were visibly initiating ingression by protruding beyond the boundary separating the ectoderm from the mesoderm. To better define this observation, we separately plotted the interfacial tensions within apicobasal columns centered on visibly ingressing cells that were 90 μm (or about 8 cells) wide versus the remainder of the non- or preingressed neural crest (Fig. 3C). Tensions were indeed higher within cell columns associated with ingression (Fig. 3 D and E and SI Appendix, Fig. S3B). Where cell ingression had advanced beyond initiation such that neural crest cells had just entered the mesoderm, we noted that interfacial tensions within the neural crest trended downward (Fig. 3D), raising the possibility that either adhesion energies become more favorable or cortical tensions diminish upon ingression. Since we cannot directly measure adhesion energies in this setting in vivo, we examined a proxy for cortical tensions in a subsequent section below.

Fig. 3.

Interfacial tensions among neural crest cells. (A) A three dimensional multimesh view of the neural crest cells of a 5 somite, E8.5 Wnt1:Cre;mTmG embryo showing the total volumetric segmentation constructed from z-stacks that were segmented using Ilastik (SI Appendix, Fig. S3A). The approximate boundary between the neural crest and the neural epithelium is given by the dotted line. (B) An exploded view demonstrates the distribution of interfacial tensions inferred from interface morphologies using foambryo ranging from low (blue) to high (red). The neural crest generally exhibits higher interfacial tensions than the adjacent neural epithelium and patches of higher tensions are observed. (C) Neural crest mesh formed by joining individual cell meshes showing three early ingressions (arrowheads). Cylinders of 90 μm (around 8 cells) in diameter around cells protruding into the mesoderm were taken as environments near early ingressing cells. (D) Plot of inferred interfacial tensions of each interface as a function of distance from the apical surface from one embryo. Data are separated into two populations based on whether the area weighted average position of the interface is within or outside of the red cylinders shown in (C). Each population is binned into 10 equally spaced apicobasal depth regions to show the average tension and SE. Data from two additional embryos are plotted in SI Appendix, Fig. S3B. (E) Box plot of the distributions of spatially binned mean interfacial tensions near or away from early ingression sites across 3 embryos at the 5 somite stage. [P = 0.007, uncorrelated t test; box plots: centerline is median, box is first and third quartile, whiskers are furthest point within 1.5 inner quartile range (IQR)].

With respect to the entropic term in the free energy equation, we examined changes in configurational entropy as a function of packing density which influences the collective behaviors of athermal materials (90) and living tissues (76, 91) alike. By measuring their densities in 3D, we found that cells in the preingressed neural crest were more densely packed than those in the neighboring pseudostratified columnar neuroepithelium (NE) or in the underlying mesoderm (Fig. 4A). Interstitial volumes were substantially greater in mesoderm relative to the NE and preingressed neural crest (Fig. 4B). Together, these data indicate that preingressed and ingressed neural crest cells represent the most and least physically constrained populations, respectively. To define cell shape and neighbor relationships, we segmented 3D volumes (SI Appendix, Fig. S3C). The nonparametric cell shape index (S/V^2/3) was progressively lower, indicating that cells were rounder as they progressed from the NE to the preingressed neural crest to ingression (Fig. 4C). The number of cell neighbors increased from a median of 7 in the NE, to 10 among preingressed neural crest cells and 12 among ingressed cells (Fig. 4D). Therefore, ingressing neural crest cells transition to a progressively unconstrained state.

Fig. 4.

Geometric packing of neural crest cells during delamination. (A) Cell density of the NE, preingressed neural crest (NC), and ingressed neural crest within mesoderm (Meso). Cells within the 3 regions of Wnt1:Cre2;mTmG embryos were rendered using a semiautomated algorithm in IMARIS: n = 8 embryos for NE, 10 embryos for NC, 4 embryos for NCI. ANOVA F(2, 20) = 5.63, P < 0.05. Tukey HSD post hoc: NE–NC Tukey CD = −85.7, P < 0.05; NC–NCI Tukey CD = 99.0, P < 0.05. (B) Fluorescence intensities of rhodamine-dextran dye in 1 × 10⁶ to 4 × 10⁶ µm³ optical tissue volumes within NE, NC, and NCI were quantified in IMARIS n = 3 embryos per region. ANOVA F(2, 9) = 7.83, P < 0.05; Tukey HSD post hoc: NE–NCI Tukey CD = −99061.0, P < 0.05; NC–NCI Tukey CD = −86490.0, P < 0.05. (C) Shape index of cells within the neural fold. Each dot represents a rendered cell. n = 54 NE cells from 6 embryos, 73 NC cells from 5 embryos, and 55 ingressed NCI cells from 3 embryos. ANOVA F(2, 176) = 30.9, P < 0.05. Tukey HSD post hoc: NE–NC Tukey CD = −0.83, P < 0.001; NC–NCI Tukey CD = 0.48, P < 0.01. Insets represent clusters within each region highlighting a central cell (white) surrounded by its neighbors (in color). Box plots: Centerline is median, box is first and third quartile, whiskers are furthest point within 1.5 IQR. (D) Distribution of number of cell neighbors within the neural fold. Cells were segmented in 3D using IMARIS. Values for entropy (S_est) were derived from fitting the number of neighbors to a normal distribution. n = 67 NE cells from 4 embryos, 62 NC cells from 5 embryos, 55 NCI cells from 6 embryos. Wilcoxon test: NE–NC W = 3,344.5, P < 0.001. NC–NCI W = 957.5, P < 0.001.

System entropies of collective granular materials (92) and living cell clusters (91, 93) have been estimated based on the volumes and aspect ratios, respectively, of individual components that follow a k-gamma distribution. We adapted such a statistical mechanics approach to estimate the entropies of the NE, pre- and postingressed neural crest based on cell shape index and packing configurations. Using a nonlinear least squares method, we fit the cumulative distribution function to our data and found that cell shape indices closely follow a k-gamma distribution, and numbers of cell neighbors follow a normal distribution in all 3 compartments. By both methods, estimated configurational entropy was highest among cells that had ingressed (Fig. 4D and SI Appendix, Fig. S4A). This entropy parameter was not simulated in our vertex model, and we propose that it contributes to making ingression thermodynamically favorable and less likely to be reversible.

Neural crest cells exhibit cell shape fluctuations (Movie S10) that represent effective temperature in our model. As a proxy for the complex, 3D nature of cell shape changes, we examined fluctuations using a transgenic Förster resonance energy transfer (FRET)-based vinculin tension sensor (VinTS) (76). Vinculin links cortical actomyosin to cell–cell and cell–ECM adhesion complexes at the cell membrane. Tension upon the sensor presumably reflects the combined effect of cortical and adhesion tensions and is responsive to actomyosin contraction (76). To measure tensions in vivo, we employed fluorescence lifetime imaging microscopy (FLIM). By that method, fluorescence lifetime of the donor fluorophore correlates directly with VinTS tension.

VinTS was expressed conditionally in the neural crest using Wnt1:Cre2 and individual cells were examined in 2D by live confocal microscopy of intact embryos (Fig. 5A). We used FLIMfit (94) and FLIMvivo (95) (available at https://github.com/HopyanLab/FLIMvivo) to generate segmentation masks and to fit the fluorescence decay curves (e.g., Fig. 5B). The degree of fluctuations, rather than static values, was most relevant to the effective temperature parameter in our model. For individual cells in the neural crest, fluctuations that we measured using VinTS correlated moderately with those of cell shape changes in 2D (Fig. 5C). We chose to quantify experimental changes in fluctuation using VinTS.

Fig. 5.

Manipulations of cellular fluctuations. (A) FLIM intensity image of Wnt1:Cre2;VinTS^fl/+;Myh9^fl/+ neural crest showing preingressed and ingressed cells. Color map is shown of FLIM lifetimes computed for each segmented cell. (B) Example of a FLIM decay curve for the cell in the white box in (A). In orange is the greatest likelihood fit of the curve to a biexponential convolution model (using Poisson statistics for fluorescence decay). (C) Cell shape fluctuations versus FLIM lifetime fluctuations were computed for segmented preingressed cells and fit at 8 time points at 2-min intervals (Wnt1:Cre2;VinTS^fl/+;Myh9^fl/+, n = 30 cells from 3 embryos at 6 somite stage). For each tracked cell, shape index (perimeter over square root of area) and FLIM were calculated from the same segments. Plotted are the SD of those two quantities for each cell. The two are moderately correlated with Pearson R statistic 0.455 and P-value 0.9%. (D) Cortical cell membrane fluctuations within the neural fold of Wnt1:Cre2;VinTS^fl/+;Myh9^fl/+ (red curves) and Wnt1Cre-2;VinTS^fl/+;Myh9^fl/fl (blue curves) mouse embryos. Each trace is the lifetime fluorescence of VinTS over time and represents the cortical tension of individually segmented cells within the NE (Wnt1:Cre2; VinTS^fl/+;Myh9^fl/+ n = 48 cells from 3 embryos), preingressed neural crest (Wnt1:Cre2; VinTS^fl/+;Myh9^fl/+ n = 109 cells from 9 embryos; Wnt1:Cre2;VinTS^fl/+;Myh9^fl/fl n = 34 cells from 3 embryos), and ingressed neural crest cells (Wnt1:Cre2;VinTS^fl/+;Myh9^fl/+ n = 182 cells from 11 embryos; Wnt1:Cre2;VinTS^fl/+;Myh9^fl/fl n = 39 cells from 3 embryos). (E) Average cortical tension represented as VinTS lifetime values for cells in the NE, preingressed neural crest (NC), and ingressed neural crest (NCI). Two-way ANOVA with significant main effect of cell position F(2, 183) = 42.26, P < 0.001; Tukey HSD post hoc: NE–NC Tukey CD = −0.05, P < 0.001, NC–NCI Tukey CD = 0.03, P < 0.01. The main effect of genotype was not significant F(1, 183) = 0.18, P = 0.67. (F) Fluctuation amplitudes of each trace, calculated as the deviation (root sum of squares, RSS) from the average lifetime over time for cells within the NE, NC, and NCI. Two-way ANOVA with significant effect of interaction between cell position and genotype F(1, 181) = 41, P < 0.001. Tukey HSD post hoc: Myh9^fl/+ NC–NCI Tukey CD = 0.04, P < 0.001; Myh9^fl/fl NC–NCI Tukey CD = −0.003, P > 0.05. (G) Fluorescence intensities from 7 somite stage Wnt1:Cre2;myrPercevalHR^fl/+;Myh9^fl/+ neural crest. ATP channel is 488 nm while ADP channel is 405 nm. The example cell in the yellow box exhibits a higher ATP:ADP ratio than that in the magenta box. (H) ATP:ADP ratios of preingressed, NC, and ingressed, NCI, neural crest cells of Wnt1:Cre2;myrPercevalHR^fl/+;Myh9^fl/+ (red) and Wnt1:Cre2;myrPercevalHR^fl/+;Myh9^fl/fl (blue) mice. Each dot represents one cell. (Wnt1:Cre2;myrPercevalHR^fl/+;Myh9^fl/+ n = 56 NC cells from 4 embryos and n = 87 NCI cells from 5 embryos; Wnt1:Cre2;myrPercevalHR^fl/+;Myh9^fl/fl n = 98 NC cells from 7 embryos and n = 106 NCI cells from 8 embryos). Two-way ANOVA with significant effect of interaction between cell position and genotype F(1, 343) = 69, P < 0.001. Tukey HSD post hoc: Myh9^fl/+ NC–NCI Tukey CD = 0.86, P < 0.001; Myh9 ^fl/fl NC–NCI Tukey CD = −0.20, P > 0.05.

Mean vinculin tensions of preingressed neural crest cells were significantly higher than those of neuroepithelial cells, and higher still among ingressed cells (Fig. 5 D and E). The amplitudes of fluctuations, calculated as the deviation from the average lifetime across timepoints (root sum of squares, RSS), were also significantly higher among neural crest cells that had ingressed into mesoderm (Fig. 5F). These data suggest that effective temperature, like configurational entropy, increases during cell ingression, presumably reducing free energy and rendering spontaneous ingression more favorable (as depicted in Fig. 2A).

Since the neural crest is obviously a thermodynamically open system, we surmised that overcoming an energy barrier to move the system to a different metastable state likely costs energy for individual cells. We acknowledge that the total energy state of a cell includes contributions that cannot readily be measured. Since glucose uptake is higher in migratory neural crest cells relative to premigratory cells in tissue explants (50), it may be possible to estimate the metabolic difference associated specifically with ingression movement. In vitro, active cell migration correlates with high ATP:ADP ratios (96, 97). To determine ATP:ADP ratios in vivo, we generated a transgenic mouse strain, myrPercevalHR^fl/fl, that harbors a floxed, myristoylated PercevalHR sensor (98). In vivo, the sensor reported appropriately high or low ATP:ADP ratios in response to the presence or absence of glucose and to glucose competition (SI Appendix, Fig. S4 B and C and Methods). Within the neural crest of Wnt1:Cre2;myrPercevalHR^fl/+ embryos, ATP:ADP ratios were significantly higher among ingressed cells relative to preingressed cells (Fig. 5 G and H). We infer that energy input by cells is associated with their transitions from one metastable state to another, a process akin to overcoming an energy barrier. Taking these observations together, we propose that raising effective temperature promotes the initial transition and increased packing entropy effectively traps cells within mesoderm, rendering the new state practically irreversible.

Manipulations of Thermodynamic Parameters In Vivo.

Our model predicts that decreasing cortical fluctuations would hamper the exit of cells from the epithelium. To test this hypothesis, we dampened cortical fluctuations by conditionally deleting Myh9 that encodes the nonmuscle myosin IIA heavy chain, a regulator of the pulsatile nature of actomyosin contractions (99). The mean tensions and, importantly, amplitudes of cortical fluctuations were reduced in Wnt1:Cre2; VinTS^fl/+;Myh9^fl/fl conditional mutants relative to Myh9 heterozygous littermates (Fig. 5 D–F). Unlike the wild-type situation, ATP:ADP ratios did not increase during ingression in Myh9 mutant embryos (Fig. 5H). That finding suggests that ATP turnover is driven by demand imposed by greater myosin-driven fluctuations that are diminished in the mutant. We interpret the similarity of ATP:ADP ratios among wild-type and mutant noningressing cells as a reflection of basal activity, implying that Mhy9 is especially important for high amplitude fluctuations.

To determine the necessity of fluctuations for cell ingression, we examined conditional Myh9 mutants. Those mutants exhibited subtle gross dysmorphology (SI Appendix, Fig. S5A) with an exaggerated neural crest bulge reflecting inappropriate pooling of cells (SI Appendix, Fig. S5B). We measured the fraction of SOX9-positive neural crest cells that had ingressed in whole-mount immunostained embryos ranging between 4 and 8 somite stages. Applying a custom script, we used E-cadherin levels (100) and physical location to distinguish cells within the epithelial neural crest that had not ingressed from those within mesoderm that had ingressed (Methods). The proportion of ingressed cells was significantly reduced in Myh9 mutant embryos (Fig. 6A), indicating that high amplitude fluctuations are required for efficient cell ingression as predicted by our model.

Fig. 6.

Manipulations of fluctuations or BM affect ingression rate. (A) Proportion of ingressed cranial neural crest cells in intact, stage-matched 6-7 somite Wnt1:Cre2;Myh9^fl/+ and Wnt1:Cre2;Myh9^fl/fl embryos. Three-dimensional projections of nuclei were identified using Nuclear Plotter. Purple label: Sox-9 positive nuclei of noningressed cells within the epithelium that are surrounded by high E-cadherin staining. Green label: Sox9 positive nuclei deep to the epithelium surrounded by nil or low E-cadherin staining. The green label in the midline arises appropriately from the notochord (101). Gray label: DAPI-stained nuclei. Right panel: Quantification of ingressed cells within intact embryos. Each dot represents one embryo: Wnt1:Cre2;Myh9^fl+l n = 6 embryos, Myh9^fl/fl n = 3 embryos; t(6) = 2.7, P < 0.05. (B) Ingressed cranial neural crest cells in intact, stage-matched 4-7 somite T:Cre;FN^fl/+ and T:Cre;FN^fl/fl embryos. Whole-mount immunostaining for fibronectin in T:Cre;FN^fl/+ and T:Cre;FN^fl/fl embryos. White arrowheads: neural crest (NC), red arrowheads: fibronectin-labeled BM. Right panel: Quantification of ingressed cranial neural crest cells in intact T:Cre;FN^fl/+ and T:Cre;FN^fl/fl. Each dot represents an embryo: T:Cre;FN^fl/+ n = 7, T:Cre;FN^fl/fl n = 3; t(8) = −4.5, P < 0.01. Box plots: Centerline is median, box is first and third quartile, whiskers are furthest point within 1.5 IQR. (C) In these summary plots, the fate of a neural crest cell is represented by the green or gray circle in the energy well. 1) Myosin-driven cell shape and tension fluctuations probe the energy landscape of neighboring cell configurations. 2) Changes in cell adhesion molecules and BM integrity change the energy landscape to make ingression favorable. 3) Energy gained by moving to a favorable configuration is dissipated so that the process is unlikely to be reversible. 4) In myosin mutants, fluctuations are reduced such that cells are less able to probe new configurations and overcome energy barriers. As a result, cells pile up in the neural crest rather than ingress. 5) In fibronectin mutants, weakening of the BM reduces the barrier to ingression, allowing cells to ingress sooner.

The ability of neural crest cells to remodel the BM has been shown to promote cell delamination and migration (36, 45, 102). Those observations suggest the BM represents a component of the system’s energy barrier. Based on our model, we expected that weakening the BM would lower the energy barrier and enhance cell ingression. In mouse neural folds, immunostaining revealed that fibronectin (FN) is a significant component of the BM that can be partially depleted by conditionally knocking out its mesodermal source (T:Cre;F1^fl/fl). Quantification revealed that loss of fibronectin significantly increased the proportion of ingressed Sox9-positive cells in T:Cre;Fn1^fl/fl embryos (Fig. 6B), suggesting that the height of the energy barrier negatively influences cell ingression.

In summary, we propose that neural crest ingression can be modeled as a dissipative stochastic system. Preingression, the system rests in a local energy minimum in configuration space. Cell adhesions and geometric configurations describe the internal energy and myosin-driven cell shape fluctuations play the part of an effective temperature. Changes in these parameters can render cell ingression more, or less, favorable (Fig. 6C).

Discussion

This study provides a potential framework in which to consider the contributions of various determinants of cell ingression. The observations here place tension heterogeneity, which has been identified previously as defining ectodermal-mesodermal boundaries in Xenopus (47) and preceding delamination during cardiac trabeculation in zebrafish (46), into a thermodynamic context at least for the mouse neural crest. We propose that high interfacial tensions are counteracted by high fluctuations and rising configurational entropy to promote ingression. Upon ingression, interfacial tensions are stable or slightly diminished despite increased fluctuations that would otherwise raise tensions, implying adhesion energies are overall more favorable postingression. Interestingly, the lead cell is often rounded as observed by others as well. In addition to reflecting a live cell extrusion process in a subset of cells (49), rounding may generally reflect the relatively high adhesive incompatibility of the lead cell that is most exposed to mesoderm. Although ingression seems to relieve some interfacial tension overall, fluctuations are required to reach that state. Under permissive physical conditions, cell ingression can occur spontaneously and is unlikely to be reversed.

This study also raises the somewhat provocative possibility that cell ingression is a stochastic process. Evidence supporting that notion includes the lack of a strong directional or spatial bias in the movement, shapes, or interfacial tensions of cells within the crest. That is not to say the conditions which facilitate ingression, such as fluctuations and adhesion energies, are randomly acquired. Those attributes must still be specified at the correct spatiotemporal interval during development. For example, a key consequence of neural crest specification is likely the alteration of cadherin profiles (27, 31, 32, 36, 37) that make adhesion energies more favorable for cell ingression.

A stochastic process can coexist with potential directional cues that are summarized in the introduction. We experimentally tested only one such potential cue and readily acknowledge that others, such as chemotactic agents (103), are important although they have not been thoroughly examined in the mouse. Directional cues may tilt thermodynamic parameters toward ingression, potentially adding robustness, regulation, and variation in different contexts.

We do not know how ingression parameters vary in different physical and biochemical environments along the anteroposterior axis, such as at the truncal neural crest (103–105), or between different animals. Let us consider the physical tissue environment. In the mouse, neural crest ingression begins well lateral to the midline when the neural tube is open (SI Appendix, Fig. S5) and continues after it is closed as is the situation in the trunk region. The latter state is more analogous to the chick in which ingression initiates after neural tube closure (106). Neural tube closure and separation of the neural epithelium from the overlying ectoderm may further stabilize the neural epithelium and perhaps accentuate fluctuation differences of “restless” neural crest cells to facilitate their escape. Differences in the shape of the NE may generally result in different interfacial energies and cortical tensions. In zebrafish, where ingression takes place before and after neural keel separation from the overlying ectoderm (107, 108), we speculate that free energy kinetics will differ by stage. In these examples, we would expect corresponding variations in the efficiency of neural crest cell ingression. In contrast, we expect that ingression parameters are not affected by variations in the subsequent migratory path of neural crest cells such as along dorsal or ventral streams around somites.

Stochastic transitions between different states are central to multiple biological contexts (109–112) such as cell fate changes (113–115). Advances that equated active parameters to those of athermal systems have opened the possibility of examining thermodynamic properties of nonequilibrium, living systems (56, 77, 78, 91–93). By applying such an approach to test for a stochastic morphogenetic mechanism, this study complements others that have assessed thermodynamic parameters in whole organisms using calorimetry or thermal temperature measurements (54, 116–118). The results here suggest to us that some dissipative (52–55, 116–119) developmental processes such as mouse neural crest ingression may have evolved to allow transient disorder that generates a favorable outcome while conserving system energy. From that perspective, thermodynamic favorability can be considered a mechanism of morphogenesis.

Cell ingression among vertebrates may have been adapted from ancient invertebrate cell behaviors observed at the archenteron margin in diploblasts (2, 120, 121). As has been suggested by others, a small number of conserved physical rules may underlie basic morphogenetic motifs that were diversified by genetic variants (11, 122). If cell ingression is fundamentally a stochastic process, it is likely regulated and varied among different organisms and tissue sites by layers of biochemical or physical cues. For example, in colder environments, perhaps additional cues help to tip the energy curve in favor of ingression. Since a process like cell ingression is dissipative, the system does not readily revert to its original state, thereby stabilizing new structures (123). Examining developmental processes from the perspective of how they combine energy-intensive cues with spontaneously favorable outcomes could be useful for defining mechanisms of morphogenesis and malformation.

Methods

The following summaries are detailed in SI Appendix, Extended Methods.

Animals.

Mouse lines.

Embryos from the following mouse lines were used in experiments: Wnt1:Cre-2 (124), Pax3:Cre (125), mT/mG, VinTS^fl/fl (76), Myh9fl/fl (126), and myrPercevalHR^fl/fl.

Imaging and Analyses.

Light-sheet microscopy and image processing.

Three-dimensional images of the cranial region of embryonic day E8.5 embryos ranging from 4 to 8 somites were acquired using a Zeiss Lightsheet Z.1 microscope as previously described with modifications (76).

FLIM-FRET.

Live embryos (E8.5) were imaged using the Nikon A1R laser scanning confocal microscope equipped with a PicoHarp 300 TCSPC module and a 400 nm pulsed diode laser (Picoquant) as described previously (76, 95).

Live confocal microscopy.

To measure ATP:ADP ratios, live mouse embryos were embedded in agarose within imaging chambers maintained at 37 °C with 5% CO₂ as above and imaged using a Nikon A1R laser scanning confocal microscope. Samples were excited with 488 nm and 405 nm lasers for ATP and ADP, respectively, and emissions were captured using a 525/50 bandpass filter.

Tissue Stiffness Quantification In Vivo.

Tissue stiffness quantification was performed using a multipole magnetic device that we developed previously (72) and applied to the limb bud (71) and mandibular arch (73).

Vertex Modeling and Simulation.

Two-dimensional model of cell ingression.

We generated a simple two-dimensional vertex model as a toy model to explore the effect of cortical fluctuations in the presence of distinct cell types that segregate.

Effective temperature.

Using the same model as above, we computed the “effective temperature” previously used in nonequilibrium granular systems (77, 78) as the ratio of particle diffusivity to susceptibility.

Inference of Interfacial Tensions in 3D.

We used Ilastik (88), a machine learning segmentation program, to segment the neural crest in 3D. We then applied the foambryo (89) python package to infer interfacial tensions for all cell membranes.

3D Cell Segmentation and Neighbor Counting.

To examine three-dimensional cell shapes and neighbor relationships, CZI files were imported into IMARIS.

Quantification of Ingressed versus Preingressed Cells.

Fixed, whole embryos (E8.5, 4-8 somite stages) were immunostained against SOX9 (AB5535, mouse, EMD Millipore) and E-cadherin (610181, mouse, BD Biosciences), and imaged using a Nikon A1R laser scanning confocal microscope. The generated nd2 files were read and analyzed using called Nuclear Plotter (available at https://github.com/HopyanLab/Nuclear_Plotter), a custom python script that identifies DAPI-labeled nuclei and allows users to classify cells based on nuclear and cellular immunostaining and proximity to designated tissue landmarks (e.g., neural crest).

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Neural crest cells (green) are seen at the dorsal lips of the closing neural fold and migrating ventrally through mesoderm (red) of a Wnt:Cre2;mTmG embryo at 12somite stage.

Movie S2.

Cells ingress at different points along the cephalic neural crest. Representative cell surface-rendered cranial neural crest within the midbrain of a Wnt:Cre2;mTmG embryo at 4 somite stage showing the initial group of ingressing neural crest cells (solid colours) and their adjacent pre-ingressed neighbours (green).

Movie S3.

Apical view of a neural crest cell ingressing at 4 somite stage. The apical region of the central cell (yellow) diminishes relative to its immediate neighbours as it leaves the plane, a configuration change similar to a T2 transition.

Movie S4.

Lateral view of neural crest cell ingression. Cell surface rendering demonstrates an ingressing cell (solid white) undergoing shape fluctuations as it moves to a more basal position relative to its immediate neighbours (green) within a Wnt:Cre2;mTmG embryo (5 somites).

Movie S5.

Lateral view of a surface-rendered chain, or ‘stalactite’ of ingressing neural crest cells comprised of a lead cell (yellow) tightly associated with follower cells within the midbrain of a Wnt:Cre2;mTmG embryo (5 somites). Marked cell shape fluctuations are observed.

Movie S6.

Individual streams, or ‘stalactites’, of neural crest cells coalescing with the mesoderm of a Wnt:Cre2;mTmG embryo (7 somites).

Movie S7.

Ingressed and coalesced neural crest cells (green) within mesoderm (red) at the midbrain are observed remodelling their group into a narrower configuration prior to deeper migration in a Wnt:Cre2;mTmG embryo (8 somites).

Movie S8.

Vertex model simulation of neural crest cells (green) within neuroepithelium (blue) and the underlying mesoderm (red). When simulated with low amplitude (6%) shape fluctuations, neural crest cells spontaneously move incompletely toward mesoderm with some remaining within the epithelial layer despite having a high affinity (60% - represented as line tension) for mesoderm.

Movie S9.

Vertex model simulation of neural crest cells (green) within neuroepithelium (blue) and the underlying mesoderm (red). When simulated with high amplitude (16%) shape fluctuations, neural crest cells move completely into mesoderm with some remaining within the epithelial layer.

Movie S10.

Cell surface rendering of an ingressing neural crest cell within the midbrain of a Wnt:Cre2;mTmG embryo (5 somites) demonstrates marked shape fluctuations.

Data, Materials, and Software Availability

The simulation code is available at https://github.com/HopyanLab/Ingression_Simulation. The tool for quantifying ingressed cells is available at https://github.com/HopyanLab/Nuclear_Plotter. The previously published tool for fitting fluorescence lifetime imaging microscopy curves (95) is available at https://github.com/HopyanLab/FLIMvivo.