The Biophysics of Volvox

Robinson Libman, Meryem Louni, Ryan McGibbon, Ali Najjar

Abstract

Volvox are microscopic colonial algae that are heavily studied as they are one of the simplest examples of multicellular organisms. Composed of hundreds to tens of thousands of individual cells moving in harmony, they are the source of several mesmerizing phenomena, each of them serving a purpose for the survival of the colony. This essay describes the main biophysical properties of Volvox that have allowed this organism to thrive in its aquatic environment for so long without having to undergo considerable evolutionary change. First, the modalities of the colony’s flagellar synchronization through hydrodynamic interactions will be studied and pave the way to explain how Volvox can perform hypnotizing dances. Then, its ability to orient its motion towards light and collect it efficiently will be covered. Lastly, we will complete our analysis with how a Volvox colony’s life cycle starts, by exploring the dynamics of the process by which its embryo turns itself inside out. The complexity of these characteristics makes their understanding challenging. Therefore, it is crucial to determine the benefits they bring to the organism to convert assumptions into explanations, and possibly into design solutions.

Introduction

The Earth’s earliest life forms were unicellular, and after nearly four billion years of competition and evolutionary development, the majority of organisms remain so. This underscores the great adaptability and resilience of the unicellular body plan in terms of survival. Consequently, it leads us to consider the origin of the diverse multicellular lifeforms that currently live on our planet (Kirk, 1997, p. 7). Multicellular organisms in the plant, fungi, and animal kingdoms branched off from their eukaryotic ancestors over a billion years ago, but the scarcity of fossil records leaves gaps in our understanding of how this diversification occurred (Kirk, 1997, pp. 10-12). In contrast, Volvox experienced a relatively recent transition to multicellularity, which occurred approximatively 200 million years ago during the Triassic Period (Umen & Herron, 2021). During this period, the multicellular Volvox carteri diverged from its unicellular relative algae Chlamydomonas reinhardtii (Herron, 2016). The volvocine algae group, which includes Volvox and its relatives, consists of a range of closely related organisms with varying sizes and complexity, ranging from unicellular species to undifferentiated colonial species to even larger multicellular differentiated colonies, as depicted in Fig. 1. These distinctive characteristics make Volvox and its relatives within the volvocine green algae group an excellent model system to study the pivotal evolutionary shift from unicellularity to multicellularity (Umen & Herron, 2021).

Fig. 1. Approximate phylogeny (chronogram) of the volvocine algae group, including time periods and developmental changes (Umen & Herron, 2021).

Despite more than 200 million years of independent evolution, the gene content of the unicellular Chlamydomonas reinhardtii and the multicellular Volvox carteri remain almost identical (Herron, 2016). Some of the most important differences in gene content include the gene families involved in DNA repair, the formation of the extracellular matrix, protein kinase activity, and cell adhesion, all of which either originated or expanded during the course of colony formation. In the initial stages of multicellularity evolution within volvocine algae, changes in cell regulation likely played an important role by adapting the cell cycle to accommodate increasing cell numbers per colony and achieving a more standardized cell division pattern. Additionally, the formation of physical connections between cells via cytoplasmic bridges marked another milestone in the progression toward multicellularity. Finally, a crucial step in the transition to multicellular complexity was the differentiation of formerly uniform, generalist cells into distinct types responsible for specific functions, as previously discussed (Umen & Herron, 2021).

Volvox are multicellular organisms comprised of large spheroid colonies ranging in cell numbers from 500 to 50,000 (Herron, 2016). The spheroids are several hundred micrometers in diameter, radially symmetric, and all the cells are embedded into the extracellular matrix (ECM). The ECM, a translucent structure abundant in glycoproteins, occupies nearly 99% of the colony’s total volume (Umen, 2020). Remarkably, the ECM evolved from the simple cell wall of their unicellular ancestor, Chlamydomonas. Most glycoproteins in the ECM consist of modules or functional subunits that form modular mosaic proteins. They can be combined in numerous orientations, granting them high flexibility and adaptability. Several ECM proteins are cross-linked, which provides structural support that helps maintain the shape of each cell and the colony as a whole. Beyond structural roles, the ECM also contributes to growth, development, reproduction, and responses to environmental stresses. Numerous glycoproteins are responsible for orchestrating these biological processes (Hallman, 2003).

Volvox is commonly referred to as a colonial organism, but the cellular differentiation observed during its evolution implies that these colonies are in fact a single multicellular organism. Cellular differentiation in Volvox involves the complete division of labor into two distinct cell types: somatic and reproductive. Typically, there are a few (usually 12-16) large reproductive germ cells and numerous small, biflagellate somatic cells (Herron, 2016). The smaller somatic cells are positioned around the outer surface of the spheroid, with mostly even spacing between cells, each adorned with two flagella extending beyond the boundary of the extracellular matrix to facilitate efficient motility (Matt & Umen, 2016). In contrast, the larger reproductive cells, known as gonidia, are positioned in the posterior region, just beneath the surface of the spheroid (Umen, 2020). Micrographs showcasing these two distinct cell types are shown in Fig. 2. Since Volvox colonies possess differentiated cells (more than a single cell type), they qualify as multicellular.

Fig. 2. Lightfield micrograph of an adult vegetative Volvox spheroid with two cell types: ~2000 small somatic cells (enlarged on the right) and ~16 large asexual reproductive cells (enlarged on the left) called gonidia. The anterior (A) and posterior (P) poles are labelled (Matt & Umen, 2016).

Volvox follow a facultatively sexual reproductive strategy, and they have a haplontic life cycle, meaning that they are haploid during all active phases, except when producing dormant, desiccation-resistant diploid zygotes capable of surviving harsh conditions. Typically, Volvox engage in asexual reproduction, with the sexual cycle triggered only by environmental stressors such as heat shock or nitrogen deficiency (Umen & Herron, 2021). When one of these conditions arise, a glycoprotein hormone known as the sex inducer (SI) triggers the sexual cycle. Vegetative (haploid) gonidia undergo a modified embryogenesis to develop into males (with V chromosomes) or females (with U chromosomes). The male then fertilizes the female (syngamy), producing diploid zygotes that later mature into dormant, thick-walled zygospores. The zygospores can remain viable for years in a frozen or desiccated state and will only start germination and meiosis upon exposure to light and nutrients. This will then produce three polar bodies and one haploid progeny that re-enter the vegetative life cycle, as shown in Fig. 3.

The asexual cycle can be synchronized with a 16-hour light to 8-hour dark regime. One full vegetative reproductive cycle takes 48 hours to complete, with two light-dark cycles (Umen, 2020). The asexual cycle starts with the mature, pre-cleavage adult Volvox colony, when each germ cell (gonidia) undergoes embryogenesis to form four-cell stage embryos that undergo cleavage and inversion. Each embryo then produces a new miniature juvenile spheroid (still residing inside the mother spheroid) through cell differentiation. These juveniles then mature within the mother spheroid before finally hatching. The somatic cells of the mother spheroid will then deteriorate until they experience a drastic decrease in viability about 100 hours following embryogenesis, after which they will die (Matt & Umen, 2016). Since this loss of viability can be delayed by treating a Volvox population with a protein synthesis inhibitor cycloheximide, cell death seems to be an active process. The total lifetime of Volvox can vary from 4 days to 5-7 days after initial formation during embryogenesis (Desnitskiy, 2021).

Fig. 3. Schematic of the Volvox vegetative (haploid) and sexual life cycle phases (Umen, 2020).

Volvox typically live in freshwater habitats, although they have also been found in soil, rivers, ice, and snow (Herron, 2016). They exist in shallow puddles, ponds, and ditches but can be most easily found in deeper lakes, reservoirs, lagoons, and ditches where rainwater is more abundant (Chamberlain, 1915). Volvocine algae are cosmopolitan, meaning they can be found all over the world, as they have been collected in all continents except Antarctica (Umen, 2020). Given their simplicity, Volvox are an ideal organism for studying the physical properties of microorganisms and how they interact with their aqueous environments. While the organism in question is simple, the biophysics behind it are quite complex, which is why we will now discuss in detail how Volvox synchronise flagellar beats, form hydrodynamic coupled bound states, swim towards light, and invert their embryo to start new life cycles. As such, we can take a very simple organism and derive sophisticated design solutions that could be useful in future applications.

Movement synchronization through waves in water

The importance of flagellar coordination and its evolution in Volvox

The flagellar beat has a crucial role in the development, motility, and sensing of Volvox. Indeed, it influences how the assembly of cells performs phototaxis to undergo photosynthesis and produce energy, as well as how the colonies interact between each other. As Rothschild discovered in 1949 with bull sperm cells, whip-like features of various biological organisms such as flagella or cilia are able to synchronize their beating (Brumley et al., 2014). Another example is the sweeping of an egg along the fallopian tube, thanks to the synchronized movement of the cilia lining the latter (Dayal et al., 2001). This remarkable property applies to volvocine algae and has been conserved through evolution, as both unicellular (Chlamydomonas reinhardtii) and multicellular organisms (Volvox carteri) of the family demonstrate the ability to coordinate their flagellar beat, to maximize propulsion efficiency (Rüffer & Nultsch, 1998).

A process only dictated by hydrodynamic laws

First, Volvox are influenced by signals originating from both cells themselves and their surroundings, which enable the colony to behave sychronously, and fulfill its required needs. Therefore, interactions with light, molecules and other cells are frequent. These external interactions could lead to flagellar coordination through chemical signals or direct physical contact (Gilles et al., 1985). However, while the underlying mechanisms remain complex, it has been proven that this phenomenon is solely dictated by hydrodynamic coupling. Indeed, experiments have been designed for cells to communicate only through the fluid waves around them. For instance, Brumley et al., in 2014, held somatic cells of Volvox with different micropipettes and analyzed movements with high-speed imaging, as seen in Fig. 4. The outcomes showed that two cells, whose initial beatings are distinct, end up taking on the same beating frequencies through the common fluid. This process can be modeled with two hydrodynamically coupled oscillators. In conclusion, the beating of one flagellum creates a wave which induces an elastic response to the shear stress of neighboring flagella, and the phenomenon spreads across the entire assembly of cells (Brumley et al., 2015).

Fig. 4. Experimental apparatus employed to prevent chemical signaling and physical contact between two Volvox flagella (Brumley et al., 2014).

The relationship between flagellum separation and their ability to coordinate

Flagella beats are well described as waves and are therefore modeled with equations of the form $A(x, t) = A_0 \cos(kx - \omega t + \varphi)$ , with w being the angular frequency, k being the wave number and φ the phase shift. Synchronous pulses allow flagella to create metachronal waves, whose wavelength is constant and guarantees coordination of neighboring flagella.

These waves form by merging of different regions where phases differences between waves spontaneously happen to be the same. It is crucial that flagella are not too rigid, otherwise hydrodynamic interactions would not be sufficient for this process to occur (Niedermayer et al., 2008). Nevertheless, why does this process occur? In fact, as it is often the case, nature has chosen the most efficient path! This wave configuration is the most favorable in terms of energy dissipation towards the surrounding fluid (Taylor, 1951). Consequently, flagellar coordination through metachronal waves is the most efficient manner to organize all cells’ beatings, thus providing them with a consistent relationship in their phase differences.

For close cells, a phase lock phenomenon is put into place, where their respective flagella gradually take on the same beating frequency through the diminishing of the phase differences between them over time. Moreover, if the distance between two cells is small enough, the synchronized state remains permanent and is called phase lock. Nevertheless, the higher the distance between cells, the higher the likelihood of developing a phase difference. Quantitatively, taking $L = \frac{d}{l}$ , the cell-cell separation (d) divided by the average flagellar length of each pair (l), it appears that with L ≥ 2, hydrodynamic coupling becomes negligible (Brumley et al., 2014). This means that the phase difference Δ, defined as \Delta = \frac{\phi_1 – \phi_2}{2\pi}, increases almost linearly with time with a steeper slope as d increases, as shown in Fig. 5.

Fig. 5. Evolution of the phase difference at four different interflagellar spacings, as a function of the time s. Over time, all flagella develop a phase difference, but in various proportions depending on their respective distance (Brumely et al., 2014).

However, for small values of d, long phase lock periods have been observed to be punctuated by phase defects caused by various bias in the environment surrounding a cell. In 2015, Brumley et al. characterized this process by defining C, a parameter controlling the extent of frequency bias. For instance, in their experiment, C_volvox= 0.0724. They found that for C > 0.007, phase slips emerge and alter waves, as seen circled in white in Fig. 6. It is important to note that these phase slips are temporary, and that for close enough cells, the influence of metachronal waves eventually overcomes these defects, allowing flagellar beatings to synchronized again.

Fig. 6. Experimental simulation of the appearance of phase defects, depending on the importance of external biases. The higher the value of C, the more biases, and the more phase slips which can be identified on both graphs by wave irregularities. (Brumley et al., 2015).

Finally, thanks to spontaneous similarities in phase differences of distinct flagella, colony-wide metachronal waves arise, thus enabling all other filaments to be coordinated, and Volvox to sustain its needs. Due to interactions between the colony and its medium, phase defects regularly emerge and temporarily interrupt phase locks.

Flagellar coordination in antiphase beating patterns

In addition, an exceptional characteristic of Volvox is that they can synchronize in phase, or in anti-phase. This process is not fully understood yet, but researchers believe it could optimize nutrient uptake and increase swimming efficiency. By having some cells rotate clockwise and other counterclockwise, the colony is able to control its motion more accurately, and limit inertia (Leptos et al., 2013). The coupling strength of both in-phase and anti-phase beatings can be compared with the constant $k = \frac{\varepsilon}{\omega}$ , which still depends on the cell-cell separation, as evidenced by Fig. 7. ε is the coupling strength, and dividing it with the average beat frequency ϖ makes it nondimensionalized, thus obtaining k. Researchers found that for in phase beating k = 0.016, while for antiphase beating k = 0.014 (Brumley et al., 2014). On the one hand, this difference is slight and could result from uncertainties inherent in the experiment. On the other hand, it confirms the hydrodynamic coupling theory and its space dependency! Indeed, in-phase pairs of flagella are on average closer than anti-phase pairs, which explains why their coupling strength is higher. Moreover, despite being different, these κ values are close, which disproves flagellar coordination through chemical means. If that was the case, their respective coupling strengths would not be similar, as their associated flows show substantial differences.

Fig. 7. Relationship between the coupling strength κ of two flagella depending on their separation , for both phase and anti-phase beating patterns. Notice how although the inversely proportional relation is the same, the coupling strength is consistently higher for in phase beating. (Brumley et al., 2014).

Dancing Volvox

Artistic swimmers, or people who enjoy water aerobics are not the only ones who tend to dance in water. In fact, Volvox colonies do this all the time, though not as a hobby, but as a lifestyle. The movement patterns of Volvox in water were first analyzed by Drescher et al. (2009), who have identified two unique dances, or stable bound states between colonies, which they have adequately referred to as the “waltz” and the “minuet”. These hydrodynamic motions have also been reproduced through simulations by Ishikawa et al. (2020) to further analyze their stability. It is now our turn to dive in and dance with Volvox.

How Volvox move in water

We have already discussed how Volvox colonies beat their flagella in a synchronous manner to swim in water, but we have not touched upon the movement patterns induced by this phenomenon. Dreschner et al. (2009) observed the movement of Volvox carteri in a glass chamber filled with water. They saw that synchronous beating led a spherical colony to spin in a clockwise manner, while always having the intention of going vertically upwards. This is because the colony is constantly in the process of sinking due to its density being slightly larger than that of water, a process called sedimentation. The colony is also slightly bottom heavy, meaning that there are more internal daughter colonies located towards the bottom of the sphere, which helps with stability. In the case where a disturbance would shift the colony from its main axis, Dreschner et al. (2009) saw that it would adjust its flagellum beating to reorient itself towards the vertical once again. In other words, it would engage in self-righting before continuing its upswimming.

Volvox colonies tend to increase in radius during their lifetime, not because the number of their somatic cells increases, but rather because new extracellular matrix is added, affecting everything related to their movement in water, as analyzed by Drescher et al. (2009), including their rotational speed, upswimming speed, self-righting, and even sinking. The changes in Volvox movement in relation to the radius of the colonies were plotted and are shown in Fig. 8. As can be observed, the greater the radius, the lower the upswimming speed, and the greater the sedimentation speed. Smaller Volvox can swim upwards faster than their sedimentation speed, but larger ones are not as lucky. There is a critical size upon which a Volvox colony can no longer go upwards, as it will no longer be able to overcome the sedimentation. This critical radius has been found by Dreschner et al. (2009) to be of about 300 μm. It is whether a Volvox colony swims to the surface or sinks to the bottom that determines what dance it will perform.

Fig. 8. The effect of colony radius R on (a) upswimming speed U, (b) rotational frequency ω₀, (c) sedimentation speed V, and (d) self-righting time τ(Adapted from Drescher et al., 2009).

The waltz

When a colony swimming upwards eventually reached the top wall of the glass chamber put in place by Drescher et al. (2009), it would get drawn towards another one close by and they would start spinning around each other in the clockwise direction in a motion similar to a “waltz” (see Fig. 9a). The researchers came to the following conclusion: when Volvox beat their flagella, they are thrusting the water down to push themselves up, but at the top, there is no more water above the colony to thrust from, dragging the water spread along the surface towards them instead, as demonstrated in Fig. 9b. This ends up attracting another colony along the surface since it gets pulled by the flow of water induced by the first colony. The waltz can then begin.

Fig. 9. The waltz in action. (a) Top view of two colonies waltzing represented as a superposition of still images taken 4s apart. The scale bar represents 200 μm. (b) Top view visualization of the flow of water generated by a colony at the surface (Adapted from Drescher et al., 2009).

Based on the experiments of Drescher et al. (2009), the orbiting of colonies around each other is not due to chemical interactions, such as chemotaxis, but rather to hydrodynamic effects from the colonies spinning around themselves, since the flagellum beating has not stopped. Given that both colonies rotate in the same clockwise direction, if one were to look at the thin layer of water between them, the flagella of each colony would be beating in opposite directions, creating an area of about zero velocity, which Drescher et al. (2009) estimated to be a no-slip wall. This factor and near-field effects coming from lubrication forces between the colonies[1] induce a viscous torque upon them. Adding the shear stress upon the colonies generated by their flagellar beat as well as repulsive forces between the flagella of the two colonies, as described by Ishikawa et al. (2020), we get the full set of forces that interact and help form the waltzing bound state. Ishikawa et al. (2020) have also mentioned that there is a small decrease in the rotation speeds of the colonies due to the viscous effects from the thin layer between them slowing down the flagellar beating.

[1] Forces induced by a fluid pushing an object away as a gap between it and a wall or other object gets narrower.

Fig. 10. Forces diagram of two Volvox colonies in a waltzing bound state, where U is the orbiting direction, T is the torque induced by the flagellar beating, q is the traction force induced on a colony surface by the shear stress f_s from the flagellar beat, and F is the net force generated by the partial traction forces. The subscripts of U, T, and F represent the axis on which they are acting and the superscripts in parentheses represent the colony they are acting upon. Notice how $f_s^{(1)}$ and f_s^{(2)}are in opposite directions in the A’ box, reducing the net traction induced at the bottom of colony 1 and at the top of colony 2 (Ishikawa et al., 2020).

The element that contributes to the orbit velocity the most has been calculated by Ishikawa et al. (2020) to be the torque of the flagellar beat on the colonies, while the one that hinders it the most is the force generated by unequal tractions. Indeed, there is a traction force that applies tangentially to the colony surface in the opposite direction of its rotation. However, in the area between the colonies, the traction forces are opposite, in a similar way to how the fluid velocity is near zero, meaning that there is a much greater traction force on the opposite side of a colony than on the side where it is close to another one. This net force is opposite in direction to the orbiting motion, reducing the speed, but not being enough to entirely cancel it. Fig. 10 clarifies this idea.

Due to the presence of this hindering force, the orbiting velocity of the waltz is much smaller than the rotational velocity of the colonies. In fact, it has been measured by Drescher et al. (2009) that two similar sized colonies (with radii of about 150 m) would spin around themselves with an angular velocity ω of about 1 rad/s, and would waltz around each other with an orbiting frequency Ω of about 0.1 rad/s. They came up with the following equation:

$\Omega \approx 0.069 \ln\left(\frac{d}{2R}\right)\omega$

where d is the distance between the colonies and R is the radius of each colony. Of course, the colonies are assumed to be of same size in this case.

The bottom heaviness of the Volvox colonies is key for maintaining the waltz, as it provides stability. In fact, Drescher et al. (2009) theorized that if not for this characteristic of the colonies, the dance could get disrupted if another colony were to reach the top and start pulling the water towards it. Ishikawa et al. (2020) went further with this idea and found critical values for bottom heaviness for which the waltz would be stable. They also considered the sedimentation speed of the colonies in their stability calculations. They determined that there is a cutoff bottom heaviness amount required for a stable waltz under which the colonies would simply end up repelling each other due to instability in the motion.

Fig. 11. Center-to-center separations between two colonies (rescaled by their radius α) as a function of time t until collision time t₀. The experimental curve comes from the observations of Drescher et al. (2009), while the simulation curve is provided by Ishikawa et al. (2020).

Observing the case where the waltzing motion is stable, Drescher et al. (2009) noticed an increase in the speed at which the colonies would approach each other as the distance between them decreased, until they would eventually collide. Ishikawa et al. (2020) came to the same conclusion through their simulations (see Fig. 11.) and explained that lubrication forces, which would usually prevent collision between two spherical objects from occurring, do not stop the colonies from coming into contact since they have flagella and are therefore not smooth spheres.
Drescher et al. (2009) have theorized that a possible evolutionary advantage of the waltz could be an increase in fertilization opportunities, as one male colony’s sperm packets could come into contact with a female colony. They have also observed that the waltzing bound state can even involve long chains of Volvox, all orbiting around each other (see Fig. 12.), largely increasing the chances of fertilization.

Fig. 12. A long chain of waltzing Volvox (Adapted from Drescher et al., 2009).

The minuet

Let us now fast forward a couple of hours into the Volvox colony’s lifetime. It has now surpassed 300 μm in size and is too large to fight its sedimentation speed through upswimming, so it sinks towards the bottom. As the colony approaches the seafloor, its sedimentation is balanced out by the water pushing back on it from the bottom surface, creating a hovering state. How low the colony hovers depends on its mass, or how much greater the sedimentation speed is compared to its upswimming speed. Ishikawa et al. (2020) have found the limit distance from the bottom to be of about $\frac{R}{8}$ (R being the radius of the colony). This distance is said by Drescher et al. (2009) and Ishikawa et al. (2020) to be too large for lubrication forces to be considered in their calculations.
Given that the hovering height depends on colony mass, two colonies of different mass can end up in a state where one hovers above the other. Ideally, they would be perfectly aligned, in a relaxed state, but the situation gets much more complex, as described by both teams of researchers. One colony’s rotation induces vorticity (swirling movement in the fluid), which creates a downward and outward flow pushing the other one away, while the other’s bottom heaviness allows it to stabilize by providing it with restoring torque and allowing it to swim back towards the center. This constant back and forth creates an oscillating pattern between the colonies, resembling a “minuet” dance (see Fig. 13.).

Fig. 13. Side view of the minuet between two colonies in action. p₁ and p₂ represent the anterior-posterior axis of the colonies, showing that the minuet disrupts their alignment with the vertical. The scale bar represents 600 μm (Adapted from Drescher et al., 2009).

Ishikawa et al. (2020) have done many simulations to determine in which situations this dance is stable. In all cases, it is the level of bottom heaviness of the colonies that dictates whether the minuet motion will persist, deviate, or stabilize. This bottom heaviness has been quantified by them through a G_bh parameter that is defined as a ratio between two timescales: that required for the colony to swim a distance equivalent to its radius, and that required for its self-righting in the case where a disturbance shifts it from its vertical axis. Given that a colony that is more bottom heavy will be able to realign with the vertical faster, the denominator of the ratio will be smaller and the G_bh will be larger. In other words, the greater the bottom heaviness, the larger the G_bh value.

Ishikawa et al. (2020) first analyzed the motions of the colonies when they are not too far apart in the vertical direction and saw that the greater the bottom heaviness of the colonies, the smaller the amplitude of motion of the minuet, and therefore the more stable the bound state. The colonies have larger restoring torques allowing them to more easily counter the repulsion of the other colony and return to the center. If the bottom heaviness of the colonies is large enough (G_bh ≥ 5), they end up completely stabilizing into an aligned position. On the opposite side of the spectrum, if the bottom heaviness is too low (the boundary is less well defined, but G_bh is somewhere below 2), they end up deviating from each other and breaking out of the bound state.

The simulations also showed the emergence of three-dimensional movement in the minuet motion after a couple of seconds have passed, as oscillations in only two dimensions proved to turn unstable. The three-dimensional movements were observed to eventually turn into an orbiting motion (if the minuet had not broken off or completely stabilized). However, this new orbit differed from the one observed in the waltz as the direction of orbit was opposite and the colonies were not at the same vertical height.

In the case where the Volvox colonies have an even greater hovering height difference but are still close enough to engage in the minuet, the simulations have shown the formation of a stable pattern at smaller bottom heaviness values. The height between the initial positions of the simulated colonies had been set to double that of the first round of simulations, yet the induced minuet motion had half the amplitude and the G_bh was equal to 1, where the motion would have been too unstable to persist in the former cases. Smaller G_bh values were also tested in this increased vertical separation model. With G_bh = 0.1, the motion diverged, while with G_bh = 0.3, the minuet motion diverged, then reconverged at another center before diverging once again. Refer to Fig. 14. for a comparison between both sets of different simulated motions.

When the researchers attempted to mathematically define this more ‘far field’ hydrodynamic model[1], they observed discrepancies between their calculations and the simulations, concluding that there were still near field effects[2] involved in the minuet motion, even when the colonies are quite distanced vertically. However, while they realized that these near field effects are important for the bound state to occur (or not), they could not describe when exactly they come into play for a given minuet bound state.

[1] A model where the fluid dynamics are observed at a distance from the source of disturbance.

[2] Effects occurring close to the source of disturbance.

Fig. 14. Simulated movement patterns of a minueting pair of Volvox for different height separations and G_bh values. In each case, the starting points are in black, and the ending points are in white. Each shape represents one of the two colonies. The letters in yellow represent the experiments where the Volvox had a small height separation, where (a) G_bh = 2, (b) G_bh = 3, and (c) G_bh = 6. The letters in green represent when the colonies were further apart vertically, where (a) G_bh = 0.1, (b) G_bh = 0.3 and (c) G_bh = 1 (Adapted from Ishikawa et al., 2020).

Light-regulated photoreceptors and light-depending movements

It has been observed that a Volvox colony can coordinate itself to move towards light. Volvox shares the same quality with all other living organisms in that it needs energy to survive. As well, it is an autotroph so it can create its own energy through photosynthesis. However, it requires sunlight to do so. Therefore, Volvox has evolved to be able to move towards the sunlight when needed to survive. The Volvox colony’s somatic cells and their flagellar beating has amply been discussed, though these cells also possess an eyespot containing photoreceptors in order to recognize light and adapt the colony’s movement to proceed in that direction. As described before, Volvox always turns clockwise during its movement. When illuminated on a single side, the constant rotation of the cells during movement creates the effect that cells are always going in and out of light, creating dark and light phases. Therefore, cells are always changing the movement of their flagella and it is this change that will direct the cells in the correct direction to move towards the light. To add, another important fact worth noting is that the eyespots of the anterior cells are more sensitive to light and larger than those of the posterior somatic cells (Edeltraud Ebnet et al, 1999).

Physical observations

A helpful article written by S.J. Holmes provides an explanation to the movement of Volvox on a more physical level. As mentioned before, the cells of Volvox differ around the colony producing an anterior and a posterior end. Therefore, we can define forward movement as movement in the direction the anterior side of the colony is pointing. This paper elaborates on the dark and light spots described earlier. The path of Volvox was shown to be extremely precise and almost always in a directly straight line following the path with the highest intensity of light. The way in which it does this is that the flagella on the posterior side of the colony are highly stimulated when in the dark. This creates the effect that once the colony is in the light it will stop moving but when in the dark it will move away from the areas of no light (S. J. Holmes, 1909).

Another observation that can be made on a larger and more physical scale is the tilting of the colony. Volvox makes strong use of the function of gravity to tilt the anterior side upwards. Whenever the colony comes to a rest and stops swimming, the back end starts to sink rapidly and more deeply than the front end. This is due to the fact that the somatic cells of the posterior end are more tightly packed together than the anterior end giving that side more mass, which goes back to the concept of bottom heaviness discussed earlier. A denser object will sink more quickly, thus the colony’s anterior end is tilted slightly upwards. This is advantageous as the anterior eyespots are stronger and will detect light much better. Not only will this bring the anterior cells closer to the light, but it will also help match the angle of the light rays making detection easier as well (Noriko Ueki et al, 2010).
A final physical observation that can be made is how Volvox controls its speed when it comes into contact with light. Since the anterior cells possess more capable eyespots, they will detect light first and thus react by changing the beat of their flagella to oppose motion (Fig. 15.). The posterior cells will continue propelling the colony forward, but with the anterior cells now opposing the movement, the colony will come to a stop and rest in the light (Noriko Ueki et al, 2010).

Fig. 15. Schematic representation of the phototactic movements in V. rousseletii (Noriko Ueki et al, 2010).

Light detection

As with most living organisms that can detect light, Volvox makes strong use of opsins, which are G-coupled protein receptors capable of detecting light and signaling it to a cell. There are two types of opsins commonly found in cells: cones and rods (Akihisa Terakita, 2005). The opsin used in Volvox is a rod type called rhodopsin. Opsins make use of the fact that light behaves both as a wave and a particle, meaning that light carries energy and can transmit it through an interaction with a molecule as proven by the photoelectric effect. Therefore, opsins react by using the energy from light once it comes into contact with the protein to lower the potential energy barrier for converting from a cis to a trans isomerization (Fig. 16.). This change in conformation allows the protein to now be in an active state for the G protein to bind and can thus start a signaling chain reaction alerting the cell of the presence of light (Zhou et al, 2012). The previous paragraph contained very important information for understanding the first step of light recognition by Volvox. An article published by Franz-Joseph Braun and Peter Hegemann demonstrates the process by which Volvox continues the signal produced by the use of photo currents through voltage gated channels. One interesting piece of information is that the study of this system is made possible by the fact that the process is significantly lower in Volvox than other organisms. This is because the rotation and movement speed are much slower than other organisms since it functions as a large colony as opposed to a single cell and therefore does not require faster signals since it would not move fast enough to make use of it. As such, the following discoveries have been made much easier. The photocurrent is carried by the movement of Ca²⁺ and H⁺ ions. During their tests, Braun and Hegemann discovered a huge cation influx during times when light pulses were directed at the cell. It should be assumed that Cl^– ions are present in order to maintain equilibrium when needed but tests show there is an absence of anion influx (Franz-Josef Braun et al, 1999).

Fig. 16. Ligand binding and conformational changes in rhodopsin (X Edward Zhou et al, 2012).

Light activation signal processing

Now onto the most important connection of the process. Just like action potential used by the human nervous system, Volvox has shown to also use ion influx and efflux to control movement. In Volvox, this action potential is driven and controlled by rhodopsin, the protein which has proven to be able to detect light through changes in conformation. The link between the two was proven in the study of Braun and Hegemann. To begin, the photocurrent rise as described using a mathematical function follows with what would be predicted if the current was activated by rhodopsin (using an intermediate). This takes into account that the function is modified by depolarization, which is also expected. As well, Braun and Hegemann found that the correlation between the rise in photocurrent and the number of rhodopsin molecules affected by the light pulse were directly proportional, once again confirming the assumption that photocurrent is controlled by the activation of the rhodopsin protein. The rhodopsin molecule controls the current of Ca²⁺ by direct contact during the protein’s active state using cation conductance. As well, it controls the flow of H⁺ through the use of inductance (Fig. 17.). However, it is important to note the contact as well as the inductance are not applied directly to the voltage channels but instead to an intermediate where they will enter an amplification chain containing G-proteins to deliver it to the channels (Franz-Josef Braun et al, 1999).

Fig. 17. Model of rhodopsin-triggered conductances (Franz-Josef Braun et al, 1999).

Dynamics of a Volvox embryo turning itself inside out

As briefly mentioned in the vegetative (asexual) cycle of Volvox, these algae have spherical embryos that need to turn themselves inside out in a process called inversion in order to achieve the adult configuration. After undergoing a series of 11 to 12 cell divisions (cleavages), which takes from 6 to 8 hours (Matt & Umen, 2016), the embryo consists of several thousand cells arranged in a spherical monolayer (i.e., a cell sheet at the surface) contained in an embryonic vesicle or fluid bag. The cells are linked to each other by cytoplasmic bridges, thin tubes composed of cell membrane that result from incomplete cell division (Höhn et al., 2015). At the sixth cell division, 16 of the anterior cells divide asymmetrically to produce small and large daughter cells. The large cells will then complete a few more asymmetric cell divisions before halting cleavage. These larger cells will later become the next generation’s gonidia. As for the smaller cells, they will continue with cell division until the end of embryogenesis. They will become the somatic, flagellated cells. At the end of cell division of the embryo, the 16 larger cells protrude outward, as shown in Fig. 18., and the flagella of the somatic cells point inward. Consequently, the embryo will need to turn itself inside out to allow the flagella of the somatic cells to protrude outward and the future gonidia to be placed on the inside (Matt & Umen, 2016). The total duration of inversion in Volvox is usually between 45 and 80 minutes (Höhn et al., 2015).

Fig. 18. Volvox embryogenesis showing the anterior hemisphere of a gonidium undergoing cell division (cleavage). The 16 cells marked with a yellow dot show the daughter cells resulting from asymmetric cleavage after the 6^th and the 7^th divisions Scale bar, 10 μm. The black arrowhead in the post-cleavage embryo indicates the phialopore (Matt & Umen, 2016).

By the end of embryogenesis, a cytoplasmic bridge network connects the cells together in a coherent cell sheet, except at the cross-shaped opening at the anterior pole called the phialopore, shown in Fig. 18.. These cytoplasmic bridges are important during inversion as they provide a stable structural framework against which cells can exert forces to complete the process of inversion. They are also flexible, allowing the cell sheet to bend backwards (Matt & Umen, 2016).

There are two types of deformation sequences during the inversion process, depending on the specific Volvox species: type-A and type-B inversion. In type-A inversion, four lips open up at the anterior pole of the embryo through the phialopore, curling outward, and peel back to achieve inversion (Höhn et al., 2015). On the other hand, type-B inversion begins with a circular invagination (folding back on itself) near the equator of the embryo. The posterior hemisphere then moves inside the anterior hemisphere while inverting itself. Throughout this process, the phialopore widens to allow the anterior hemisphere to peel back over the already inverted posterior hemisphere (Haas & Goldstein, 2018). These two types of inversion sequences are illustrated in a few steps in Fig. 19.c.

In both inversion types, the widening of the phialopore is crucial to allow the cell sheet to pass through it. In type-A inversion, a simple sequence of cell shape changes is sufficient to invert the cell sheet since the process is facilitated by the four outward-curling lips. At the beginning of inversion, the somatic cells of the embryo elongate to become spindle shaped, as illustrated in Fig. 19.b. This results in a 40% reduction in their diameters, reducing the size of the embryo by 10%. This contraction results in the separation of the embryo from its fluid vesicle, decreasing tensile stress enough to allow the lips at the phialopore to start curling outwards. Then, a few cells near the phialopore become flask-shaped, with long, thin stalks (Fig. 19.b). This cell shape change localized at the opening allows for additional curling of the lips. Finally, the cells become columnar and more compact (Fig. 19.b). This sequence of cell shape changes propagates in a wave-like manner from the anterior pole towards the equator until half the embryo is inverted. By this point, tensile stress has accumulated at the equatorial bend region. This would cause the posterior hemisphere of the embryo to undergo an “elastic snap through”, meaning that it would flip inside-out very rapidly. In fact, the posterior hemisphere inversion occurs approximately 7 times faster than the inversion of the anterior hemisphere. This second part of the inversion process is therefore not driven by cell shape changes but by mechanical forces acting to dissipate the accumulated stress at the bend region (Matt & Umen, 2016).

In comparison to type-A inversion, the sequence of cell changes of type-B inversion is much more complex as the type of cell shape changes involved varies depending on the parts of the cell (Haas & Goldstein, 2018). The type-B embryo inversion dynamics are shown in Fig. 20. The driving forces of the type-B inversion are local changes in intrinsic curvature, and active contraction and expansion (Höhn et al., 2015).

The distance between the posterior pole and the bend region (illustrated in Fig. 20a) continuously decreases over time during the inversion process, at a speed that abruptly increases by five times after = 0 (around when the bending ring is at the equator). Shortly after this time, the surface area of the embryo peaks due to the bending ring, which adds to the total surface area. It then decreases as the embryo finishes inverting. The minimum (most negative) curvature κ peaks at the same time as the speeding up starts ( = 0), once again due to the bending ring at the equator, which is the most curved part.

Fig. 19. The inversion of Volvox embryos. (a) Volvox spheroid showing somatic cells and gonidia, which become embryos. Scale bar, 50 µm. (b) Median plane cross-section of the cell sheet showing the sequence of cell shape changes during inversion, starting from spindled-shaped [S] to flask-shaped [F], to columnar [C]. The red line shows the cytoplasmic bridges, and the arrow shows the direction of propagation of bend regions. (c) Top row shows type-A inversion, and bottom row shows type-B inversion. The phialopore (P), lip (L), and invagination (I) as well as the anterior pole (ant) and posterior pole (post) are indicated (Haas & Goldstein, 2018).

Fig. 20. Dynamics of type-B inversion in Volvox. The red lines show the data for one representative embryo. (a) Distance e from posterior pole to bend region (normalized by its value e₀ at t ≈ -10 min) plotted against time. (b) Surface area A of the embryo (normalized by its value A₀ at t ≈ −20 min) plotted against time. (c) Curvature κ in the bend region (Höhn et al., 2015).

Conclusion

The physics governing Volvox colonies’ movement and interactions with their environment make this species a biological marvel and have revealed many design solutions to the problems they face. Since Volvox are autotrophs, they use photosynthesis to convert sunlight into the energy they need to survive. However, this raises the problem that Volvox must have access to said sunlight for significant periods of time, which may not always be easy, given their aqueous environment. Thus, they have found a way to orient their movement towards areas of light through the use of the conformational change of a certain molecule activated by interaction with light that allows them to locate its source and swim towards it. However, Volvox colonies face an additional problem: they tend to sink due to their bottom heaviness, which is not ideal, as being closer to the surface would benefit them by allowing for higher light intake. Therefore, they are forced to constantly beat their flagella to swim upwards and counter their sinking in addition to them already having to swim towards light. This sounds like quite a disadvantage, with each flagellar pair seemingly being forced to waste energy on maintaining upswimming, though it has been shown that a phenomenon of flagellar synchronization occurs, where certain flagella that are coincidentally beating at the same frequency will generate a metachronal wave across the colony surface that will influence all other flagella to beat at the same frequency. Not only does this minimize the amount of energy required for the upswimming process, but flagellar coordination also makes colonies resilient against the numerous external disturbances coming from their surroundings that could affect their direction of movement. Unfortunately, the colonies eventually become too large to maintain upswimming and sink to the bottom, though they do not stop their flagellar beating, as it has another very useful function: the synchronized beating leads to colony rotation. This has been observed to create hydrodynamic bound states between two or more colonies, both at the surface and at the bottom of water, where they attract each other and start dancing, allowing for contact and possible mating, which is particularly useful in dire situations (such as a lack of light at the bottom), are observed to make the colonies prioritize sexual reproduction over asexual reproduction. A colony’s bottom heaviness also has its advantages, since it provides stability to the dances, prolonging them and further improving chances of mating. Another design solution to the everyday conundrums of Volvox is how the embryos turn themselves inside out to achieve their adult, functional configuration. After undergoing cell division, the circular embryo is arranged with the larger cells (future gonidia) protruding outward and the smaller cells (somatic, biflagellate) protruding inward. This is a problem because the flagella of the somatic cells should be on the outside to effectively provide motility. To remedy this issue, the Volvox embryo undergoes the inversion process: after a series of cell-shape changes propagating from the anterior pole to the equator, half of the embryo is inverted. This creates an accumulation of tensile stresses at the equatorial bend region. To dissipate these stresses, the solution is for the Volvox embryo to finish the inversion process with an elastic snap through that quickly inverts the anterior hemisphere using mechanical forces.

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