Exploring Thermally Stressed Symbiosis Formation

Stephen Williams¹, Kyra Ruiz¹, 

Elizabeth Heath-Heckman², Erica Rutter¹, Shilpa Khatri¹

¹University of California, Merced.
²Michigan State University.

Abstract

For many organisms, symbiosis plays a pivotal role in their survival. By forming associations with symbionts, host organisms can be provided with supplemental nutrition, environmental management support, and bioluminescence. In an age where global environments are changing at an unprecedented rate, it isn't clear whether these systems will be resilient or even present hidden vulnerabilities to the future of these organisms. One key symbiosis model is in Euprymna scolopes, the Hawaiian Bobtail squid. By hosting the bacteria Vibrio fischeri, the squid is provided with the bacteria's light. This light allows the squid to perform counter-illumination, which provides light during the night to act as active camouflage. Several complex mechanisms co-evolved between the two organisms to initiate their association soon after the squid's hatching. In particular, scolopes have evolved an internal specialized organ, the light organ, which captures and stores the Vibrio. As the squid breathes, debris, particles, and bacteria move through its internal passages. Along the way, they come into contact with the light organ. The light organ is a highly ciliated structure, and through the collective motion of its cilia, the squid can generate local biophysical flows. These flows can direct particles and even filter for particle size, allowing the squid to capture its symbiont. To study this system, we have developed computational fluid dynamics models of these flows using the Method of Regularized Stokeslets. As a result, we have simulated the motion of bacteria within this space and have been working to quantify how the variations driven by changing climates impact the colonization process.

The spiel 

Boxes in poster read left to right, row by row...
1, 2, 3, 4, 5,

6, 7, 8, 9 ,10,...

1. Symbiosis is when two (or more!) organisms coexist. This coexistence is defined by its neutral, one-way beneficial, or two-way beneficial impact on the organisms. Crucially, neither of the pair is net-losing out. Symbiosis has become an area of interest in modern scientific research, particularly since Lynn Margulis (then Lynn Sagan) first introduced the endosymbiont [1]. This is because we have increasing access to complex data and tools to decode the relationships present in this data.  Our research focuses on the symbiotic relationship of the bobtail squid Euprymna scolopes and its bioluminescent partner, the bacteria Vibrio fischeri. Squid can use a special organ they evolved, the light organ, to store and use the glowing bacteria. The glowing allows them to perform counter-illumination, which helps them to offset shadows they cast under moonlight when viewed from above and prevents them from obscuring the night sky when viewed from below. This is very important to avoid predators, particularly when night hunting. Importantly, they acquire their symbionts horizontally, which means after they are born. So, how are they able to do this?

2. When squid want to breath they do so by exanding and contracting their body cavity. This cavity is called the mantle (in red). As a result of this, fluid is drawn inside the mantle from their surrounding environment, and then they are able to pump it out through a funnel-like structure to an exit area. Luckily for the squid, Vibrio bacteria are ubiquitous throughout global marine environments, particularly near other squid, which vent ~90% of their light organ's own stored bacteria each morning. As a result, before long a newly hatched squid will inhale a bacteria from its environment. 

3. Once inside the mantle, a number of key structures are encountered. Crucially, this space contains the gills, from which the squid extracts oxygen from the fluid that passes over them. Further downstream, the flows encounter the light organ. In juvenile squid, this is a bilobed structure, from which two pairs of finger-like protrusions stretch out into the path of the fluid.

4. These appendages from the light organ hold the key to the selective recruitment of bacteria from the flow. The appendages themselves are covered in many hair-like cilia, which can beat. There are two key types of cilia on the surface: (1) near the tips of the appendages, the cilia are long, beating in monochromatic waves to drive the fluid around the appendage surfaces. (2) close to the base of the appendages, the cilia are shorter and beat randomly, offsetting the flows from other regions. The classification of these distinct types of cilia and their role in particle capture was first explored in 2017 [2].

5. Which brings us to the driving question behind our study. How can we model the processes in this system, particularly those present in pre-colonised juvile squid? And, how then can we encode in these models the impacts of thermal stress, to understand how these external stresses can impact the processes effectiveness?

6. As with any fluid-dynamics problem, the first thing to be asked is what approximations are valid for the problem. This allows us to select the appropriate tools to capture what is driving the system. To do this, we can calculate the Reynolds number, a non-dimensional quantity that looks at the relative effect of inertial to viscous forces [6]. In the case of the squid, by considering experimental characterisations of the space and the fluid flow moving through it, we can estimate the Reynolds number to be roughly 0.0001 [7]. That means for our system, we can neglect all those effects driven by viscosity. 

7. What this means is that, in order to capture the fluid flow in our space, we need to solve the Stokes equation [8]. This is a simplification of the famous Navier-Stokes equation where  all the inertial terms have been set to 0. Here, u is the flow of the fluid, and f is some set of external forces which act on our system. Since the Stokes equation is linear it can easily be solved in the following way: using the method of Green's functions, we can solve the flow for a point force. In essence, from this calculation we get a matrix S,  which tells us how a force, located at position X0, translates to fluid located at another point X. We can then add together a superposition of these point force solutions (since the problem is linear) to capture more complex geometries.

8. It seems like at this point that our problem is solved. We can take the geometry of the squid, which we saw in panel 3, determine some forces which capture the boundary conditions and then we have the flow in the whole system. However, it is at this point we need to look at the structure of the solutions. It contains terms of the sort 1/r. That means, very close to point force solutions we will have a flow speed which diverges. This, of course, is not realistic and when applied numerically will lead to all sorts of problems. Instead, we dont use point-forces but regularized force "blobs". These blobs are regions over which a sharp, but bounded force, is spread. As a result of the boundedness we dont get the diverging flow speeds, but we can still find regularized solutions to the Stokes equation. Moreover, if we take the limit of these regularizations to be infinitely small, we return to our "true" solutions.

9&10. We can now look at solution for our system to see what they are like. Here the while lines represent the trajectories of particles moving through the system. The systems is composed of stokeslet at the red positions, distributed uniformly along these surfaces. In the first panel we consider a squid which is constantly exhaling. The boundary conditions on the sides of the funnel are no-slip, on the inlet we get a static Poiseuille flow, and on the surface of the appendages we have the flows measured in [2]. Emerging from this we see a number of regions of interest emerge. Interestingly, we see the region SZ or the stagnant zone. This is an area of relatively slow flowing fluid, which was observed in the experiments, and is just what bacteria in this space need to arrive at the light organ successfully. Next, we have regions VZ1 and VZ2, in order to understand these we turn our attention to panel 10. The Poiseuille flow is now time-dependent, this is our attempt to model breathing dynamics in our squid. We see they move along these jerky motions, until reaching the light organ where the trajectories become somewhat chaotic. However, if we look closely, we see that particles are actually being moved altenately between VZ1 and VZ2... the vortical zones. This means bacteria in certain areas are recirculated multiple times close to the light organ (and so have multiple chances to reach their target region!).

11. So, now we have our model, we can discuss how exactly we can capture the impact of the stress present under heating. [3, 4, 5] are previous studies of cephalopods, which link the breathing patterns of several cephalopods to thermal stress. Within our system obvious parameters which allow us to capture this are U0 (the strength of the breathing, encoded by the magnitude of the Poiseuille flow), Uc (the strength of the ciliary beating), omega (the number of breaths per second), and Y0 (the distance particles start upstream, a proxy for squid size). In our example we see extremes of the phase-space of omega, with all other parameters being held at the nominal.

12. In order to quantify what is happening in our system, we need to come up with some sort of metric, which captures whether bacteria are doing "well" or "badly". Two are proposed. The first is the total time which particles spend at some critical distance from the light appendages. For a particle along the curved red line shown, the sum of the time spent on the yellow parts of this line then tell us this total time. Another possible choice is the maximum time which particles spend in this area without leaving. Again, represented in the example by the yellow portion of the line. These two metrics account for two different effects in the system, the first being the number of chances the bacteria has to be deposited in the right place by chance. The second, the time bacteria have in which to perform taxis. Our model currently doesn't include this effect, but the rational here is that large maximum times represent large times where the bacteria would be able to swim toward the correct location.

13. Now that we have our metrics it remains to determine the impact of variation of the parameter on them. We selected a range of biologically relevant parameters for each case. Here we can see a bi-variate scatter plot of the total time spent in the region close to the light organ in terms of the breathing strength and the initial position. Here, blue points in our scatter represent parameter sets which resulted in simulations with higher than nominal times, while red resulted in lower times. It is clear already that the relationship between our parameters and the metric is non-linear. As such, we must determine some way of looking at their relative importance that is able to account for this non-linear relationship.

14. So, we turn to Sobol sensitity analysis. Here, we have some set of data Y, the idea is to come up with some set of functions, f(x_i), which depend on the different parameters. With f_i(x_i) depending only on x_i. f_i,j(x_i,x_j) depending only on the co-operative effect of x_i and x_j, and so on. We can then use these functions to decompose the variance across some large collection of parameter sets, in terms of the 1st order, 2nd order, etc. effects of the parameters which underpin the data.

15. And, we did just that! For our squid system, we performed N=81920 simulations for different parameter sets. We then decomposed the resulting data's variance. The blue present here shows the variance coming from single parameters. We can see that only the breathing frequency in the case of our system that significantly impacts the system alone. However, if we look at the parameter and all its pair-wise interactions, we then get the total-variance contribution of that parameter; this is the red bar. Here, all parameters can impact the system's output when working with the others. The scatter plot and the trends highlighted in [3, 4, 5] show that the parameter changes corresponding to thermal stress (harder, faster breathing) result in shorter bacterial occupation times. Suggesting that such conditions are adverse to the colonisation process.

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