Self organizing cells (slime molds)

# Question and background:

 Fig. 1: Physarum polycephalum; photo by © Martin Livezey (CC BY-NC-SA 3.0) via eol.org.

How do cells organize themselves? Plasmodial slime molds (e.g. Physarum: Fig. 1) provide an opportunity for anyone to study how cells organize complex structures. Despite their unattractive name, slime molds can be quite beautiful, are harmless, and can be grown cheaply, easily and safely at home. They eat bacteria and — because they dislike dry conditions — they won't escape to colonize your kitchen.

Physarum and other plasmodial slime molds are giant unicellular organisms1. These cells create intricate self-organizing networks of veins through which they pump cytoplasm back and forth to find food patches [1]. Previous studies suggest that the flow through the veins controls the diameter (and therefore conductivity) of the veins, optimizing transport of cytoplasm [1].

The veins have an inner fluid cytoplasm flowing through them ('endoplasm'), surrounded by a tube of gel-like cytoplasm ('ectoplasm') [2][3]. It is not hard to see how the inner diameter could be controlled by the flow: perhaps the cytoplasm fluidizes at high shear rates, so that the inner core expands if the flow is strong and shrinks if the flow is slow. But how is the outside diameter controlled?

# Hypothesis 1:

The inner diameter is determined by the shear rate at the boundary between the gel layer and the flowing fluid; the outer diameter is determined by the internal pressure. When the pressure increases, the gel-like outer layer stretches, widening the vein. For a constant volumetric flow rate, this reduces the shear rate at the boundary between the gel and fluid cytoplasm. Therefore more gel can form, and the gel layer thickens. The expansion stops when the gel layer is thick enough enough to resist the pressure. The reverse response - narrowing the vein and thinning the gel layer - could occur if the gel layer is contractile. Note that the structure of the gel-like layer is actually more complex, with a network of membrane and pores, as well as cytoplasm [2]. The pores may be responsible for transfer of material between the inner fluid 'endoplasm' and gel-like 'ectoplasm' [2], resulting in changes in thickness of the wall of the vein [3].

## Predictions for Hyp. 1:

1. The outer diameter of the veins and the thickness of the gel-like layer will increase as internal pressure increases, for a constant internal flow rate.
2. The outer diameter and thickness will increase with inner diameter (because the tension on the wall for a given internal pressure increases with diameter, based on the Young-Laplace relation).
3. The inner diameter will be independent of internal pressure.
4. The average shear rate at the boundary of the gel-like and fluid-like cytoplasm will be constant, independent of inner diameter.
5. Increasing/decreasing the flow will increase/decrease inner diameter.

## Possible tests of Hyp. 1:

1. Because hydrostatic pressure varies linearly with height in a water column, predictions 1 - 3 could be tested by placing slime mold plasmodia on a vertical surface (in air) and observing the correlation between the inner and outer vein diameters and the height. Only a good macro lens (or dissecting microscope) and a plasmodial slime mold would be required.
2. Predictions 4 - 5 could be tested by observing flow through the veins and manipulating it by cutting the veins. A microscope mounted video camera (or a camera with a strobe) could be used to measure the flow rates, and therefore the shear rates.

## A simple model:

Quantitative predictions for these hypotheses require mechanical models. However the veins of the plasmodium are mechanically complex. There is a material that sheaths the cell {citation needed}, the plasma membrane, a gel-like layer of cytoplasm forming the wall of the vein, and finally there is the fluid cytoplasm ("sol") flowing through the vein. The pressure at any given point will oscillate as the flow goes back and forth during a contraction-relaxation cycle.

 Fig. 2: A model (model 1) of regulation of wall diameter in the veins of the plasmodium. Top: force balance on the wall. Bottom: model predictions of variation in wall thickness with height.

To make a useful model we need simplifying assumptions about the mechanical behavior of these components. One simple model (model 1; Fig. 2) is that the gel is under a constant state of uniform tensile stress (time averaged over the contraction-relaxation cycle). Such a state might involve a constant level of actomyosin contraction {citation needed}, for example. This model assumes we can ignore short-time-scale elastic/viscoelastic behavior. In model 1 the tension in the wall balances the pressure in the fluid core so that the left and right halves of the vein together. The gel exerts a force (fG) per unit length equal to two times the product of the thickness of gel wall (w) and the tensile stress (s). Assuming that the pressure can transmit through the pores of the membrane in the wall to the outer cell membrane [2], the pressure exerts a force (fP) per unit length equal to the time averaged pressure (PA) times the vein outer diameter (or the inner diameter plus twice the wall thickness). Hence, $ws=(R_{I}+w)P_{A}$.

PA has to be greater than zero to balance the tension. Therefore there must be an internal pressure not due to vertical height of the portion of the vein: the instantaneous pressure (P) should be the sum of the pressure due to the contraction-relaxation cycles that drive fluid flow (Pt) [4]2, a pressure (P0) that keeps the vein from collapsing at the top of the plasmodium, and the hydrostatic pressure due to the vertical height (-g*z*d; where g is the acceleration of gravity, z is distance from the top of the plasmodium {z<0}, and d is the cytoplasm density (assumed equal to water). By definition, the time-averaged value of Pt is zero, so $P_{A} = P_{0}-gzd$. Note that there would be some non-linearity to the fluctuations in pressure (and presumably diameter) as the flow goes back and forth in the vein, so this is really only valid if the magnitude of the fluctuations in Pt are small relative to PA. Furthermore, this is assuming that one can time average over the thickness of the wall, which oscillates substantially over the contraction cycle [3]. Entering this into the equation above, and solving for w/RI gives w as a function of inner radius and vertical height:

(1)
$$w/R_{I} =(P_{0}-gzd)/(s-(P_{0}-gzd))$$

Note that in this model, there is a definite height limit for the plasmodium determined by the pressure at z=0 (P0) and the tensile stress in the wall (s). The results for a reasonable guesstimate of s (based on cytoskeletal stress in other cell types {citation needed}), z max, and range of values for P0 are shown at right.

One should consider other models, but most depend on factors that are difficult to account for without more data or constraints. For example, if one assumed the gel behave as an elastic (i.e. spring-like) material, one has to factor in the rest length of the material (it's dimensions when unstressed). This would depend on how the vein formed, and how it was modified over time. One might take special cases to simplify things, for example starting with the vein horizontal and then tipping it. One start from a state with RI(z) and w(z) equal to RI(0) and w(0). Then (assuming that PA(0)=P0 is a constant, independent of the vertical extent of the slime mold (i.e. that the pressure at the upper end of the slime mold does not drop when it's tipped) one could calculate how each point in the vein stretches as it's tipped from horizontal. New, unstressed material would be laid down on inner surface of the wall at the lower end. But, if PA(0) drops when the slime mold is tipped, the vein would shrink, so material would be removed, resulting in a change the stress on the remaining material. So, one would end up with a somewhat more complex differential equation.

# Method development

 Fig. 3: Split-level slime mold. Setup with plasmodium growing down paper towel following line of oats so it sits at two heights.

An alternative approach to testing the role of hydrostatic pressure in controlling wall thickness is shown in Fig. 3. The plasmodium was placed on a strip of paper towel, with one end of the paper towel tacked down to a large petri dish with nail polish. The other end was tacked to a piece of thin plastic (from a weigh boat). The end on the weigh boat can be lifted up different distances, or left flat. In the example shown, it was lifted so that the free end (on the weigh boat) was 28 mm above the petri dish. The slime mold grew up and down a line of oats so that part of it was on the platform, and part grew down to the dish. Once some of the plasmodium grew off the dish and formed distinct veins, they could be filmed. Here it was filmed with an inverted compound microscope, but a similar setup could be used with a standard compound microscope.

There still prove to be several challenges with this setup.

• It is hard to see the flow in large veins (it is possible that a yellow filter might help).
• The curvature of the vein distorts the light at the edges (this can be reduced by putting that vein under a drop of water). 4/21/2014: Further observations suggest that adding water disrupts the flow and structure in the vein: see http://igor.wikidot.com/projects:physarum-sensation.
• Veins accumulate particles around them that obscure the edges and the flow.
• It is hard to control how much of the organism is at different heights.
• The veins change greatly with age. At the growing margin, there is webbing between them; elsewhere they can be dying back. It is not clear how to choose veins at a consistent state of development or degeneration. (Ideally one could raise and lower the pressure in the same vein and look at its response, but this presents some of its own challenges: how long should one wait between changing the pressure and filming?)
• There is not always a distinct channel (Fig. 4). In this case there appeared to be flow into the side of the vein. This fits with the observation of pores and channels within the wall of the fixed veins as described in Wholfarth-Bottermann and Achenbach, 1982 [2]. Note that these authors also argued that the flow of fluid between the inner fluid cytoplasm and the wall (the ectoplasm) could also be actively regulated by changing the diameter and number of pores3, which would greatly complicate the connection between pressure and wall thickness.
• The diameter varies so much with time that the assumption that one can consider a steady state pressure and diameter overlain by small pressure and diameter fluctuations may be weak. Figure 5 illustrates the changes in outer diameter in one vein, but in some cases they appear even more extreme. In other cases diameter fluctuations can be much less extreme as well.
 Fig. 4: The flow channels in a vein. The whole vein is indicated in blue. Red indicates channels of flow and changes in vein diameter. One can see the main channel (red) running down the vein, and side channels (red) going to the right side of the vein. The flow in the side channels was generally right-left as the vein expanded, and left-right as it contracted. The red halo around the vein is due to changes in the diameter of the vein. This image was made from a grayscale time lapse movie. The movie was taken using an Olympus microscope camera (DP71) mounted on a Leitz inverted microscope with a 25x objective lens. Image J was used to process 25 frame of the movie. To capture the vein's average shape, the 24 frames were averaged, the gray values were inverted, and set to the blue channel. To capture regions of flow, the difference between each image and the preceding one was taken. Pixels where features move between frames appear bright in the difference images. These difference images were summed and set to the red channel. Fig. 5: Changes in dimensions of a vein. Top: A still image of a vein. Bottom: A kymograph illustrates changes in the diameter of the vein over time. A kymograph takes a single line of pixels (shown in the top image) from each frame of the movie, and stacks them on top of each other. Position along the line runs along the left-right direction, and time runs top to bottom. This way you can see the large changes in vein diameter over time. The oscillations in diameter were not at a single frequency. The time lapse was taken as in Fig, 4, and the kymograph was made using ImageJ.

## Getting closer

Plasmodia do take up carbon particles when fed oats which have been ground with activated charcoal, and then boiled a few seconds in the microwave (see video in this project). Using a yellow filter made from nail polish painted on a slide (yellow nail polish mixed with clear nail polish) does reduce the difference in intensity between the vein and the background, while reducing the light levels needed to get a good image.

 Fig. 6: Variation of wall thickness/inner channel radius (w/RI) with inner channel radius (RI). Note that the veins at the two vertical heights (z) of the plasmodia were only from a single plasmodium each, so one cannot separate any possible effects of vertical height (hydrostatic pressure) from other sources of variation among plasmodia in the data shown here.

As a preliminary trial, two plasmodia were grown with a streak of oats cooked with charcoal. In one case the streak was only on the bottom of the petri dish (vertical height difference of 0 cm), in the other case the streak ran up the side and onto the top of the dish (giving a vertical height difference of about -1 cm). Each plasmodium grew along its entire streak of oats. Multiple plasmodial veins on the lower surfaces of the dishes were videoed at 24 frames per second for 5 minutes with the Leitz inverted microscope and a 25x objective lens, using an Olympus microscope camera. Only easily visible veins, which were not terminal veins (i.e. leading to a blind end), and which did not have visible webbing, were videoed. The time point of the middle of the first contraction cycle was estimated from three measurements (made by eye using kymographs). Then the outer and inner vein diameters were measured at three approximately evenly spaced locations, with the inner diameter measured based on the particle flow (Fig. 6).

The ratio of w/RI was quite variable (Fig. 6). Furthermore, the inner channel was often displaced from the center, so that the wall thickness was greater on one side or the other. Neither of these two observations fits with the model above (eqn. 1 and Fig. 2). However, further testing is needed because the model assumes long term time-averaging, and the veins are constantly changing. Therefore individual veins could deviate from the model substantially at single time points. Therefore, a larger sample size (and improved optics: see below) are needed to satisfactorily rule out the model.

There was also no obvious difference in w/RI between the two plasmodia. However at this point this is not a statistically valid comparison as far as effects of vertical height is concerned: there could be other physiological differences between the two plasmodia. A minimum of 3 plasmodia in each height will be needed, and — given the high within-plasmodium variability — it's best to aim for more. Due to practical constraints, four seems a reasonable target.

The image quality differed greatly between the two plasmodia, possibly due to differences in moisture on the bottom and the smear of material on the top of the dish. In the z=0 cm plasmodium, the images were fairly crisp (although often obscured by charcoal on the outside of the veins), however there was substantial shadowing on the walls of the vein, probably due to the curved air/vein interface. In the z=-1cm plasmudium, images were more blurry, but there was no shadowing on the walls. Liquid may have drained from the top streak of oats to the bottom, making the layer of slime/water thicker in the z=-1cm plasmodium, and reducing the curved interface between air and slime mold+media. Both the shadowing (in z=0 cm) and the blurring (z=-1cm) made the inner edge of the gel layer very difficult to see. To try to compensate for this, the next batch are being grown on thin layers of agar. This should make for more moisture overall, possibly reducing the curvature of the air interface, and therefore reducing the shadowing.

 Fig. 7: Differences in image quality between the two plasmodia for Fig. 6.

# Literature Cited

1. Nakagaki T, Yamada H, Ueda T. 2000. Interaction between cell shape and contraction pattern in the Physarum plasmodium. Biophysical Chemistry.84(3):195-204. DOI:10.1016/S0301-4622(00)00108-3.
2. Wohlfarth-Bottermann KE, Achenbach F. 1982. Lateral apertures as passage-ways between ectoplasm and endoplasm in plasmodial strands of Physarum. Cell Biology International Reports.6(1):57-61. DOI:http://dx.doi.org/10.1016/0309-1651(82)90104-7. http://www.sciencedirect.com/science/article/pii/0309165182901047
3. Grębecki A, Cieślawska M. 1978. Dynamics of the ectoplasmic walls during pulsation of plasmodial veins of Physarum polycephalum. Protoplasma.97(4):365-71. DOI:141007/bf01276293. http://dx.doi.org/10.1007/BF01276293
4. Kamiya N, Kuroda K. 1958. Studies on the velocity distribution of the protoplasmic streaming in the myxomycete plasmodium. Protoplasma.49(1):1-4. DOI:10.1007/bf01248113. http://dx.doi.org/10.1007/BF01248113