Postdoc researcher at the Institut Pasteur in Paris 🇫🇷. I work on the biophysics of avian embryos 🐣. ENS de Lyon and
Harvard Medical School alumnus 🇺🇸
Science communicator: ex Pint/taste of science in Boston, Softbites founder and ComSciCon France founder.
If you just landed on Softbites for the first time, you probably have not had the chance to read our previous posts about microfluidics (like this one, or that one, and more). If this field of science is foreign to you, all you need to know is that it studies how fluids flow at really small scales (typically tens to hundreds of micrometers). For instance, you can quickly generate tiny droplets of a solution, turning each droplet into an individual “reactor”. Or you can create microenvironments with precisely controlled chemical concentrations to grow cells in different conditions.
In addition to being a thriving field of research, I think microfluidics is simply beautiful! I have spent hours looking at the Softbites website’s banner, a movie of droplets that was shot by the Lutetium project. You can imagine my excitement when I registered to the annual MRSEC microfluidics summer course 2019 at Brandeis University. This summer course was run by four talented grad students from the Fraden lab and the Rogers lab: Ali Aghvami, Alex Hensley, Marilena Moustaka and Zahra Zarei. These labs are part of the MRSEC program at Brandeis, an important place in the New England soft matter community. Therefore, I think it was the perfect place to get started with microfluidics!
Figure 1. Me, trying to pour some resin on a silicon wafer (left). A drop maker setup (middle). A gradient maker chip, with a defect leading to a non-stable gradient (right).
Over five days, we learned the basics of one of the standard methods for making microfluidic channels, called soft lithography. The rationale is to make a mold using a UV-light sensitive resin. A 2D pattern can then be polymerized in the resin by shining UV-light through a mask. Whatever the UV light hits gets hard, while the rest of the resin stays soft. The soft resin is washed away leaving only the hard, UV-treated resin behind in the shape of the mask. The mold will finally be used to imprint the design in a soft transparent material called PDMS (a very nice video from the Lutetium project explains all this process). We experimented with this fabrication during three main sessions:
We drew our 2D design using a drawing software
We fabricated our mold in a clean room so no dust ruined our tiny features
We cast the PDMS on the mold and sealed the device with a glass slide
We were taught each of these steps through a combination of lectures and hands-on sessions. You can see a droplet maker and a gradient making device that we made in Figure 1.
In addition to learning how to make these routinely used PDMS-based devices, we were also introduced to another technique used in the Fraden lab. This technique, which was recently published (2017), uses a thermoplastic (a plastic that melts at moderate temperatures) as the main device material. This thermoplastic can be cast onto a PDMS mold by means of a thermopress (as shown in Figure 2). Unlike PDMS, thermoplastic is not permeable to water and organic solvent, and is stiffer. If the permeability of PDMS is a limitation for your microfluidics application, thermoplastic might be the way to go!
Figure 2. Ali Aghvami placing thermoplastic chips onto the PDMS mold (left). Close-up on the thermoplastic in the thermopress before being cast (middle). The final device (right), from Aghvami et al. 2017.
This week-long course introduced us to both classical microfluidics techniques that are routinely used in labs and some more advanced ones. More importantly, our instructors dedicated important time to discuss our personal projects with us. We even had a consulting session with Seth Fraden! I strongly encourage anyone to attend the next editions of this course. Each year, the dates and the call for applications are released in spring, so don’t miss out!
These were my first drops! I literally spent 45 minutes watching them!
“USE YOUR LEGS!” That’s what might have been yelled at you the first time you went climbing. We are so used to walking or running that we don’t even think about how we do it. But when we face a new environment, such as a steep slope, we realize that finding the best strategy to move through space is not so easy. Now, imagine you are as small as few dozens of microns, without legs or arms, and you live in a viscous fluid. How would you move? This is the question biologists who are interested in cell movements have been trying to solve. By observing cells under a microscope, they saw that depending on their type or their environment, cells exhibit a wide variety of motion strategies. However, one thing never changes: cells need to exert forces on their environment to move. To do so, some kinds of cells create structures called focal adhesions. These structures are made up of several proteins, assembled on the outside of the cell. Like tiny bits of double-sided tape, their purpose is to stick the cell to whatever is nearby (see Figure 1). In slightly more technical language, focal adhesions connect the molecular skeleton of the cell to a substrate.
Cells can exert forces on their environment through focal adhesions. While it is possible to measure these forces outside the cell by engineering some force-sensing substrate [1], it is much trickier to understand what happens inside the cell. Accessing these forces inside the cells is the challenge Grasshoff and colleagues tackled in their 2010 paper.
In order to measure a force, the most straightforward method is to use a spring. A spring is a stretchable object for which, after calibration, we can relate its extension to the applied force. Therefore, a force can be measured by measuring the length of the spring. To measure the forces focal adhesions apply on the cell, one would need to inject tiny springs in the cells and connect them to the exerting-force structures.
To do this, the authors had the idea of taking advantage of a silk protein, produced by a spider, which is literally a molecular spring. Thanks to genetic tools, a part of the gene of this silk protein could be inserted within a gene called vinculin. The vinculin gene produces a protein that is an essential part of the focal adhesion structure. As shown in Figure 2A, vinculin connects the protein filaments of the cell skeleton to the outside of the cell (the extracellular matrix). The researchers engineered an artificial variant of vinculin that includes a molecular spring, derived from the silk protein, right in the middle of the naturally occurring vinculin molecule (see Figure 2B).
Figure 2.A. Schematic of focal adhesion. B. Schematic of the modified vinculin under low and high tension. Under high tension, the molecular spring is stretched. Red: adhesion protein, orange: vinculin head domain, yellow: vinculin tail domain, grey: contractile filaments. Arrows represent the magnitude of the tension.
After verifying that cells that are genetically modified to include the engineered focal adhesion protein behave normally, the next step was to measure the molecular spring extension. However, measuring distances at the molecular scale is not a piece of cake. For instance, the typical extension of such a spring is 6 nanometers, which is, by far, below the resolution of the best optical microscopes [2]. To circumvent this limitation, Grasshoff and colleagues took advantage of the Förster resonance energy transfer (FRET) effect to measure the distance between the two vinculin domains. The FRET effect takes place between two fluorescent molecules very close in space. A fluorescent molecule, when excited by a light at a precise wavelength, emits a light at a longer wavelength. But if a second fluorescent molecule is close enough, the first molecule (the donor) can directly transfer its energy to the second molecule (the acceptor). Then, the acceptor will emit light at an even larger wavelength than the donor’s. Consequently, the FRET intensity can be computed by measuring the relative emissions of the donor and acceptor molecules: the closer the acceptor is to the donor, the more energy the acceptor will absorb and re-emit. Furthermore, and importantly for this application, the efficiency of this process is very sensitive to the distance between the donor and the acceptor As a result, the distance between the two molecules can be measured with great precision (sub-nanometer) by measuring the intensity of the FRET effect. Therefore, the authors further engineered the vinculin protein by placing the molecular spring between two fluorescent molecules (Figure 3, yellow and red circles) that were capable of undergoing the FRET effect to measure the extension of the molecular spring.
Figure 3. Förster resonance energy transfer (FRET) effect in the modified vinculin of a focal adhesion under low and high tension. The excitation light of the donor molecule (yellow circle) is shown in green and the emission light of the acceptor molecule (red circle) is shown in red.
At this point, the authors had a method for measuring the tension intensity across vinculin molecules just by looking at the FRET intensity. In this way, they could generate a tension map across the contacts of the cell with its environment. They saw that focal adhesion under high tension leads to a growth of the size of the focal adhesion which relieves it from its high tension. Perhaps surprisingly, they also showed that regions where the contact is extending (protruding areas) are under higher tension than regions where the contact is receding (retracting areas), as shown in Figure 4.
In this paper, the authors developed a new technique to measure forces inside cells. By conducting single-molecule experiments, they even could calibrate their engineered molecular spring and relate the FRET intensity to absolute values of forces (in the order of a few piconewtons [3]), paving the way to a whole class of new FRET-based force sensors with different stiffnesses, which can now be used in other structures inside cells.
Everything started with adding a spider silk gene in a cell. Such mutant cells have the amazing power of shading light on the cellular force machinery. But “with great power, comes great responsibility” as another spider mutant has once been told.
Figure 4. The FRET index (ratio of donor to acceptor fluorescence) reveals the state of tension through vinculin across a cell. Close-ups retracting areas (R1 and R2) show a high FRET index, ie. a low tension, and protruding areas (P1 and P2) show a low FRET index, ie. a high tension (adapted from Grashoff et al.).
[1] These techniques are called traction force microscopy. The deformation of calibrated substrate (either a gel or micropillars) is measured to calculate the forces exerted by the cell. [2] Classical optical microscopes have a typical resolution of around 200 nm. New techniques of super-resolution microscopy reach a resolution of a few dozens of nanometers. [3] To give you a sense of this order of magnitude, when you hold a pen of, let’s say 10 g, you apply a force of 0.1 N. At the cellular level, cells exert on their environment forces in the order of dozens of nanonewtons (according to this study). At the molecular level, DNA has been manipulated applying forces in the same range as the vinculin tension: 1-100 pN (according to this study).
In episode one of this series, I presented a research paper by Stéphane Douezan and his colleagues in which they studied a ball of cells (called a cellular aggregate) sitting on a flat surface. After introducing the concept of cellular aggregate wetting by comparing it to the classical system of a drop of water, today I present the main part of the paper which looks at the dynamics of spreading of the cellular aggregate. I strongly suggest that the reader reads the first post before reading this one.
As introduced previously, the spreading of a cellular aggregate is set by the surface tensions of the three interfaces: cells-substrate ($latex \gamma_{CS}$), cells-medium ($latex \gamma$), substrate-medium ($latex \gamma_{SO}$). The spreading can be controlled by finely tuning two adhesion energies: the cell-cell adhesion ($latex W_{CC}$) and the cell-substrate adhesion ($latex W_{CS}$) [1]. The authors of this paper set $latex W_{CC}$ by controlling the level of E-cadherin (a molecular glue between cells), and $latex W_{CS}$ by varying the concentration of fibronectin (a molecular glue between the cells and the substrate) deposited onto the substrate.
Figure 1. Schematic of a wetting cellular aggregate. $latex R_0$ is the initial radius of the aggregate. $latex r(t)$ is the radius of the contact line. $latex \theta$ is the contact angle. $latex \gamma$, $latex \gamma_{SO}$ and $latex \gamma_{CS}$ are the three interfacial tensions. (adapted from Douezan and colleagues.)
To characterize the dynamics of spreading, Stéphane Douezan and his colleagues measured the area of the cellular aggregate in contact with the surface with respect to time. The authors noticed two distinct regimes: at short times (the first hour) the cellular aggregate flattens, and at longer times, it forms a film which spreads completely. In the first regime, they observed a non-constant spreading speed. More interestingly, it depends on the cellular aggregate size: the bigger the aggregate, the faster the spreading (see Figure 2a).
To understand this non-trivial spreading dynamics, the authors investigated in detail what is driving and what is preventing the cellular aggregate from flattening at short times. The contact area expands because the adhesion between the cells and the substrate is more favorable than the cell-cell adhesion. So increasing the cell/substrate adhesion $latex W_{CS}$ should increase the speed of spreading. On the other hand, the process is slowed down by the friction of the cells: there is a so-called viscous dissipation, like when you pour honey, the more viscous the honey the longer it takes to flow. So increasing the viscosity, should decrease the speed of spreading. The authors expressed the energy of these two antagonist contributions to the speed.
Figure 2. Spreading dynamics of cellular aggregates of different sizes (adapted from Douezan and colleagues.) (a) The contact area A grows faster when the aggregate initial radius $latex R_0$ is larger. (b) The contact area scaled by $latex R_0^{4/3}$ dynamics follows a power law and depends on the initial radius.
First, the energy gain is the work per unit of time of the capillary force $latex F_c$ [2]:
Energy gain = $latex 2\pi r F_c \frac{dr}{dt}$ [3]
At early times, the contact angle is very small, so the capillary force $latex F_c$ can be simplified: $latex F_c = W_{CS} + \gamma (cos \theta -1) \approx W_{CS}$ . In this way, $latex F_c$ can be replaced by the constant $latex W_{CS}$ in the expression of the energy gain.
Second, the authors show the dissipation is expressed by $latex \eta (\frac{dr}{dt})^2 \frac{r^3}{R_0^2} $ where $latex \eta$ is the cellular aggregate viscosity. Per conservation of energy, the energy gain should be exactly compensated by the viscous dissipation. Thus, by equating these two energies and integrating $latex r$ over time, we have the time variation of $latex r^2$ that follows a power law [4]:
$latex r^2 \propto R_0^{4/3}\frac{W_{CS}}{\eta}^{2/3} t^{2/3}$, with $latex R_0$ being the aggregate initial radius.
This dynamics of $latex r^2$, which is proportional to the contact area, indeed depends on the aggregate size $latex R_0$ in a consistent manner with the experimental observations: the bigger the aggregate, the quicker it spreads.
So if the law is valid, rescaling the measured area by $latex R_0^{4/3}$ should remove the dependency on the size of the cellular aggregate. This is exactly what they saw: all the data points collapsed on the same curve (Figure 2b). There is something even more interesting here: fitting the spreading curve gives an estimate of the ratio $latex W_{CS}/\eta$, two variables which are difficult to measure. Of course, this is only a ratio, which does not provide absolute values for these two variables but it possible to measure relative changes by playing with some biological parameters. For instance, as mentioned above, the authors can tune the cell-cell adhesion energy using genetic tools (see the first post to understand how they measure it) and the cell-substrate adhesion by coating the substrate with different concentrations of an adhesive molecule. In this way, they quantitatively described how the viscosity decreases when the intercellular glue expression — the E-cadherin — is reduced, see Table 1. Similarly, they studied the relative change of the cell-substrate adhesion energy depending on the substrate coating.
Table 1. Relative change of the viscosity depending on the E-cadherin expression.
To summarize, Stéphane Douezan and his colleagues were able to explain what is driving the initial flattening of the aggregate at short times by showing how this dynamics depends on the aggregate size, and they were able to estimate the ratio of the cell-substrate adhesion energy over the viscosity.
Figure 3: Long-time spreading. Top: cohesive cellular aggregate (E-cadherin — the molecular glue between cells — expression = 100%), liquid state. Bottom: Poorly cohesive cellular aggregate (E-cadherin expression = 21%) liquid-to-gas transition. (adapted from Douezan and colleagues.)
After studying this short-time regime, the authors analyzed the spreading at longer times. Depending on the cell-cell adhesion energy, they noticed two behaviors: either the aggregate flows as a cohesive two-dimensional sheet of cells (like a liquid) when the adhesion is strong, or individual cells escape from the aggregate (analogous to a liquid-to-gas transition) when the adhesion is weak. These two behaviors are shown in Figure 3 and in movie 2 and 3 of the supplementary data. This phenomenon could be used to model an invading tumor for which the biological parameters that control the transition between two kinds of spreadings can be precisely tuned.
Long-time spreading of a cohesive cellular aggregate (movie 2 of the supplementary data)
In this paper, the authors successfully captured the complexity of a biological system with a predictive law of spreading. By measuring well defined physical variables, such as the viscosity and the cell-substrate adhesion energy, they were able to quantify how cells bind to each other or to their environment. These complex biological processes, which involve many different molecular actors, are often described in a qualitative way. Even more interestingly, the authors showed how they could tune these physical variables by controlling some biological parameters, which directly shows their implications in the processes mentioned above. The approach taken in this paper is very elegant as biology often fails to be predictive because of the important complexity of the processes at stake.
[1] The adhesion energy of an interface is the work that should be spent by unit of area if one were to break this interface. The stronger the energy, the more stable the adhesion. Therefore like the surface tension, it is an energy density (unit: $latex J/m^2$). As a reminder from the previous post, the two adhesion energies can be expressed by the surface tensions: $latex W_{CC} = 2 \gamma$ and $latex W_{CS} = \gamma_{SO} + \gamma – \gamma_{CS}$.
[2] The capillary force is the sum of the components of the three tensions along the tangent axis to the substrate. This force per unit of length is basically the force that pulls on the drop: $latex F_c = \gamma_{SO} + \gamma cos(\theta) – \gamma_{CS}$.
[3] The differential expression of the energy gain is obtained through the following reasoning: during an infinitesimal duration of spreading $latex \delta t$, the radius of the contact line increases by $latex \delta r$. So the infinitesimal work of the driving force $latex F_c$ is: $latex \delta W = perimeter * F_c * \delta r = 2\pi r F_c \delta r$.
[4] A power law is simply a mapping of a variable at some power. They are usually presented on log-log plots, as they appear as a straight line, for which the slope is the power of the function.
Disclosure: The second author of this paper is my Ph.D. supervisor. However, she did this work while she was a postdoc. Consequently, I have never been involved in this work.
Disclosure: The second author of this paper is my Ph.D. supervisor. However, she did this work while she was a postdoc. Consequently, I have never been involved in this work.
Have you ever noticed how drops of water have different shapes on a clean piece of glass and in a frying pan? The frying pan surface is coated with a hydrophobic (“water-repellant”) molecule so it does not stick to food, which typically contains a lot of water. As a result, a drop of water will take a roughly spherical shape to reduce as much as possible its area of contact with the frying pan. If a surface has an even more hydrophobic coating than a frying pan, the drop can even reach a perfectly spherical shape (this is called ultrahydrophobicity, or the lotus effect). At the opposite extreme, glass is said to be hydrophilic (“water-loving”) — when placed on a clean piece of glass, a drop of water tries to increase its surface of contact much more than a droplet on a hydrophobic frying pan. Depending on the hydrophilicity of the underlying surface — which is known as the substrate — the drop has a well-defined area of contact. The interaction between fluid interfaces and the solid surfaces is a very well studied field of soft matter called wetting. Researchers in this field investigate how the three different interfacial energies — interfaces between water and substrate, between water and air, and between substrate and air — dictate what shape a droplet takes, and how it spreads across the surface.
Figure 1. (a) Water drop on an ultrahydrophobic surface (public domain image). (b) Cellular aggregate after deposition (adapted from Douezan et al.)
Today’s post is the first one of a series of two (click here for the second one), which deals with the work of scientists who replaced the drop of water by balls of living cells called cellular aggregates. They deposited these aggregates onto different surfaces to carry out an experiment analogous to the spreading of water droplets. In the case of a drop of water, only the physical and chemical interactions between molecules determine the shape of the drop. When a drop sits on a substrate there is an interface between the water and the substrate. If the chemical interactions between the substrate and the water are not favorable (hydrophobic), the price to pay will be a large interfacial energy. As every system in physics, it tries to reduce its overall energy by reducing the area of contact. The drop shrinks, like the ones you can see in your frying pan. But as it shrinks, the interface area between the substrate and air increases by freeing the surface. And, as the volume of water is fixed (we consider a no-evaporation situation), the surface of the drop in contact with air changes too. Therefore, the drop shrinks or spreads up to a point for which the sum of the three interfacial energies is minimized.
Figure 2. (a) Schematic of a wetting drop. (b) Schematic of a wetting cellular aggregate (from Douezan et al.)
The shape of the drop can be described by the contact angle ?, which can be used to predict the interfacial tensions, which quantify how the interfacial energies change when the areas of contact are changed. A tension is a force divided by a length. So, we can write the equilibrium of tensions on the point where the three interfaces meet (1):
$latex S = – \gamma_{wa} (cos\theta -1)$ (by using the equilibrium of tensions) (2)
This expression shows that there is a partial wetting ($latex \theta$ between $latex 0$ and $latex \pi$) if $latex S<0$. If $latex S>0$, the drop spreads completely: that is, the droplet covers the substrate with an infinitely thin fluid film.
Now, what happens if we consider a ball of cells instead of a drop of water? Since the pioneering work of Malcolm Steinberg, we know that cellular aggregates can behave as liquids over long times. If you were to poke a piece of biological tissue, it would resist at short times (less than a dozen of seconds to a couple of minutes) but on the long run, it would start to flow. As every liquid, a surface tension builds at its interface. For instance, a rough ball of cells in suspension will round up over time to minimize its ratio area/volume. As presented in a previous post, the surface tension is a physical value that can also be defined for biological tissues even though its nature is very different from the one of purely physical systems.
To further investigate the role of surface tension in living tissues, Stéphane Douezan and his colleagues decided to study how the biological properties of these aggregates of cells can influence their liquid behavior. The first property they considered was the “stickiness” of the cells, also known as cell-cell adhesion. Cells produce a large number of molecules at their surface which allows them to sense and interact with their environment. E-cadherin is an important molecule that acts like a kind of glue between cells, allowing them to stick to their neighbors. Using genetic tools, the researchers grew cells with different levels of E-cadherin, making them more or less sticky with respect to the others. By using micropipettes to pull on two sticking cells until they broke apart, the researchers then measured the energy of cell-cell adhesion. Integrating the force exerted during the separation and dividing by the contact area leads to the cell-cell adhesion energy $latex W_{CC} $.
The authors played with a second property too.In living tissues, cells interact with the extracellular matrix — a scaffold of molecules that gives the tissue its structure. One of the important molecules of the extracellular matrix is called fibronectin. By coating the glass substrate with different concentrations of fibronectin, the researchers could finely tune the adhesion of the cells to the substrate. To measure this adhesion: the researchers define the cell-substrate adhesion energy, $latex W_{CS} $.
In order to know if an aggregate will spread, the wetting coefficient $latex S = \gamma_{SO} – (\gamma + \gamma_{CS})$ must be evaluated. However, not all these tensions can be measured directly, so they must be expressed in term of the energies we can measure $latex W_{CS}$ and $latex W_{CC}$. A classical approach to connect the adhesion energy to the tensions is to write the balance of tensions if we were to break an interface. For instance, to separate a cell-cell interface, two new interfaces (between the cells and the surrounding fluid) must be created so, by energy conservation: $latex W_{CC} = 2 \gamma$. Similarly, breaking a cell-substrate interface requires creating an interface between the cell and the surrounding fluid, an interface between the substrate and the fluid, and removing a cell-substrate interface, so: $latex W_{CS} = \gamma_{SO} + \gamma – \gamma_{CS}$.
Therefore, the wetting coefficient becomes $latex S = W_{CS} – W_{CC}$. If $latex S>0$, the energy of adhesion with the substrate is larger than the cell-cell adhesion energy, and the aggregate spreads completely. In this case, the dynamics of spreading can be monitored, as you can see in this video of wetting (video S1).
In the next post, I will present the dynamics of spreading, where the cellular aggregate literally behaves as a chunk of silly putty!
(1) In reality, the three interfaces meet at the line that circles the drop, but since the system has a circular symmetry, it makes more sense to write the tension balance on a point instead of writing the force balance all along the circle.
(2) Usually, $latex \theta$ is defined as the complementary angle ($latex \theta ‘ = \pi /2 – \theta$), so $latex S=( cos\theta ‘ – 1 ) \gamma_{wa}$ . But here I decided to use the same definition as the authors for the sake of consistency.
We all started as one single cell. This cell contains all the information to make a complex adult body. Developmental biologists try to understand how this cell will first divide to make a dull ball of cells which will then start making dramatic changes in shape to pattern the future organs of the body. One of the difficult questions is how cells that will form the same structure are able to find one another and sort from the mix of other cell types. In a famous experiment in 1955, Philip Townes and Johannes Holtfreter showed that different cell types had an intrinsic ability of sorting, even when they were completely mixed. They took two different cell types from a frog embryo and mixed them to make a heterogeneous aggregate of cells like you would shuffle a Rubik’s cube. After some time, the heterogeneous aggregate spontaneously evolved to a two-layer structure with each layer containing only one cell type. If only a Rubik’s cube could spontaneously solve itself!
Figure 1. Cell sorting of two cell types (adapted from Townes and Holtfreter 1955)
A popular explanation for this phenomenon has been brought by Malcolm Steinberg in 1970 who presented his ‘Differential Adhesion Hypothesis’: each cell type has a certain pattern of adhesion molecules at its surface. As a result, adhesion between cells of the same type is different from adhesion between cells of different types. This system seems really analogous to emulsions in which the interfacial energy drives droplets of the same phase together. When you whip oil and vinegar together, you separate the two phases into droplets that quickly coalesce to reduce the total surface of the interface. This ‘Differential Adhesion Hypothesis’ raises the question: is it an analogous process to oil-vinegar emulsions that drives cell sorting? Is there an interfacial energy which sorts the cells?
To answer this question, in 1992, François Graner and James Glazier decided to run a computer simulation of cell sorting with the interfacial energy as the driver of the sorting. Previous simulations had been run before, but the geometries of the cells were not realistic. The authors derived their model from the Potts model which was inspired by the classical Ising model used to study the behavior of magnetic materials. The Ising model relies on a discrete representation of space by a lattice with sites occupied by one of two possible numbers, usually 1 or -1, which in the case of magnetization represent atomic spin. In the Pott’s model, the sites can be occupied instead by numbers 1,2,…N, and in the application of the model in this work, the numbers are used to identify individual cells in a two-dimensional system. A region where all the sites share the same number is a single cell; the i-th cell is made of all the sites with the index i. This region needs to be simply-connected, i.e. there is no hole in the region. This idea which was introduced by Renfrey Potts in 1952 has been generalized to describe grain growth and froth systems. In other words, this model shows a pixelated image of a cellular material.
Figure 2. Schematics of the Ising and Potts models
Such a system has an energy which is minimized when it has reached the equilibrium. The Potts energy is just the total length of interfaces between cells. So, it has a simple expression:
with $latex \sigma(i,j)$ the value of the cell index on the lattice site of coordinates $latex (i,j)$.
Graner and Glazier modified the Potts model by taking into account that cells, unlike bubbles in a froth, have a characteristic size from which they cannot deviate too much. They added a second term to the energy of the system:
$latex H_{extended\ Potts} = $ interface term + area constraint (2)
The area constraint assumes a linear elasticity of the cells: they can be compressed or dilated but the difference between their area and the target area increases the energy with the square of this difference. In addition, they modified the first term so that the interface term is not constant for all the interfaces.
In order to study the cell sorting, they introduce different cell types. The interface energy between two cells of different cells type is greater than the one between two cells of the same cell type. In this paper, the authors mimic Townes’ and Holtfreter’s experiment by considering two cell types (they actually consider a third cell type which represents the surrounding media, for which they remove the area constraint). To simulate the dynamics of sorting, they use a Monte Carlo algorithm, which is a common simulation method that uses random sampling to obtain numerical results for complex problems. In short, a site of the lattice is randomly selected and its cell index value is changed to one of the neighboring cells. If this change decreases the energy, it is accepted, if it increases it is accepted with a probability depending on the energy increase. Transiently accepting unfavorable configuration allows to explore a larger region of the energy landscape and avoid being trapped in local minima. In this way, the cell interfaces move along the simulation.
They start their simulation from a cellular aggregate of cells with two cell types (grey and black). The geometry of this aggregate has been produced by running the simulation on a square aggregate with only one cell type and by randomly assigning the cell types afterward. This aggregate exhibits an overall round shape found commonly in biology, and it is made up of cells with classical cell shapes. However, the interfaces are biased towards the 45° direction due to the discretization anisotropy (because of the pixels, the length between neighbors is not equal in all directions). After running the simulation over 10000 Monte Carlo time steps, the cells are able to sort into two distinct layers, in a very similar fashion to Townes’ and Holtfreter’s experiment.
Figure 3. Monte Carlo simulation of aggregate cell sorting (adapted from Graner and Glazier 1992)
Graner’s and Glazier’s model has therefore been able to recapitulate the general evolution of a mixed cellular aggregate by just using the interface properties and the cell deformability. Although this model cannot handle a realistic time and its discretization leads to anisotropic artifacts, it is still widely used to model biological tissues. This paper has been an important step to show that a surface tension exists in biological tissues and can drive morphogenetic processes. Surface tension in biology is still a hot topic today since other ingredients, like the cell contractility (see especially this paper from Carl-Philipp Heisenberg’s lab), have been shown to be involved in the surface tension.
(1) The energy is the sum over all the faces between sites. If the two neighboring sites are from different cells, they have different indices $latex \sigma,\ \sigma’$, and the Kronecker delta $latex \delta_{\sigma \sigma’} = 0$. Therefore $latex 1 – \delta_{\sigma \sigma’}$ adds one to the sum.
(2) Here is the detailed expression of the extended Potts energy:
$latex J(\tau,\tau ‘)$ is the energy of an interface between cells of types $latex \tau$ and $latex \tau’,\ \lambda$ is the strength of the area constraint and $latex A_{\tau}$ is the area target of a cell of type $latex \tau$.
![Berginski M, Vitriol E, Hahn K, Gomez S [CC BY 2.5 (https://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons](https://softbites.org/wp-content/uploads/2019/02/high-resolution-quantification-of-focal-adhesion-spatiotemporal-dynamics-in-living-cells-pone.0022025.s014.ogv.jpg)
