research – Page 8

2018-02-072018-04-14

Knotty DNA

Original paper: Direct observation of DNA knots using a solid-state nanopore

Try taking out your earphones from your pocket and, in all probability, you’ll find knots and entanglements between the ends. As it turns out, this knotting effect is not limited to macroscopic objects, but occurs on the nanoscale as well. A DNA molecule that carries the genetic information of a living organism is actually a long string-like polymer, so you can imagine that it would also get tangled up just like the cords of your earphones. In fact, scientists know that DNA does form knots when it is in the nucleus of a cell, and these knots need to be removed by specialized bio-molecules, called enzymes, so that a cell can ‘read’ the genetic information encoded in the DNA. [1] In today’s paper, Calin Plesa and his colleagues at TU Delft are able to observe and measure these knots in DNA strands. In the process, they also observe interesting knotting behaviour which has not been observed before.

Knots on DNA

DNA translocation through a solid-state nanopore — *Figure 1: This animation shows the DNA moving through the nanopore. The associated dip in current is mapped onto the graph below. (Animation created by Calin Plesa, available under* *CC BY-SA license)*

The researchers use a nanopore sensor to infer the structural properties of a DNA molecule. The sensor is made up of two reservoirs filled with electrolyte (a solution which separates into cations and anions, which can be used to conduct electricity, e.g. a salt solution), and they are separated by a membrane, or thin sheet, with a tiny hole in it. An electric field applied across the membrane generates an ionic current in the electrolyte and also pulls a negatively charged DNA strand through the tiny opening. The passage of a DNA strand through the nanopore causes a dip in the ionic current that is proportional to the volume of ions displaced—in other words, it’s proportional to the size of the molecule (a typical scenario is shown in Figure 1). Therefore, a knot in the DNA can generate a bigger drop in the current than an untangled strand. From this difference it is possible to infer the characteristics of the knot itself, since a bigger drop indicates a bigger knot.

The typical time for a DNA to pass through the pore is in the order of a few milliseconds, when the DNA is in a solution of potassium chloride (which is the typical salt solution used to carry out nanopore experiments). This makes it difficult technically, to see any features present on the DNA. Previous work has shown that it is possible to slow down the DNA passage by at least 10 times by using lithium chloride as their salt solution. [3] This increase in the translocation time (time it takes for the DNA to pass through the pore) is necessary to clearly see the additional dip in the current as the knot traverses the pore, as illustrated in Figure 2.

Translocation of a DNA molecule containing a trefoil knot through a solid-state nanopore — Figure 2: This animation shows a DNA with a knot moving through the pore. An additional dip in the current can be seen in the current trace as the knot (purple line) passes through the pore. (Animation created by Calin Plesa, available under *CC BY-SA license)*

The dip in the current signal caused by the knot passing through the pore can then be used to infer characteristics about the knot. In particular, it can be used to calculate the size of the knot, which has not been experimentally determined before. This has both physical and biological significance. Physically, it helps us understand the types of knots being formed on polymers as it can tell us whether the knot is loose or tightly formed. Biologically, it can help us understand how naturally occurring enzymes are able to disentangle knots in DNA strands, a function which is still poorly understood. The size of the knot is estimated by using

$latex d= v t$

where d is the length of the knot along the DNA strand, v is the average speed of the DNA translocation, and t is the time the knot takes to traverse the pore. Using this technique, the researchers estimate that the majority of the knots are less than 100 nm long. Previous research has shown that the DNA strand is rigid over lengths shorter than 50 nm, so considering this, the estimated knot size suggests that the knot is very tight. [2] However, this result needs further analysis, as the process of pulling the DNA through the nanopore might cause the knot to tighten, so this might not be the knot’s size in its natural state.

Slipping and sliding knots

When considering a linear (think: a thread with loose ends) DNA molecule, there is a possibility of the knot ‘slipping’ off the end of the strand before it gets pulled into the nanopore. For the knot to traverse the pore, it needs to be pulled fast enough to get squeezed to the size of the pore. If this process doesn’t happen fast enough the knot ‘halts’ at the pore entrance while the unknotted region translocates through. This allows the knot to disentangle, in case of a linear DNA molecule.

To determine if this slipping process occurs in knotted DNA strands, the researchers repeat their experiment using a circular (think: a thread joined end-to-end) DNA molecule. By using a closed loop they avoid possibility of the knot disentangling, but the knot can still slip towards the trailing end of the DNA during the translocation. The position of the knot is determined by the position of the dip in the current signal (purple line in Figure 2). They measure the probability of finding the knot at each position along the strand using two voltages, 100 mV and 200 mV. As shown in Figure 3, the knots show a preference for sliding toward the trailing end of the molecule at higher voltages, indicating that pulling too hard on the leading end of the DNA strand can indeed cause knots to slip along the strand instead of being pulled through the pore. The researchers also observe a 55% higher knotting occurrence in the circular molecules compared to linear ones. This suggests that knots may have slipped off the end of the linear molecules, thereby not detecting them at all.

Figure 3: The graph shows the probability of detecting the position of the knots along the length of DNA. At 200mV, the knots are observed to be at the trailing end of the DNA motion indicating the slipping phenomenon (adapted from Plesa et al.)

The researchers in this study have shown that naturally induced knots occur in DNA strands and they measured the sizes of those knots, which were previously unknown. This measurement showed that the knots detected are actually quite tight, which was not expected, although this result still needs to be investigated further. Additionally, these knots were seen to slide along the DNA molecules as they traversed the nanopore due to the strong pull at the end of the DNA strand. This was seen clearly by repeating the knotting experiments using circular DNA where there were no ends for the knots to slide off.

This new information about the structure of knots in DNA strands will help inform future studies of the complex topological structures formed in biomolecules such as DNA and proteins. It will also contribute to understanding the effects of topological features on the biological functions of these long, string-like biomolecules. In effect, it can help us explain the consequence of knotted DNA on the cell’s function as well as how the cell is equipped to handle these defects.

[1] http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html

[2] Baumann, Christoph G., et al. “Ionic effects on the elasticity of single DNA molecules.” Proceedings of the National Academy of Sciences 94.12 (1997): 6185-6190.

[3] Kowalczyk, Stefan W., et al. “Slowing down DNA translocation through a nanopore in lithium chloride.” Nano letters 12.2 (2012): 1038-1044.

2018-01-312018-04-14

Soft nanoparticles: when polymers meet soap

Original paper: Self-Assembly of Complex Salts of Cationic Surfactants and Anionic? Neutral Block Copolymers. Dispersions with Liquid-Crystalline Internal Structure

For more than four decades, scientists have been investigating the properties of small objects dispersed in solutions. Some of these objects – produced in laboratories – are the so called soft nanoparticles. The name soft comes from the fact that these particles are partly solid and partly liquid. One of the scientists’ aims is to design nanoparticles that will be used as carriers of medical compounds (like drugs, DNA segments, and enzymes). The nanoparticles’ role will be to protect this cargo from partial degradation through the human body until reaching the specific target cells where the nanoparticles’ structure will break up and the useful compounds will be released. This technology will allow for disease treatments using smaller amounts of drugs, which will mean fewer side effects for the patients.

The effectiveness of this treatment depends on several factors that control the nanoparticles’ properties. It turns out that one important factor is the protocol of preparation, which is the recipe used to make the nanoparticles. Today’s paper by Leticia Vitorazi, Jean-Francois Berret and Watson Loh introduces an alternative method of preparation and shows how the individual chemicals that are chosen for creating the nanoparticles can influence the nanoparticles’ properties.

The formation of the soft nanoparticles is a spontaneous process that takes place in specific mixtures of solutions because of the electrostatic attraction between two different compounds, macromolecules and surfactants. Each of these compounds usually exist as individual molecules in water (or other solvents).

Macromolecules consist of long, chain-like synthetic molecules. They can have either one long chain (called homopolymers) or two long chemically different chains connected together by covalent bonds (called diblock copolymers) (Figure 1).

figure 1.homopolymers — Figure 1. Representation of a homopolymer with a long charged chain and a diblock copolymer with a charged (A) and a non-charged (B) chain. Each sphere represents a repeated unit of the chain.

The other type of compound, surfactants, comprise short molecules with both hydrophilic (water-loving) and hydrophobic (water-hating) parts. Surfactant is what soap is made from, and its name is a shortcut to the term surface active agents. When the amount of surfactant molecules in a water solution exceeds a specific number (above the so called critical micelle concentration), the surfactants clump together and form small spheres (called micelles) with the hydrophobic parts inside the sphere to avoid contact with the water molecules, leaving the hydrophilic parts at the surface to be in contact with the water (Figure 2).

Figure 2 micelles — Figure 2. Representation of surfactant molecules with a hydrophobic tail and hydrophilic head (red spheres) surrounded by oppositely charged small ions (counterions). Above a specific concentration of surfactant in water, the surfactants molecules organize themselves in micelles with the tails inside the micelle to be protected from water and the heads in the micelle surface to be in contact with water.

Mixing macromolecule solutions with surfactants/micelle solutions causes the two compounds to come together because their opposite charges attract, and a new object is created: a nanoparticle, composed of macromolecular chains surrounded by surfactant micelles. If the macromolecule is a diblock, with a second type of non-charged chain attached chemically to the charged chain, the resulting nanoparticles usually have a specific structure with an internal core consisting of surfactant micelles and oppositely charged macromolecule chains, surrounded by an external shell. This shell consists of the macromolecules’ non-charged chains and acts as a non-stick coating that prevents the particles from clumping together (Figure 3).

Figure 3. Nanoparticles scheme — Figure 3. Schematic of soft nanoparticles with an inner core comprised of charged surfactant micelles (red), and oppositely charged polymer chains (yellow) surrounded by an external shell comprised of non-charged polymer chains (green). The red dots represent the surfactant micelles that form inside the nanoparticle core at a cubic order (enlarged scheme). Adapted from Vitorazi and colleagues.

The most popular method of creating the nanoparticles is the direct mixing method of the macromolecules and surfactants. The two compounds are dissolved separately in water (or an appropriate solvent), and then they are mixed together to reach specific amounts of the opposite charges in the solution. The mixtures consist of charged macromolecules and oppositely charged surfactants surrounded by small ions. The resulting nanoparticles are small: between 30 and 50 nm. (For comparison, HIV virus is about 120 nm large and bacteria are about 1000 nm.)

Vitorazi and co-workers used a different method named “complex salts.” The recipe of making the complex salts consists of mixing the macromolecule solutions with an increasing amount of the surfactant solutions until all acidic groups of the macromolecules bind to the hydroxide groups of the surfactants. The next step of the process is the removal of the solvent to create a single-component powder, which is freed from the small ions that surrounded the individual compounds in the solution (Figure 4A). Finally, the powder is dissolved in water at different concentrations and the electrostatic attraction between the surfactant micelles and oppositely charged macromolecules results in the formation of the nanoparticles.

Figure 4. the complex salts vs direct mixing — Figure 4. Representation of a diblock copolymer with charged (yellow parts) and non-charged (green parts) chains surrounded by surfactant micelles (red parts) in the absence of small ions (A) and in the presence of small ions (B).

The researchers used the complex salts method for different macromolecular chain lengths to explore the effect of the macromolecules’ chain lengths on the nanoparticles’ size and core structures, and they compared the results with the direct mixing method that was used in previous years for similar mixtures. They found that if the two sub-chains of the diblock copolymer (A and B in Figure 1) have roughly equal lengths then the nanoparticles were larger, compared to those made through the direct mixing method. As shown in Figure 3, these particles consist of a core with a cubic ordered structure where the surfactant micelles are positioned at specific places in the core. They also found that for unequal sub-chain lengths, the core was disordered and the nanoparticles were smaller. This is an important finding because the ordered structure and the larger size of the nanoparticles can incorporate larger amounts and different types of drugs in the core.

They also studied the effect of the addition of salt or macromolecular solutions in the nanoparticle solutions. The amount of salt strongly affects the nanoparticles’ properties because the added ions in the nanoparticles’ solution weaken the attraction between the macromolecules and the oppositely charged surfactant micelles. Therefore, it is a useful factor to be considered for the biomedical applications in the salty environment of the cell system in our body. Vitorazi and co-workers found that after the addition of salt or macromolecule solutions in the nanoparticles’ solution, the already formed nanoparticles lost their internal cubic structure (meaning that the micelles were now randomly oriented in the core) but the stability of the system was preserved.

To conclude, the method of preparing soft nanoparticles plays an important role in determining their properties and therefore affects their performance as drug carriers. Compared to direct mixing methods, the complex salts formed stable and 3 times larger nanoparticles with a core of cubic internal order. In a charged environment (created by addition of small or macromolecular ions) the final structure lost its cubic structure, but the nanoparticles were still stable, making them important candidates for the future of drug delivery technology.

2018-01-242018-04-14

How swimming bacteria spin fluid

Original paper: Fluid Dynamics of Bacterial Turbulence

The next time you’re washing your hands, start by turning the water on just a little. Notice how clear is the flow of the water from the tap. There aren’t any bubbles in the water, and when you put your finger in the stream, it smoothly flows around it. This is called laminar flow. Now keep increasing the water flow until it is very fast and rough. The chaotic nature of the flow in this stream is called turbulence, and how a flow turns from being laminar to turbulent is a popular area of research. In general, turbulent flows are very fast and are made from fluids that are not very thick.

In today’s study, Dunkel and his colleagues investigate how bacteria can make flow patterns that look turbulent – chaotic and full of vortices – even though bacteria are tiny and slow. The bacteria push the fluid around as they swim and create vortices, spinning regions in the fluid. The 5 ?m long bacteria create vortices with diameters of 80 ?m by swimming at the speed of 30 ?m/s!

To determine whether a fluid flow is turbulent or laminar, the Reynolds number, a ratio of the strength of the flow (inertial forces) to how much the fluid resists motion (viscous forces) is used and defined as [1]:

$latex Re = \frac{\rho V D}{\mu}$

It depends on the fluid density ?, flow speed V, the length of an object D (for example, the diameter of a bacterium), and flow viscosity, or thickness, ?. When the Reynolds number of a flow is high, the inertia of the fluid (how powerfully it flows), is much higher than its viscosity (how thick the fluid is). In this case, the resistance of the fluid to small fluctuations in its motion is not enough to prevent the fluctuations from growing and spreading. Before you know it, your previously smooth, easily predictable flow is chaotic and full of vortices – it has transitioned to turbulence.

Typically, flow in a pipe like that in a tap becomes turbulent at a Reynolds number of 2300; the flow of air over an airplane begins transitioning when the Reynolds number is about a million. Bacteria are so tiny and slow that their Reynolds number is very small – on the order of $latex 10^{-5}$.

The researchers in this study investigate turbulent-like fluid behavior caused by swimming bacteria, B. Subtilis, in a fluid. B. Subtilis is a rod-like bacteria that swims by pushing its surrounding fluid with its flagellum (or tail). The researchers grew the bacteria in a nutritious fluid medium, and they added small, 1 micrometer beads to the fluid to act as tracer particles. When tracer particles are added to a fluid, you can see them being pushed around by its flow. If the particles are small enough, tracking them can be used to show how the fluid is moving. Thus, the experiment consisted of a suspension (a mixture in which particles do not dissolve) of bacteria and particles in the fluid.

The researchers loaded this suspension into sealed microfluidic cylindrical chambers with a 750 ?m radius and 80 ?m height (Figure 1). Since the chamber was sealed, the bacteria grew tired and did not move as much as they ran out of oxygen, so their activity levels decreased during the 10 minutes they spent in the chamber. Thus, by waiting several minutes between sets of measurements, the researchers were able to test how the bacteria affected the fluid at different energy levels.

The researchers took high-speed 2D videos of slices of the suspension containing the bacteria and the particles in the middle of the cylinder (even though the motion of the bacteria was in 3D). They used visible light to illuminate the motion of the bacteria and fluorescent light to view the motion of the tracer particles through a microscope.

setup — Figure 1: Bacteria and tracer particles in the field of view of the camera. Although the entire test chamber is filled with fluid, only the 2D cross-section in the center of the cylinder is in focus.

The researchers used two methods to measure the motion of the suspension of bacteria and particles. First, by monitoring the motion of the bacteria, they obtained vectors showing how the bacteria were moving. Trajectories of bacterial movement were calculated by comparing the position pattern of the bacteria from one image frame to another. In the second approach, they tracked the motion of the particles in the flow by comparing the actual position of the particles from one image frame to the next. The researchers used both methods (monitoring patterns of bacterial motion and tracking individual particle positions) to make sure the flow fields measured from bacteria were an accurate representation of the flow. Since the tracer particles used by the researchers were very small, with diameters of 1 micrometer, they were small enough to be reliably pushed by the flow.

The typical results the researchers got are shown in Figure 2, with visible vortices.

The researchers measured the velocity distributions of the bacteria. They found that the results from the tracer particles and the bacteria had the same distributions – the flow of the bacteria and solvent were very similar, and the flow of the bacteria could be used to represent the fluid flow. However, it is possible that the particles were pushed around by the bacteria, and not by the motion of the fluid itself, which is not mentioned in the paper.

To analyze the average motion of the bacteria, the researchers calculated various properties of the velocity field. They calculated the vorticity, $latex \omega_z$, or how much the bacteria rotated the fluid. The average of the square of the vorticity throughout the 2D experimental plane is called the enstrophy, $latex \Omega_z$. They then calculated the kinetic energy of the flow, $latex E_{xy}$, the energy a fluid has because of its speed, and also calculated its average throughout the space (1). Although the instantaneous kinetic energy and vorticity fluctuated as the bacteria moved, the average kinetic energy and enstrophy over time were approximately constant throughout the 50 seconds of recording.

The researchers then measured two properties of the flow with functions called the VCF (“equal-time spatial velocity autocorrelation function”) and the VACF (“two-time velocity auto-correlation function”). The first function, the VCF, measures how the velocity of the fluid changes throughout the space (the 2D slice of the cylinder). If this function goes from positive, velocities in the same direction, to negative, velocities in different directions, then it indicates that there is a vortex in the fluid (Figure 3a). The results of the VCF are shown in Figure 3b. The researchers calculated that the radius of a vortex in the fluid was $latex R_v = 40 \mathrm{\mu m}$ from the VCF. The vortex radius did not change with the kinetic energy of the flow.

The average enstrophy was found to be linearly proportional to the time-averaged kinetic energy by about half the vortex radius ($latex \Lambda = \mathrm{24 \mu m}$) over all the energy scales tested:

$latex \overline{\Omega_z} =\frac{\overline{E_{xy}}}{\Lambda^2}$

So when bacteria have more energy, the fluid has a higher tendency to form vortices and rotate (but the vortices will be the same size).

The second property the researchers calculated was the “VACF”. The VACF measures how the velocity changes over time. The VACF represents the memory of the fluid. If it decays to 0 slowly, that means the velocities stay similar for a long time; if it decays quickly, the velocity changes in a very short time. The results of the VACF are shown in Figure 3c. The researchers found that at higher energies, the system has a shorter “memory”. The VACF shown in black has the highest energy, and decays much faster than the VACF shown in purple, which has the lowest energy. In bacterial turbulence, bacteria add energy to the system by swimming to make small vortices, which then lead to larger vortices. This is the opposite of how vortices form in a non-active turbulent fluid, where the energy is added to the larger vortices that create smaller vortices because of the fluid’s viscosity.

correlation — Figure 3. (a) Diagram of a vortex. The blue and green vectors are correlated (positive VCF), showing that there is no vortex between them; the blue and red vectors are anticorrelated (negative VCF), showing the presence of a vortex. VCF (b) and VACF (c) of the bacterial motion. The minimum value of the VCF indicates a vortex with a 40-micron radius. The VACF varies with energy when plotted as a function of the time lag. Inset: energy as a function of time; the black spheres show the highest energy and data in purple is at the lowest. Figure modified from the original paper.

Finally, the researchers present a recently developed theory for the flow caused by bacteria. This theory is a continuum model equation – it treats the suspension of fluid, bacteria, and particles as if it were a continuous material, and accounts for the effects of the fluid flow and the forces the bacteria apply to the system.

The equation in the theory can be modeled to predict how the suspension will move. If the parameters of the equation in this theory are chosen to represent flow without bacteria, it simplifies to a theoretical fluid flow model. The researchers chose coefficients in the equation known to represent bacteria that push the fluid, like B. Subtilis in this experiment. They found that the model was accurate to within 10%-15%, making it a good candidate for a quantitative description of bacterial turbulence.

In today’s paper, Dunkel and his colleagues made significant contributions to the understanding of bacterial turbulence. The researchers developed a method to show that the motion of the bacteria swimming in a fluid can be used to measure the motion of the fluid itself. They developed a mathematical model of the motion and tested it with their experimental results to create a method for quantitatively studying how bacterial motion affects fluid flows. The observations they made can be used to compare bacterial turbulence to traditional turbulence in fluid mechanics, and give insight into how other fluids with active particles might behave.

[1] Fox, Robert W., Alan T. McDonald, and Philip J. Pritchard. Chapter 2, Introduction to fluid mechanics. Vol. 5., New York: John Wiley & Sons, 1998.

[2] Dunkel, Jörn, et al. “Fluid dynamics of bacterial turbulence.” Physical review letters 110.22 (2013): 228102.

(1) The vorticity, $latex \omega_z$, is calculated by taking the difference between the change in the y-component of the velocity, $latex v_y$, in the x-direction and the x-component of the velocity, $latex v_x$ in the y-direction:

$latex \omega_z = \frac{\partial v_y}{\partial x} – \frac{\partial v_x}{\partial y}$

The average kinetic energy (represented by angle brackets is calculated as:

$latex E_{xy} = \frac{v_x^2+v_y^2}{2}$

As the mass of the bacteria was constant, the energy depends only on the measured velocities of bacteria, and mass was not included in the calculation. As you may remember, a range of energies was tested by varying the bacterial activity.

The enstrophy, $latex \Omega_z$, is calculated as the average of the vorticity as:

$latex \Omega_z = \langle\frac{\omega_z^2}{2}\rangle$

2018-01-172018-04-14

The living silly putty, episode 2: the spreading!

Original paper: Spreading dynamics and wetting transition of cellular aggregates

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.

E-cadherin level (controls cell-cell adhesion energy)	21%	48%
Relative viscosity to the 100% expression	42%	57%

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.

SM02 — 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.

2018-01-112018-04-14

Rebuilding hard matter with soft matter

Original paper: Composite Colloidal Gels Made of Bisphosphonate-Functionalized Gelatin and Bioactive Glass Particles for Regeneration of Osteoporotic Bone Defects

The skeleton is the backbone of the body, both literally and figuratively. Healthy bones protect soft organs from injury and enable the body to move. Starting from childhood, staying active and following a healthy diet helps the body maintain healthy bones. However, as people age, their bones can start to weaken. There are often no early symptoms to weakening bones, and as a result the first indication of a problem may be a painful break once the weakening has already significantly progressed.

Although it may seem like bones are made of a hard material, they are actually an elegant combination of hard and soft materials. A primary component of bone is collagen, which forms a soft protein network. This network then provides a scaffold for calcium phosphate, a mineral that provides bone with its hardness and strength. This mixture of hard and soft material enables bone to be flexible enough to withstand impacts, but rigid enough to maintain its structural integrity.

Bone tissue, like other tissue in the body, is alive and therefore able to grow and heal. The material in bones is always being resorbed (or removed) by the body and simultaneously replaced. Around the age of thirty, the rate of resorption overtakes the rate of replacement, causing bones to slowly weaken. When these rates become too disparate, osteoporosis can develop, causing bones to become weak and porous [1].

Bone fractures are more common in patients with osteoporosis, and when they occur, treatment is often needed at the site of the break to promote bone regeneration. The typical treatment in these cases is an autologous bone graft, which involves taking bone from another area in the patient’s body and transferring it to the fracture site to help promote bone regrowth. This technique has two clear downsides: there is limited availability of bone tissue for grafting since it has to come from somewhere else in the person’s body, and removing bone tissue for a bone graft can cause damage to the donor site.

As a result, researchers are working to develop synthetic materials that can provide a better alternative to autologous bone grafts. A promising material would promote bone growth and be strong enough to sustain bending and support weight. Additionally, it needs to be non-toxic and not cause an immune system reaction. Synthetic bone graft materials already exist, but they do not work as well as bone grafts taken from the patient’s own body because they are not as mechanically strong and they do not promote as much bone growth. Hence, the search for better synthetic bone graft materials continues.

In today’s paper, Mani Diba and co-workers investigate a new synthetic material for use in the regeneration of bone tissue in osteoporotic patients. The material in question is a colloidal gel, which is a disordered network structure made of microscale particles suspended in a liquid. This network allows the material to resist applied forces and behave like a solid, even though it may be mostly made up of liquid. Colloidal gels are different from chemically covalent bonded gels [2], like jelly or agar, because their building blocks are microscale particles instead of polymers. These particles bond to each other mostly because they are hydrophobic, or water-repellent, so they would prefer to be next to each other than surrounded by water. The bonds between the colloidal particles are reversible, meaning they can break and reform more easily than covalent bonds in polymer gels, which allows the colloidal gel network to be more adaptable and reform after being broken apart.

The behavior of the colloidal gels is similar to that of toothpaste, which acts like a fluid as it is being forced out of the tube, but once it stops being squeezed it becomes more solid again and doesn’t flow off your toothbrush. For a bone graft material, this means that the colloidal gel can behave as a liquid as it is being injected into a bone defect, and then harden as the network reforms once it’s in place. While this is not a requirement for bone graft materials, it does make the material easier to put in place at a bone defect site.

The researchers prepare a colloidal gel by mixing gelatin particles and glass particles in water (see Figure 1). This choice of particles mimics the structure of bone tissue by using gelatin- a soft material- with glass, which is hard and provides mechanical strength. In order to be a good replacement for a bone graft, the gel must satisfy two main requirements. First, it needs to be mechanically robust to serve as a load-bearing scaffold for bone growth. Second, it needs to be biocompatible, meaning that it should support the growth of new bones.

fig1_all — Figure 1: (a) A schematic showing the formation of a gel by mixing glass and gelatin particles (b) Electron microscopy images of colloidal gelatin (left) and glass (right) particles (adapted from Diba et al.)

The first set of experiments in this paper look at the mechanical properties of the colloidal gel by measuring its storage modulus, which characterizes how strong the gel is. The researchers find that increasing the ratio of glass to gelatin particles or increasing the total number of particles in the gel increases the storage modulus by a factor of more than 100, from about 0.1 kilopascal to tens of kilopascals. The gel is also able to recover its initial storage modulus after being broken apart by shearing, similar to how silly putty can recover its mechanical properties after being stretched. This indicates that the network is able to reform in the bone and become solid again, as expected.

After characterizing the gel’s mechanical properties, the researchers investigate whether it can promote new bone growth. The growth of new bone starts with the multiplication of osteoblasts, or bone-forming cells, that produce bone matrix material. A signature of this process is an increase in the levels of certain enzymes. Once the matrix is well formed, it undergoes mineralization, which is the deposition of inorganic material (calcium) onto an organic matrix (collagen). This process can be monitored by measuring the amount of calcium added to the area [3]. The researchers track these two indicators, enzyme levels and calcium deposition, to measure the biocompatibility of the gels.

Diba and coworkers study the biocompatibility of the gels both in test tubes and in living animals. In the test tubes, they only find significant mineralization at a glass to gelatin ratio of 0.5 (the highest investigated), which also corresponds to the largest peak in enzyme levels. For testing in animals, the researchers therefore opt for a composite gel with a glass to gelatin ratio of 0.5 and compare the bone growth to that with a single-component gel with no glass particles. The researchers implant these gels in bone defects in the femurs of osteoporotic rats and measure the amount of bone growth after 8 weeks.

Surprisingly, in the rats, the addition of glass particles to the gel did not increase the amount of bone mineralization beyond that seen in the single-component gel as the researchers hoped. However, the bone growth in the composite gel did show more blood vessel-like structures than in the single component gels (see Figure 2), which is important because bone—like other living tissues—needs blood flow to supply oxygen and nutrients, as well as to remove waste products.

fig3_bone_2 — Figure 2: Images of bone regrowth from the composite gel in a rat femur. Left: Bone regrowth in a defect that was originally the size of the black circle. Center: Higher magnification image of the small green rectangle on the left. Black arrows point to blood vessel-like structures. Right: Higher magnification of the red rectangle in the center. Red arrows point to cells observed in the center of the original defect. (Adapted from Diba et al.)

Though the researchers in this study did not find the desired increase of bone mineralization in live rats by using a composite gel instead of a single-component gel, they did see other indicators of improvement. Including glass particles increased the storage modulus of the gel, indicating more mechanical strength. They also saw indicators of improved biocompatibility. The bone growth in the composite gel showed an increase in blood vessel-like structures, and they found test tube results which suggested that including the glass particles may still improve mineralization if a higher ratio of glass to gelatin is used. Considering these improvements over a single-component colloidal gel, this composite colloidal gel is a promising development in the search for better bone graft materials.

[1] https://www.bones.nih.gov/health-info/bone/osteoporosis/overview#b

[2] Covalent bonding is the sharing of valence electrons, which are in the outer shell of electrons, between atoms to make a full valence shell. Any time two non-metals come together they will share their valence electrons.

[3] http://www.promocell.com/fileadmin/knowledgebase/pdf-xls/Osteoblast_Differentiation_and_Mineralization.pdf

2018-01-012018-04-14

Kepler’s New Year’s Gift — On the Six-Cornered Snowflake

Original Booklet: On the Six-Cornered Snowflake: A New Year’s Gift by Johannes Kepler

Some things never change. In winter 1610, Johannes Kepler was stressing out about holiday gifts — in particular, one for his friend and benefactor, the rather grandly-named Johannes Matthaeus Wacker von Wackenfels. Kepler, at the time employed as Imperial Mathematician at the court of Holy Roman Emperor Rudolph II, records his musings on the problem in the opening pages of his now-famous discourse, The Six-Cornered Snowflake.

Kepler sets a high standard for himself. His gift should be of the intellectual variety: an amusing idea or a clever argument. After all, that’s why Wacker keeps him around. Kepler considers several potential topics, dismissing each in turn as being either too serious or too light. Kepler’s intellectual respect for Wacker — who was an accomplished scholar and amateur philosopher in his own right — renders other topics off limits. (An example: given the impressive size of his patron’s zoological library, Kepler jokingly complains that writing a treatise about animals would be “like bringing owls to Athens.”) Wandering around town, feeling guilty about his procrastination, Kepler notices some snowflakes landing on his coat, “all with six corners and feathered radii” [1]. Kepler immediately identifies the perfect topic for his essay:

“‘Pon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year’s gift… the very thing for a mathematician to give.”

The pattern Kepler alludes to here is the six-fold shape of the snowflake [2]. Considering this shape, Kepler alights on what will become the central puzzle of the piece,

“Our question is, why snowflakes in their first falling, before they are entangled in larger plumes, always fall with six corners and with six rods, tufted like feathers.”

In other words, why should snowflakes be six-sided, rather than five-sided, seven-sided, or anything-else-sided? Kepler’s attempts to answer this question are a treasure trove of condensed matter physics: they include the first observation in print that regular shapes can arise from close-packing of identical objects [3] and the famous conjecture that hexagonal packing is the densest way to fill space with spheres [4]. But perhaps the most interesting thing about The Six-Cornered Snowflake is how Kepler sees nature and how he wants the reader to see it.

Untitled.001.jpeg — **Left:** ***Kepler’s sketches of sphere packings.*** **Middle:** ***Thomas Bentley’s classic snowflake photographs, taken around 1902.*** **Right:** ***Pyramidal packing of colloidal spheres gives the beetle*** ***P. argus its characteristic iridescence. (Scale bar is 1 micron.)***

Intellectually and personally, Kepler straddled the gradual transition away from the medieval era of alchemy and astrology [5], and towards the modern age of empirical observation, mathematical models, and testing ideas by experiment. Despite some asides that strike the modern reader as somewhat mystical in character — for instance his numerological obsession with the properties of the natural numbers — we can immediately recognize The Six-Cornered Snowflake as a scientific work. While the whole chain of reasoning is somewhat convoluted [6], The Snowflake includes the following:

1. Kepler imagines matter as being made of tiny, discrete “pieces” that are all identical to one another;

2. He considers (and draws!) how the arrangement of those pieces might influence the material properties of an object, in particular, its shape. In modern parlance, Kepler defines a crystal: a macroscopic object made out of identical pieces arranged into a regular structure called a lattice [7];

3. He tries to articulate a physical principle — such as close-packing — that might explain why the small pieces arrange themselves in such a way as to produce a crystal lattice.

By Kepler’s own admission, none of his arguments adequately explain the shape of snowflakes. (For one thing, he can’t figure out how a three-dimensional process could possibly create two-dimensional crystals.) But, despite this failure, Kepler still manages to suggest a productive direction for future research. Noting that different substances crystallize into different 3D shapes [8], Kepler finishes his essay by kicking the problem over to another branch of the natural sciences: “I have knocked at the door of chemistry and see how much remains to be said before we can get hold of our cause.” In other words, Kepler correctly intuits that understanding the form of crystals necessitates understanding their chemistry.

Today, we know that macroscopic objects are indeed made of tiny identical pieces — atoms and molecules — and that those pieces often arrange themselves in structures that are highly reminiscent of Kepler’s sphere packings. We have learned how to accurately describe the forces that bind atoms together or push them apart. In addition to the shape of crystals, we know that many important material properties — most strikingly rigidity, the ability of a solid to resist deformation — arise because of the regular arrangement of atoms on the micro-scale. We understand (some) statistical physics, which explains how, at high enough temperature, thermal motion overcomes the forces holding the atoms in place, destroying the lattice and melting the solid. Our knowledge of the physics and chemistry of solids has allowed us to engineer with precision the technologies — in particular silicon-based semiconductors — that underpin the modern world.

Although Kepler couldn’t have begun to imagine all this, scientifically speaking, the world of The Snowflake is very modern: a world of material cause and material effect, of microscopic bodies in motion and in contact, a world of forces that are invisible yet comprehensible, and where the properties of the whole can be understood by considering its parts. As physics students, we learn that all the fundamental physical laws can be written on the back of a napkin. And yet, the materials in the world around us exhibit an amazing variety of properties: solid and liquid, conductive and insulating, magnetic and not. How can such a zoo of behaviors and properties arise from physical laws that are fundamentally simple? Kepler’s essay gives us a framework to understand the apparent contradiction. Kepler says: look inside. Look at the pieces. Look at the structure and the symmetry.

A pretty good New Year’s gift for a soft matter enthusiast, even in 2018.

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[1] In today’s world of snowflake wrapping paper, snowflake ornaments, and snowflake emoji, it’s hard to imagine that there was a time when people didn’t know what snowflakes actually look like, but Kepler was apparently the first European to write about the hexagonal symmetry of snow crystals. (The observation is, however, recorded in much older Chinese documents, from the 2nd century BC.)

[2] In fact it isn’t: there are triangular snowflakes too, and how they form is still an active area of research.

[3] Prior to the publication of The Snowflake, English scientist Thomas Harriot privately communicated his ideas on the efficient stacking of cannon balls to Kepler.

[4] This conjecture was only formally proved in 2015, by a group of mathematicians led by Thomas Hales.

[5] Kepler, who was at times employed as an astrologer, thought that astrology, as practiced in 17th century Germany, was mostly nonsense. However, he himself cast horoscopes that he claimed were correct and argued with scholars who wanted to dismiss the discipline entirely.

[6] Despite Kepler’s assurances in the introduction that his piece is “next to nothing,” The Snowflake is 21 pages long — evidence, I think, of the modern tyranny of page limits and copy editors.

[7] Interestingly, this argument correctly predicts the form of so-called “complex materials” like opal (colloidal silica), where the pieces really are (relatively) uniform hard spheres. The water molecules in an ice lattice have much more complicated, directional interactions arising from the hydrogen bonds between them, and so their crystal structure is harder to understand or predict. In fact, it seems that Kepler generally has something like colloidal particles in mind throughout The Snowflake, rather than modern atoms or molecules.

[8] “But the formative faculty of the earth does not take to her heart only one shape; she knows and is practiced in the whole of geometry. I have seen… a panel inlaid with silver ore; from it, a dodecahedron, like a small hazelnut in size, projected to half its depth, as if in flower.”

2017-12-282018-04-14

Drop deformation in miniature channels under electric field

Original paper: AC electric field induced droplet deformation in a microfluidic T-junction

Microfluidics is the science and technology of manipulating small volumes of fluid in channels with dimensions as small as the size of human hair. You can think of a microfluidic system as a plumbing network composed of miniature pipes. Microfluidics has the potential to advance biology, chemistry, and medical diagnostics by allowing many operations such as mixing of fluids, and synthesis of materials, as well as lab analysis to be miniaturized and integrated into a single device. Such a device is typically only a few cm² in size and is called a lab-on-chip platform. Many analyses that are done using lab-on-chip devices use droplets. For example, instead of growing cells on a flat surface, such as Petri dishes, it is possible to grow cells inside a droplet. The advantage is that one can better control the microenvironment, allowing high throughput single cell manipulations.

Electric fields are often used in lab-on-chip systems to control droplet generation, sorting, merging, and mixing. These active droplet manipulation methods often involve deformation of droplets. Although there are other techniques for droplet manipulation such as thermal, magnetic, and acoustic, electric fields are often preferred as they provide a faster response time. However, the interaction between electric fields and droplets in lab-on-chip systems is poorly understood. Thus, it is of vital importance to have a better understanding of an electric field induced droplet deformation.

In general, when a uniform electric field is applied to a conductive drop, for example, a water drop containing salt, suspended in an insulating liquid such as mineral oil, charges will accumulate at the drop interface due to a mismatch between the electrical properties of the water and oil. The accumulation of charges at the drop interface (shown in Figure 1) will induce electric stresses that will deform the drop. Today’s paper focuses on the on the effect of electric fields due to alternating current (AC), which is a current that periodically reverses its direction. AC current has many potential biological applications, and its effect on droplets has not been studied extensively.¹

Figure 1 post 2 — **Figure 1.** Schematic of electric field induced polarization (movement of charge carriers to the interface) of a conductive drop suspended in a non-conductive medium. Electric stresses deform the drop and result in its elongation in the direction of the electric field.

The researchers used a microfluidic device to make water-in-oil emulsion droplets. The droplets are generated using a T-junction geometry as shown in Figure 2. Water is injected into a flowing stream of oil where it is sheared off into individual droplets. An electric field is applied using two electrodes (shown in black and red in Figure 2) that are positioned on both sides of the droplet channel. The electrodes are not in contact with fluids to prevent electrolysis of water (a process where electricity is used to break apart water into hydrogen and oxygen).

The deformation of the droplet is imaged as it passes through the electrodes using a microscope and a high-speed camera at 5000 frames per second. Before entering the electric field, the droplet takes the shape of a horizontal ellipse due to deformation by the flow. When the droplet enters the electric field, the shape of the droplet changes from the horizontal ellipse to an ellipse with a flattened back side, illustrated in Figure 2, as electric stresses act mainly in the direction of the electric field.

figure 2 post 2 — **Figure 2.** Schematic sketch of the microfluidic device and the electric field induced deformation. The droplet deforms as it passes through the electrified region. The blue and gray streams are water and oil phase, respectively. Adapted from Tan *et al.*

The effect of field strength and AC frequency on droplet deformation is shown in Figure 3 and is captured by a dimensionless parameter D which represents the change in the droplet aspect ratio (the ratio of the droplet length along the electric field direction to the length in the flow direction) after deformation. At low electric field strength², D Is directly proportional to the electric field strength and is independent of AC frequency. One possible explanation for this lack of relationship between D and AC frequency is that at low electric fields, viscosity and interfacial tension (a measure of the tendency of liquids to resist deformation by an external force) are the dominant factors determining the droplet deformation.

figure 3 post 2 — **Figure 3.** Deformation of droplets expressed as D (change in droplet aspect ratio after being subjected to an electric field) as a function of electric field amplitude at different AC frequencies. Images of the droplet at different field strengths are also presented to show the shape change of the droplet (AC frequency remains at 40 kHz). Adapted from Tan et al.

A promising feature of this setup is the ability to induce controlled oscillation in droplets. The authors used amplitude modulation to induce oscillation in droplets. Amplitude modulation is a technique commonly used in communication applications, such as radio, to transmit information.

In this technique, we start with a wave of high frequency – a carrier wave and add to it a wave of low frequency – a signal wave, as shown in Figure 4. The combination of both will give a wave with the same frequency as the carrier wave and an amplitude which is higher than the original carrier wave.

figure 4 post 2 — **Figure 4.** Amplitude modulation: the combination of a carrier and signal wave to result in a wave with characteristics of both carrier and signal wave. Image adapted from https://www.tutorialspoint.com/communication/amplitude_modulation.asp

Tan and his coworkers used amplitude modulation as a periodic on-off signal. The droplet deformed when the signal is on (100% amplitude) and goes back to its original shape (0% amplitude) when it is off, as shown in Figure 5. Here, higher frequency of the AC field corresponds to faster switching of the electric field amplitude.

Figure 5 post 2.gif — **Figure 5.** Periodic droplet deformation under different AC frequencies. Adapted from Tan *et al.*

In summary, the research group led by Tan have provided a unique platform to deform and oscillate the deformation of microdroplets in microfluidic channels. This result has tremendous potential in many future applications including drug screening, cell study, chemical reaction and any other applications for which enhanced mixing conditions are preferred.

1 Both AC and direct current (DC) field can induce drop deformation in a similar physical mechanism as described in the article. However, the use of DC fields often involves heating of the sample due to the use of large direct current. Such heating might be undesirable in biological samples that are heat sensitive. In the case of AC fields, the current is smaller compared to DC, thus there is less heating.

2 At higher field strengths, the droplet aspect ratio depends non-linearly on the field. In fact, high enough fields can even break the droplet apart completely.

2017-12-132018-04-14

The living silly putty

Original paper: Spreading dynamics and wetting transition of cellular aggregates

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.

fig-01 — **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.

fig-02 — **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):

$latex \gamma_{ws} = \gamma_{as} + \gamma_{wa} cos\theta$

To predict if the drop will spread on the surface the wetting coefficient $latex S$ can be defined as:

$latex S = \gamma_{as} – ( \gamma_{wa} + \gamma_{ws} ) $

$latex S = – \gamma_{wa} (cos\theta -1)$ (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} $.

fig-03 — **Figure 3. W_cc measurement (adapted from** Chu Y.-S *et al.*)

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.

2017-12-062018-04-14

Dripping, Buckling and Collapsing of a Droplet

Original paper: Mechanical stability of particle-stabilized droplets under micropipette aspiration

Most of us have had the childhood experience of blowing bubbles. But have you ever wondered how bubbles form and what keeps them stable? The key to making bubbles is surface tension, the tension on the surface of a liquid that comes from the attractive forces between the liquid molecules. Water has a very high surface tension (that’s why bugs can walk on water) making it difficult to stretch to form a thin water layer that we see when bubbles form. By adding soap to water, we can lower the surface tension of the water, allowing us to stretch this water-air interface to form a thin water sheet. As you blow more and more air into a bubble, the bubble will grow larger and larger as the thin layer stretches. Eventually, you’ll reach the limit of the added stretchiness, and the bubble will burst, engraving in your memory its fragile nature.

Fig1-1 — A typical air bubble made out of a water-soap mixture (Figure courtesy of Gilad).

In soft matter, sometimes scientists utilize materials such as solid macroscopic particles instead of soap molecules to reduce the surface tension of an interface. Using particles to stabilize an interface allows them to tailor the mechanical and chemical properties of the interfaces to fabricate capsules. For instance, if a capsule needs to travel in blood-stream for therapeutic purposes, it must be tough enough to withstand blood pressure without rupturing. But if we make such a capsule how can we measure its mechanical response?

In this post, we’ll look into the work by Niveditha Samudrala and her colleagues on measuring the mechanical properties of a particle-stabilized interface. They utilize a direct approach of applying force on such a stabilized interface to study its mechanical response that has eluded earlier techniques. Knowing the stiffness of these particle-coated interfaces, say in the form of capsules, would enable us to use them for different controlled-release applications such as treating a narrowing artery [1] as well as tune them to have different flow properties.

The authors use tiny (smaller than a micrometer!) dumbbell-shaped particles with different surface properties to stabilize an oil-in-water emulsion (see note [2]). Here instead of a thin layer of water sandwiched by the soap molecules, the water-oil interface has been stabilized with micron-sized particles. This stabilization technique will render higher mechanical properties to the interface. Droplets stabilized in this way, known as colloidosomes, have been shown to be capable of encapsulating a wide variety of molecules.

The researchers characterized the particle-stabilized droplets using the micropipette aspiration technique. To understand this technique, imagine picking an air bubble with a straw. What you need to do is to approach the air bubble and then apply a gentle suction (or aspiration) pressure. When the suction pressure becomes larger than the pressure outside of the droplet, then the droplet gets aspirated into the straw forming the aspiration tongue (Figure 1A). Similarly, in the micropipette aspiration technique, a glass pipette (the straw) with an inner diameter of $latex R_p$ is usually used to aspirate squishy stuff, such as cells, vesicles, and here droplets.

To obtain the tension response, therefore the toughness of an aspirated interface, we need to consider the pressures applied to the interface. Let’s consider an aspirated droplet as shown in Fig 1A at mechanical equilibrium (which means the sum of all the forces is zero). We know that each interface has a surface tension acting on it (See Fig 2a). In our bubble example, I mentioned that the soap molecules tend to gather at such interface to decrease the tension (See Fig 2b). But when there are other forces acting on the interface in addition to the presence of the molecules, such as the suction pressure in our case, the tension of the interface now comes from both the surface tension and the suction force. We call this total force the interfacial tension (See Fig 2c). The Young-Laplace equation can be used to relate this interfacial tension to the pressure applied to the interface (Fig 1-B3).

Fig1. Schematic representation of the aspiration technique (A) and the Young-Laplace equations obtained at both interfaces of the outer edge of droplet and tongue inside the pipette (B). $latex P_{atm}$ is the atmospheric pressure set to zero, $latex P_{droplet}$ is the pressure inside the droplet. $latex P_{pip}$ is the suction pressure. $latex R_{v}$ is the radius of droplet outside the pipette and the $latex R_{p}$ is the pipette radius.

When the molecules, or particles in our case, are forced to pack tightly together they oppose the compression force. This opposition is felt at the interface by a pressure called surface pressure (see Fig 2c). Under the interface tension and the surface pressure, the new net interfacial tension is defined as:

$latex \tau=\gamma_{0} – \Pi$.

where $latex \Pi$ is the surface pressure, $latex \gamma_{0}$ is the interface tension which is constant for a given interface.

In this study, Samudrala and her colleagues show that there are two critical pressures after which instabilities form at the interface resulting in droplet dripping into the pipette and buckling respectively (Fig 2d). They conclude that the dripping happens due to the transition of the interface from a particle-stabilized interface to a bare oil-water interface resulting in a sudden suction of tiny oil droplets (basically the droplet drips at this point, see Fig 3B, blue and 3C).

The second instability is the buckling which the researchers propose happens when $latex \tau$ tends to zero. Now let’s see how buckling happens.

The dripping at the first critical pressure decreases the volume of the particle-coated droplet, but note that the surface area is constant because neither particles leave the surface nor the free ones join the droplet (the latter argument is assumed). The continuation of the increase in suction pressure plus the volume lost in the dripping step results in the buckling of the interface (Fig 3b red and 3E, also see note [3]). When the authors aspirate the bare oil droplets as well as droplets stabilized by small molecules, they only see the sudden droplet disappearance with no shape abnormalities due to the fluid nature of the interface rather than solid-like nature for the particle-stabilized case. But why does the buckling happen?

Screen Shot 2017-11-06 at 17.55.45 — Fig 3. Evolution of instabilities of a particle-coated droplet under tension. (a) shows the schematics of the particle-coated droplet being aspirated. (b) Change in aspiration length as a function of suction pressure. Blue line remarks the capillary instabilities. Red line shows the elastic failure of buckling process. (c & d) are the images of capillary and buckling instabilities respectively. (e) shows the case when the suction pressure is above buckling pressure at which the particle coat fails (the figure is adapted with no further change from the original paper).

Recall how we defined the net interfacial tension above; $latex \tau=\gamma – \Pi$. The authors hypothesize that upon suction of a particle-stabilized droplet, particles jam at the interface of the droplet outside of the pipette, creating a high surface pressure. When this surface pressure approaches $latex \gamma$, the net tension becomes zero ($latex \tau=0$, see fig 2d and note how the interface tension is opposed by the surface pressure due to repulsion between particles). When an interface possesses no tension, it means that the interface can no longer bear any loads. Considering any sort of defects or irregularities due to nonuniform particle packing, for such interfaces deformations such as buckling will form. Now, let’s see how the authors test their hypothesis.

The authors observed that at the tip of the tongue, there is a very dilute packing of particles in such a way that the interface to a good approximation resembles the Fig 2a, a bare water-oil interface. With this observation, one can safely assume that the interfacial tension, the $latex \tau$ is equal to the oil/water interface tension, the $latex \gamma$ and write the Young-Laplace equation across the tip of the tongue (see Fig 1B-(1)):

$latex P_{droplet} – P_{pip} = \frac{2\gamma_{0}}{R_{p}}$

where $latex R_{p}$ is the radius of the pipette and is fixed. The authors experimentally show that for a range of droplet size ($latex 10\ \mu m < R_{droplet} < 100 \ \mu m$), the droplet pressure right before buckling varies very close to zero (in above equation all parameters are known except the $latex P_{droplet}$, which is calculated when we put $latex P_{pip} = P_{buckling}$). Therefore, considering the equation (2) in Fig 1B, the net tension would be zero (see note [3]) and with this, the authors correlate that the reason for the formation of buckling is the net-zero tension of the interface.

Taking it all together, we saw that for a droplet with solid-like thin shell, the mechanical response is completely different from the bare or the molecule-stabilized interface. A fairly rigid interface undergoes buckling due to its net tension tending to zero and knowing the threshold of buckling will enable us to tune the mechanical properties of such droplets for different applications from load-caused cargo release (see note [1]) or emulsions with varied flow properties. Imagine if we encapsulate a fragrance in our air bubble, which upon rupturing will release the scent. Now, wouldn’t it be nice if we could control the toughness of this bubble or similar architecture to rupture under a specific condition that we desire (see note [1])?

[1] In a disease called atherosclerosis, the arteries narrow down due to plaque buildup. In this narrow region, the blood pressure is higher than the normal region of the artery. So one can use this pressure difference to crack release the relevant drug from the capsule only in the narrow regions of the artery to dissolve the plaques away. Neat!

[2] If we apply a shear force on a mixture of two or more immiscible liquids in the presence of a stabilizing agent, we produce an emulsion and the stabilizing agent is called an emulsifier. The particles show a significantly higher tendency to gather at an interface in comparison to amphiphilic molecules. Thus, particles are strong emulsifiers. If we mix lemon juice and oil, soon after stopping the mixing, the two solutions will separate. Now, if you add eggs, you stabilize this mixture (egg works as an emulsifier) and you get Mayonnaise!!

[3] The authors report that for particle-stabilized droplets they observed different deformation morphologies such as wrinkles, dimples, folds and in some case complete droplet failure. They attribute this diversity to the non-uniformity of particle packing at the interface. But what is interesting to me is when they decrease the suction pressure, the droplets go back to their original spherical shape and then upon the second aspiration, the deformations happen at the same exact location as were for the first aspiration. This means that during the suction, there is limited particle rearrangement (Watch here).

[4] We can easily set the atmosphere pressure to zero before aspirating the droplets, thus here the $latex P_{atm} = 0$.

2017-11-292018-04-14

When espresso evaporates: the physics of coffee rings

Original paper: Capillary flow as the cause of ring stains from dried liquid drops

fig1a — Figure 1. A 2-cm dried drop of coffee with a stain around the perimeter, forming a coffee ring. Adapted from Deegan *et. al.*

I’ve spilled a lot of coffee over the years. Usually not a whole cup, just a drop or two while pouring. And when it’s just a drop, it’s easy to justify waiting to clean it up. When the drop dries on the table, it forms a stain with a ring around the edges (Figure 1), giving it the look of a deliberately outlined splotch of brown in a contemporary art piece (when I say “coffee ring” I mean the small-scale, spontaneously formed stain around the edge of the original drop, rather than the imprint left on a table from the bottom of a wet coffee cup). But the appearance of these stains is simply a result of the physics happening inside the drop. Coffee is made of tiny granules of ground up coffee beans suspended in water, so the ring must mean that these granules migrate to the edge of the droplet when it dries. Why do the granules travel as they dry? Today’s paper by Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greb Huber, Sidney R. Nagel, and Thomas A. Witten provides evidence that coffee rings arise due to capillary flow–the flow of liquid due to intermolecular forces within the liquid and between the liquid and its surrounding surfaces.

Figure 2. Diagrams of contact angles for different droplets. From left to right, the first is exhibits poor wetting, with a large contact angle. The next has good wetting, with a smaller contact angle. The last has perfect wetting, with a contact angle of zero, and coffee grains suspended in this solvent would not be able to form a ring upon drying.

The researchers found that these rings don’t just form in coffee. Their observations showed that the rings form in a wide variety of solutes (the suspended coffee granules), solvents (the water), and substrates (the table you spill on) as long as a few conditions are met. First of all, the droplet has to have a non-zero contact angle[1] (See Figure 2). In other words, the droplet doesn’t spread out into a completely flat puddle on the table. Second, the contact line has to be pinned. This means that the surface has irregularities or roughness that cause the edge of the droplet to get stuck in place. Last, the solvent has to evaporate; the ring won’t form if the droplet never dries.

So now we know the conditions required for rings to form, but we want to know how they form. Deegan and his colleagues found that the rings are caused by a geometrical constraint. Here’s how it works: The pinning of the contact line means that the perimeter of the droplet cannot move, so the diameter of the droplet has to remain constant. But if the water in the droplet is evaporating, the droplet’s height will be reduced at every point (Figure 3a). Along the edges, where the droplet is thinnest, the height would be reduced to zero, and the droplet would shrink.

But the contact line pinning means that droplet can’t shrink. To prevent this shrinkage, liquid must flow out to replenish the liquid at the droplet edge as it evaporates. This flow brings with it the suspended coffee granules (or whichever solute is suspended in the solvent), pushing them outward until they pack at the edge of the droplet to form a ring (Figure 3b).

droplet cross sections — Figure 3. (a) Diagram showing the cross-section of a droplet on a surface. The shaded region shows how the droplet will shrink due to evaporation after a small amount of time if the contact line is not pinned. (b) Now, a black line is added to show how the droplet will shrink if the contact line is pinned. The arrows indicate that more liquid must flow to the outside of the droplet to replace what is lost to evaporation. Adapted from Deegan *et. al.*

By calculating how quickly water evaporates from the surface of a droplet, the researchers derived an expression for the mass of the ring as a function of time. It takes the form of a power law, which can be shown as a straight line on a log-log plot. Equipped with a quantitative prediction, the researchers set about performing experiments to test their model. Instead of using coffee, they opted for plastic microspheres suspended in drops of water. They placed the drops on glass slides and used a video microscope to image the droplets as they dried, recording the particles moving to the edges of the droplet (Figure 4).

video — Figure 4. Particles flowing to the edge of a droplet during evaporation to form a ring. Video from [2] and produced by Deegan *et. al.*

The researchers knew the mass of the individual particles, so they were able to calculate the mass of the ring as a function of time by counting the particles as they traveled to the edges. The results were shifted by an offset time t₀ to account for early times where the power law prediction doesn’t hold and were shifted by mass M₀ to account for the particles deposited during this initial stage. From the plot comparing the data and theory (Figure 5), we can see that the prediction shows good agreement with the data.

M vs T — Figure 5. Plot of mass in the ring as a function of time. The mass is plotted in units of particle number, so the plot shows how the number of particles grows over time. The three lines correspond to three different droplets. The upper curve overlapped with the middle so was shifted up for clarity. The circles show data and the solid lines show the theoretical prediction. The slope of 1.37 is the exponent of the power law predicted by the theory; On a log-log plot, a power law is a line with the exponent as the slope. Adapted from Deegan *et. al.*

In the twenty years since this paper was published, the study of drying droplets has continued in full force [3]. Scientists have discovered various particle patterns that can form under different drying conditions. Why do we care so much about these drying droplets? If the beauty of the physics isn’t motivation enough, then maybe the applications will convince you. The physics of drying is essential to inkjet printing, and a better understanding of the drying process could help make more precise printers [4]. Drying patterns can be used to identify the presence of certain proteins, making this a potential tool for disease detection [5]. Maybe next time you spill some coffee, you’ll take a moment to think of the physics of the drying droplet before you wipe it away.

[1] The contact angle is the angle where a liquid-gas interface meets a solid surface. The smaller the contact angle, the better the wetting of the surface.

[2] https://mrsec.uchicago.edu/research/highlights/coffee-ring-effect

[3] https://www.nature.com/uidfinder/10.1038/550466a

[4] Soltman, D. & Subramanian, V. Langmuir 24, 2224–2231 (2008).

[5] Trantum, J. R., Wright, D. W. & Haselton, F. R. Langmuir 28, 2187–2193 (2012)