Mechanical Model of Globular Transition in Polymers

Understanding polymers in solution, in a wide range of environments – from DNA and proteins in cells to long chain polymers in gels – is important throughout science. In complex, multi-component systems, polymers often undergo phase transitions between distinct conformations; examples include the folding of proteins, or the coil-to-globule transition of homo-and hetero-polymers. This paper demonstrates a millimeter-scale granular model of coil-to-globule transitions: one “polymer” chain – a cylinders-on-a-string “pearl necklace” – and many spheres, all shaken on a horizontal surface. This model includes short-and long-range interactions and is more complex than most granular models of molecular systems. It is possible to describe the behavior of this granular system using formalisms generally used in statistical physics of polymers , i.e. first and second order coil-to-globule transitions. This designed granular system represents another kind of approach to the study of polymer phase transitions and might inspire future designs of polymer-like mesoscale systems.


Main text Introduction
Although a homo-polymeric molecule experiencing Brownian motion in solution, and a "pearl necklace" shaken on a horizontal flat surface in an assembly of spheres, are very different systems, they exhibit surprisingly similar statistical-physical behaviors. The problem we have explored is the extent to which systems topologically analogous to polymers, but with a scale of components large enough to be visible and with numbers of components small enough that all can be accounted for explicitly, exhibit complex phenomena such as phase transitions. Such systems could provide a tool for improving intuition about molecular phase transitions, and guide the design of molecular and nanoscale systems that exhibit such transitions.
The folding of proteins to the native state, or the coil-to-globule transition of homo-and hetero-polymers are examples of complex behaviors of long chain molecules in solution. [1][2][3][4] Numerous theoretical and experimental studies are actively performed to understand these transitions, and often involve intensive computations, or sophisticated experimental methods. [5][6][7][8][9] In parallel, solid-liquid granular phase transitions have been reported, and described in detail for layers on horizontally vibrating devices. [10][11][12] For example, Clerc et. al. reported a granular solidliquid phase transition in a vibrating granular monolayer in one dimension, 13 and Castillo et. al. extended this description to two dimensions. 14 Here we designed a granular system to simulate a phase transition that is ubiquitous in homo-polymers in solution: the transition from an expanded fluctuating coil state to a collapsed globule state, which depends on the intermolecular interactions between the polymer and the surrounding solvent, and on intramolecular interactions between different regions of the polymer. 1,15 If the solvent is a "good" one, the polymer chain will expand as the temperature of the system increases; if it is a "poor" solvent, the chain will collapse. This transition is usually second-order (continuous), but there is evidence that it could occur as a first-order (discontinuous) transition as well. 5,16 We previously introduced "mechanical agitation" (MecAgit) as a method for physical simulation of the behavior of microscopic systems, using a two-dimensional arrangement of millimeter-sized components that move under a pseudo-random agitation. 17,18 Even though MecAgit systems are macroscopic, out-of-equilibrium, and dissipative, and even though their motions are not completely random, we used them to mimic experimentally the folding of a short chain RNA, 19 the statistics of conformations of worm-like chain (WLC) polymers, 17 and the development of a Boltzmann distribution. 18 Other groups have used chain models as well: Ben-Naim et. al. studied topological constraints such as knots, 20 Safford et. al. the effect of confinement, 21 Zou et. al. glass transition in polymers, 22  Here, we use polymer statistical physics to describe the collapse of macroscopic cylinders-on-a-string shaken in an assembly of free spheres. We shook millimeter-sized cylinders-on-a-string (the "polymer chain") on a flat horizontal surface within a region bounded by vertical walls, in the presence of a variable number of free spheres (Fig. 1a). We changed the filling ratio (FR) of free spheres: that is, the ratio between the number of spheres in the system and the number of spheres required to completely fill the available area with a single layer of spheres having compact hexagonal packing (for a closed-packed packing, FR=100%). Unlike The novelty of our approach is the focus on designing granular models of phase transitions in polymer physics; our models have multiple phenomenological similarities to the behavior of polymers. We achieved this purpose by engineering inter-particle interactions and tailoring mechanical agitation.
other horizontally shaken granular systems, 12,25 ours has a pseudo-random agitation motion, which results from a combination of an orbital shaking of the surface, and randomization by agitation of a pendulum hanging underneath, with a linear actuator. 19 This type of agitation randomizes some aspects of the motion of the chain (although the motion of the spheres is strongly correlated) such that this motion could be described to a useful approximation by a "macroscopic temperature": that is, a parameter that had a simple experimental relationship to the frequency of the orbital shaker, f, and described the steady-state behavior of the chain. 17,18 This granular system is a model of polymers in solution: the cylinders-on-a-string represent a polymer molecule, and the assembly of free spheres a thermal bath of solvent molecules. To characterize the folding state of the chain we used its radius of gyration, R g , defined as the root-mean-square distance between the monomer units and the center of mass of the polymer: 26 where N is the number of cylinders, r k are the positions of the cylinders and r CM is the position of the center of mass of the chain.

Modeling the second-order coil-to-globule phase transition
First, we studied a chain made of ten cylinders threaded on a string and separated by small spheres that allowed the chain to bend (Fig. 1b). 19,27 The contour length of the chain was equal to 23.9 cm; in absence of free spheres, the chain had an extended (but not completely stretched) conformation under agitation that was described by a worm-like chain (WLC) behavior with a persistence length equal to 12.2 cm. 17 Under certain experimental conditions, the chains collapsed after collisions with free spheres (Fig. 1c). In a three-bead chain, the collapse, as we described previously, 18 is a consequence of imbalanced rates of collision with spheres between the open and the closed sides of the chain -the spheres exert a "pressure" that can close the chain if it overcomes its intrinsic rigidity. Extrapolating this mechanism, the collapse of longer chains will occur if the "sphere pressure", which increases at larger filling ratios, is strong enough. 25 Once the chain reached the steady state, we took multiple snapshots of the system at equally-spaced time intervals to determine the more likely conformations and their probability of occurrence. The steady-state distribution of R The collapsed conformation was not static, though: the chain reached a steady state in which it continuously stretched and shrunk, but never returned to a completely extended conformation ( Fig. 1d).
g is shown as a function of f and FR in Fig. 2a to 2e.
At a low filling ratio of free spheres (FR = 40%), the chain always stayed in an extended (but not stretched) conformation: the filling ratio of free spheres was sufficiently low that it did not perturb the behavior the chain displayed in their absence. When we increased f at FR = 40%, the distribution of R g broadened and shifted continuously towards lower values -a WLC behavior that we also observed in an analogous system without spheres. 17 From FR = 50% to FR = 70%, the mean value of the R g distribution, denoted as <R g >, first increased with f, and then decreased, as we varied f from low to high values (Fig. 2f). Since, in our system, f plays the role of temperature, the change in the sign with which the system responded to variations in f indicates that the chain transitioned between two states with different "temperature" dependencies. In analogy with the coil and globule states of polymers, in which the radius of gyration has different temperature dependencies, we identified a coiled state that corresponded to a chain with <R g > increasing with f, and a globular state that corresponded to a chain with <R g > decreasing with f. We calculated the agitation frequency at which the transition occurred, conditions between FR=50% and FR=70%, is another signature of a second-order transition. 1 The coil-to-globule transition of MecAgit model described here is thus analogous to a coil-toglobule transition in polymers, induced by a change in temperature. The assembly of spheres is analogous to a "poor" solvent with a lower critical solution temperature (LCST), as observed for some polymers and proteins. [6][7][8] When f increased, the model chain exhibited the gradual compaction characteristic of second-order phase transitions of polymer chains when the temperature rises above the T-point (i.e., where the second virial coefficient changes sign from positive to negative). 9,26,28

Modeling the first-order coil-to-globule phase transition
First-order phase transitions are encountered in some polymer systems -such as charged polymers -but experimental proofs of such transition at the molecular scale remains controversial, and their mechanistic origin is still a subject of investigation. 1,5,16 Under agitation, the chain had two primary conformations: one ring-shaped, and one extended (Fig. 3b). The ring-shaped conformation was the more compact conformation; the rigidity of the thread prevented compaction below the ring size for the extended chain. The The MecAgit approach is novel for three reasons: 1) We used a more complex system than all the previously reported ones, with the introduction of two antagonistic forces at different scales, which led to a first order phase transition; to our knowledge, this effect has never been reported before with granular models (most of them were simply vertically vibrated, leading to a fundamentally different behavior).
2) The origin of a first-order phase transition is still a subject of debate in the polymer physics community; we proved, by a physical experiment, that the hypothesis of using two antagonistic forces at different scales is a valid hypothesis. conformations, just as monomers assemble to form polymers. 32 Our work, which shows that the engineering of inter-particle interactions is a valid approach to control phase transitions in millimeter-scale granular systems, might thus inspire the design of future polymer-like mesoscale systems.

Summary.
We prepared a circular mixing area with a diameter of 0.48 m using an aluminum rim, and we covered the area inside the rim with paper to generate an area with a constant friction coefficient on which the objects would roll, but not slide, when the plate was agitated. To avoid any possible electrical charging by contact electrification within the experimental setup, we maintained a relative humidity of more than 60% RH using a humidifier connected to the enclosed space above the plate. 19 In all experiments, we filled the mixing area with simple (spheres) and composite (cylinders connected by a string) polymeric objects. We varied the filling ratio (FR) of spheres between 40 % and 90 % (see the definition of FR in the main text).
To simulate molecular phenomena in the presence of thermal agitation, we shook the system with a combination of orbital translation (with frequencies f from 80 rpm to 150 rpm) and randomly timed flicks to make the movement of particles aperiodic. The movement of the cylinders-on-astring was followed by taking snapshots with a picture camera. For each (f, FR) condition, we took one hundred pictures at regular intervals to probe all the possible conformation taken by the chain. We extracted the R g values for each picture with a custom-made program.  19 We maintained a relative humidity higher than 60% using an air humidifier (Vicks V5100NS). To record snapshots of the system during agitation, we used a photo camera (Nikon D40) that was suspended ~1 m vertically above the plate. To achieve pseudo-random agitation, we applied two shaking drives: one from an orbital shaker attached to the plate, and one from a linear actuator which kicked the weight of the pendulum under the plate. The orbital shaker (Madell Technology Corp., ZD-9556-A) was attached to the plate via an elastic polyurethane cord, and the motion it imparted to the plate was a combination of orbital translation with a small-amplitude angular oscillation. The radial amplitude of the orbital shaker was 5.1 mm, and its frequency was variable; we used orbital frequencies ranging from f = 80 to f = 150 revolutions per minute (rpm). The second shaking motion had the role of randomizing the orbital motion. We kicked with a linear actuator (LinMot, Inc., P01-23x80) the weight under the plate; the linear actuator moved at a fixed frequency of 4 Hz in all experiments, but the times at which it impacted the pendulum were not periodic due to the complicated motion of the pendulum. The frequency of kicks was higher than the frequency of orbital shaking, and the system did not complete a full orbit between kicks to the pendulum. The overall motion of the plate was aperiodic but not entirely random, and we estimated that the maximum horizontal acceleration of the plate was approximately 5 m/s 2 . The area of agitation of the components (spheres and cylinders-on-astring) was defined by an aluminum rim with a diameter of 0.48 m. We covered the surface of the board with paper to control the movement of the spheres and the cylinders; the roughness and softness of the paper was such that the objects consistently rolled but did not slide.
Preparation of the Cylinders-on-a-string. The flexible links in the chain were weakly elastic and were made from a nylon string on which we strung three 3.18-mm diameter PMMA spheres.
Aluminum crimps fixed the cylinders on the string and controlled the elasticity of the chain, as described previously. [17][18][19] We chose a system of ten cylinders with a contour length L = 23.9 cm.
For the non-magnetic chain, we used a Nylon-6,6 threads with a diameter of 75 µm and Nylon cylinders. The persistence length was previously determined and was equal to L p = 12.2 cm. 17 When we used several chains, the system jammed in few minutes -because of a high number of inelastic collisions, which are known to lead to clustering. 33 For the magnetic chain, we kept the same sequence, but we chose a Nylon-6,6 thread with a diameter of 600 µm and magnetic cylinders. To fabricate the magnetic cylinders, we enlarged the holes of the Nylon cylinders to a diameter of 3.2 mm and glued hematite tubes inside with epoxy glue. for the two types of chains used (magnetic and non-magnetic), resulting in two distinct imageprocessing procedures. In both cases, since the processes were not entirely reliable, each processed image was shown to a human viewer, with the detected cylinder center locations marked on the image. The viewer could then edit the locations if they were incorrect. This system allowed the cylinders to be detected more accurately and quickly, and with less work, than a purely manual system, while maintaining high accuracy. After the verification of the processed images, we calculated the radius of gyration from the coordinates of the centers of mass of the cylinders for each frame in the image sequence.