Mechanical catalysis on the centimetre scale

Enzymes play important roles in catalysing biochemical transaction paths, acting as logical machines through the morphology of the processes. A key challenge in elucidating the nature of these systems, and for engineering manufacturing methods inspired by biochemical reactions, is to attain a comprehensive understanding of the stereochemical ground rules of enzymatic reactions. Here, we present a model of catalysis that can be performed magnetically by centimetre-sized passive floating units. The designed system, which is equipped with permanent magnets only, passively obeys the local causalities imposed by magnetic interactions, albeit it shows a spatial behaviour and an energy profile analogous to those of biochemical enzymes. In this process, the enzyme units trigger physical conformation changes of the target by levelling out the magnetic potential barrier (activation potential) to a funnel type and, thus, induce cascading conformation changes of the targeted substrate units reacting in parallel. The inhibitor units, conversely, suppress such changes by increasing the potential. Because the model is purely mechanical and established on a physics basis in the absence of turbulence, each performance can be explained by the morphology of the unit, extending the definition of catalysis to systems of alternative scales.


Introduction
In the biochemical realm, enzymes (E) help substrates (S) yield products (P) by catalysing the activation potentials of the transition paths [1]. In a typical microscopic view of catalytic reaction E acts on S, configures an enzyme -substrate complex (ES), induces a conformation change of the substrate (EP) and carries off with the product, P E þ S À k1 ! ES O k2 EP À k3 ! E þ P: (1:1) Albeit an individual molecule involves complex kinematics and is difficult to engineer, each transaction phase can be regarded as a logical operation [2], and the macroscopic view of the temporal dynamics can be characterized by the corresponding reaction speeds (k 1 , k 2 and k 3 ). While there is a discrepancy between the microscopic (mechanics) and the macroscopic (chemistry) perspectives, it has been generally considered that the key to this remarkable achievement lies in the addressability of individual molecules in representing discrete states, hidden in the morphology that rules the individual reaction order in a bottom-up manner. Lately, a process (inspired by chemistry) in which components spontaneously organize into complex structures, (self-assembly), has gathered attention [3]. A typical operation is, as described in chemical engineering studies, to control a global state of a system consisting of many components by regulating an environmental agitation, inducing a composition as a product. Such a synthetic process provides a new perspective for understanding biochemical reactions, and a promising path towards new manufacturing methods for complex non-molecular composition engineering (e.g. self-assembling electronic circuits) [4]. To date, various attempts at creating artificial self-assembly systems have performed simple aggregationbased assemblies, characterized by the direct forward reaction. The major attempts explored in the field can be represented by a reaction equation in which the components S 1 and S 2 are configured into S 1 S 2 , i.e. S 1 þ S 2 ! S 1 S 2 : The components form a lattice structure after interacting mechanically [5], magnetically [6][7][8][9][10], electrostatically [11,12], via capillary forces [13][14][15][16], hydrophobic/hydrophilic forces [17,18], fluid dynamics [19] or through configuring circuitry [20][21][22][23]. Hereafter, we use the term reaction in a broad sense, including those obtained mechanically.
In contrast to the capability of assembly, disassembly or the so-called backward reaction, (S 1 S 2 ! S 1 þ S 2 ), has attracted less attention, although it is critical for reconfiguration processes, catalytic reactions or regrouping components. These processes regularly combine with external forces to realize disassembly [24], or change the surrounding medium to alter the interaction between the components [25,26]. A unique approach focusing on the asymmetry of a membrane and its influence on the diffusion speeds of molecules is found in [27].
The engineering challenge here is to orchestrate an ordered assembly/disassembly down to individual components to globally attain a high yield of products, where the component has limited capabilities, as the available physical forces such as the electric, chemical or magnetic interactions provide limited interaction channels for the involved parts. Thus, for example, magnetism and capillary forces support only binary binding (either attraction or repulsion). A few notable attempts exist in which the emphases are placed on the logical responses of the components with respect to their possibilities of combining with the neighbour components, performing template replications [28][29][30][31], efficient crystallization [32] or exclusive-or (XOR) calculations [33]. These approaches actively exploit physical 'states' of the components (e.g. S versus P), whereby the two states are realized by changes in the mechanical and/or magnetic configuration of the involved components. A state change (implicitly or explicitly) alters the terrain of the system's potential energy and, thus, has the effect of accelerating the transition from one state to another. However, little reasoning has been conducted to quantify the cause of a transition, and proposed explanations have instead been based on phenomenological descriptions with heuristically designed components. One of the reasons for this could be that the presence of environmental agitation in the system complicates these analyses (in other words, these systems are essentially open to the environment). Then, the amount of kinetic energy delivered from the environment to a component contributing to a transition over time is difficult to assess, and, thus, the component's mechanical role is difficult to evaluate in a continuous parametric space. Beyond these approaches, we expect one component to function like an enzyme, that is, to act as a third agent and enable a state switch of a targeted component (S ! P) using magnetic potential energy only, where almost no environmental agitation is applied (thus, the system is essentially closed).
Here, by demonstrating that a simple enzymatic process, described by equation (1.1), can be obtained mechanically by passive magnetic units on the centimetre-scale, we focus on deriving the mechanical design principle of this chemical reaction and show that the concept of catalysis from chemistry can be generalized to alternative fields such as engineering. The proposed model, which consists of water-floating units equipped with permanent magnets, exhibits behaviours analogous to biological enzymes, and a comparable method of energy employment that levels out the potential barrier.

Design principle of magnetic catalysis
This section provides a theoretical reasoning on how the magnet motion must be coordinated in space in order to attain catalytic behaviour. We assume a physical set-up where magnets with a cylindrical shape slide on a horizontal plane, guided by physical walls.

Magnetism
To realize catalytic behaviour with magnets, the trajectories of the magnets must be designed at each reaction phase, which requires knowledge of the relationships between different intermagnet distances. The magnetic potential energy between two magnets M 1 and M j (treated as dipoles) with magnetic moments m i and m j ([ R 3 , i = j [ N) separated by a position vector r ij ([ R 3 ) connecting their centres, is given by where m 0 ¼ 4p Â 10 2 7 H m 21 is the permeability of free space, and r ij ¼ jjr ij jj ) the magnet diameters. When the magnets have an axially magnetized cylindrical shape, placed vertically on a frictionless two-dimensional plane in either parallel or anti-parallel configurations (the magnet directions are denoted by N or S in the following figures), they interact laterally. Assuming that the magnets feature the same magnetic moment magnitude jjm i jj ¼ jjm j jj ¼ m, the potential and resulting forces are simplified to and where s ij ¼ 1 if the magnets are anti-parallel, i.e. the two magnets are attracted along the line that connects them, and s ij ¼ 21 if they are parallel, i.e. repelling. We can determine the potential energy of the system, considering all involved magnets, from If the magnets are free to move, they will move such that the total energy is reduced (dU total =dr , 0), and, consequently, their relative distance is reduced (for an attractive configuration). This behaviour is the basis for designing the motion of the magnets in this work.
Equations (2.1)-(2.4) hold strictly for r ij ) the magnet diameters, and they gradually lose accuracy as r ij becomes comparable to the diameters. However, because our model mostly needs to take the relative distances of magnets sets into account, they satisfy our requirements; in equation (2.3), the magnitudes of magnetic force strength are related to the relative distances, cancelling out the inaccuracy of their values.
The proposed magnetic catalysis is phenomenologically described in figure 1a-d, which provides the incremental design of the paths for the three magnets involved in the enzymatic behaviour. The horizontal dimension is the reaction coordinate, and the vertical dimension is the distance between neighbouring magnets. Note that all the magnets maintain the same horizontal coordinate positions. The paths are illustrated as straight lines for intuitive apprehension, even though the reaction speed along the horizontal axis is nonlinear. This maintains the generality of the path descriptions, because a curved function can be approximated by a combination of linear lines. We illustrate the profile of the system's potential energy U total on top of each transition path. The state of the system is characterized by the motions of the involved magnets.

Sliding motion and conformation change
In figure 1a, if the paths of magnets M 1 and M 2 (anti-parallel) converge by a distance x (R1 . R3; x :¼ R1 2 R3 . 0), the magnets slide owing to the increasing magnetic attraction force, which, in turn, reduces the relative distance (R2 is not transition leve llin g out acti vat ion p o te n ti a l R2 R1 listed). The energy release can be used to perform work, which enables a kinematic reconfiguration.

Activation potential
The rate of physical convergence of the paths is a regulative parameter in the design. This is shown in figure 1b, where the choice of R1 , R2 ( y :¼ R2 -R1 . 0) and hence R2 . R3 create an outward wedged path acting as a 'threshold', which can suppress the sliding motion. Consequently, the magnets must overcome a potential energy barrier-which can be interpreted as the magnetically created activation potential-before the reaction can proceed. Note that, owing to the quick spatial decay of the magnetic force, the net force on M 2 is always dominated by the closest distance to any another magnet, and we neglect the magnetic crosstalk of the non-neighbouring magnets M 1 and M 3 . By designing the distances in the paths as R5 . R6 and R6 . R7, we can ensure that M 2 reaches an endpoint where the distance from M 2 to M 1 is again closer than to M 3 (R3 , R7). In the end, incorporating all the distance relations above, we obtain the condition for designing paths for catalysis

Catalysis
which draws the magnitude relations anticlockwise in the figure, starting from R3. Given R1, R2 and R3, we show the passable region for M 3 in pink, which certifies that as long as M 3 transits in this region, the reaction will proceed. The opposite happens in figure 1b, where the reaction stops. In this case, the system proceeds with the reaction implying that the terrain of U total is levelled out (see the mathematical proof in appendix B). Note that the condition derived in equation (2.5) holds even if the interaction force depends on a different power of the distance, when Figure 1d shows the complete enzymatic reaction, which is composed of five distinctive phases. Each magnetic reaction phase can be viewed in correspondence to the three reaction phases, k 1 , k 2 and k 3 , in equation (1.1). Phase k 1 is when M 3 is far away, approaching M 2 . Phase k 2 is further divided into two subphases, where phase k 221 represents the event when the activation potential is levelled out, and phase k 222 represents the event when the energy is used for work, i.e. conformation change and distancing M 3 . From a mechanical standpoint, phase k 221 can be divided into two subphases, which correspond to the two sectors of the original activation potential, i.e. the uphill and downhill sectors, respectively. Phase k 222 is similar to phase k 221 , in that all three magnets are moving, except the driving force is now between M 1 and M 2 . M 1 and M 2 attract each other, decreasing the relative distance and performing the conformation change (R3 . R4). Eventually, the distance between M 2 and M 3 is sufficiently large to reduce the net magnetic force on M 3 considerably (R7 , R8). Phase k 3 is the stage when M 3 is magnetically repelled. In our case, we designed the physical path of M 2 such that M 2 flips and changes polarity (see details in §3).   path represents the movement of M 3 embedded in a circular unit E called enzyme. Two paths, the green path for M 1 and blue for M 2 are mechanically arranged and embedded in the small S S and the large S L subunit as guiding walls for the magnets. Together, S S and S L compose the substrate S: As the motion of the conformation change can be arbitrary, it is implemented such that S L and S S , which rotate through a relative angle 908, switch contact facets and in so doing mechanically simulate a protein's folding motion, forming a different configuration (P; product). In addition, expecting to realize autocatalytic behaviour, we placed another E encapsulated in S L (this second E is called the docked E in contrast to the mobile E). The docked E is positioned far from M 2 , and, hence, has a small effect on the interactions of the other magnets; the existence of the docked E is not necessary for S S and S L to maintain the configuration of S.

Enzymatic reaction
The behaviour of an enzymatic reaction is illustrated in figure 2b-d, where the phases k 1 -k 3 comprise distinguishable stages represented by the positions of the mobile E (tagged with positions red-1 to red-8). In brief, a mobile E approaching from the left triggers a conformation change of S, releasing the docked E, whereas it itself eventually moves away from S after a short contact. More concretely, the mobile E sits on the long-arc edge of S L , further rolls along it to a certain position where the distance between M 2 and M 3 (R5) becomes shorter than between M 1 and M 2 (R1; phase k 1 , positions red-1 to red-3). Note that the magnets, just as well as E, can reduce the friction with the side walls by rolling. Then, E drags M 2 , by continuing to roll along the edge of S L ( phase k 221 uphill and downhill, positions red-3 to red-5), until the distance between M 1 and M 2 becomes shorter than that between M 2 and M 3 (R3 , R7). Then, the attraction between M 2 and M 1 initiates a conformation change by sliding along their respective paths, decreasing the relative distance ( phase k 222 , positions red-6 to red-7). Eventually, M 2 falls into a hole and connects to the bottom of M 1 by flipping upside-down, in the process binding the floors of S L and S S . These longitudinally coupled M 1 and M 2 repel the mobile E as well as the docked E from S ( phase k 3 , position red-8). When the docked E is expelled, it can subsequently act as a new mobile E: Hence, the number of mobile E is doubled after a conformation change, inducing a cascade reaction when multiple S S exist. The geometry R1 -R8 is reflected by the paths in figure 1d, drawn in a proportional scale for this substantiation.

Inhibition
Inhibition or at least the partial suppression of a chemical reaction is also a basic biochemical function primitive, realized by highly specific molecules forming complexes with other molecules. Such molecules, called inhibitors, often dock to the binding sites of enzymes via non-covalent bonding, or prohibit conformational changes of such molecules via steric hindrance. Inspired by this fact, the inhibition of the conformation change of a substrate is realized by making the shape of the mobile unit rectangular, but keeping the same magnetic arrangement as for E (this new unit is called an inhibitor, I , whose role is described in figure 2e-g). In contrast to the case of E, the system with I inhibits a conformation change by hindering the rotational motion of I : Owing to the angular shape, the mobile I cannot roll along the edge of S L or it requires separation of two attracting magnets M 2 -M 3 (thus, the barrier would indeed be regarded as an activation potential by I ), and restricts the conformation change by trapping M 2 midway in its path (position yellow-4 in figure 2f ). The docked I is nevertheless released because it is now in a repulsive region, and the number of mobile I is preserved to continue reactions (see the change in the attractive region in the electronic supplementary material, figure S4). Thus, S is magnetically inactivated; it cannot change its conformation nor attract another mobile E or I : Note that reactions can proceed in parallel, because E and I act on S, and vice versa, whereas Es and I s repel each other, as do the Ss:    figure 3a, a mobile I , which has the same magnet arrangement, is attracted to S (-0.619 and -0.071 s). When I makes contact with S L , it attracts M 2 , but because I itself cannot roll along the edge of S L , it holds M 2 at the midpoint of its path, suppressing a conformation change (1.483 s). When this entrapment occurred, the docked I entered a repulsive region created by M 1 , M 2 and the mobile I , and, thus, was released from S L (2.152 s). This way, the released I can continue the inhibition process. Figure 3c displays the transitions of the system's magnetic potential energy U total , derived by analysing the spatial positions of the involved magnets (for experimental plots with E and I ), and by the design in figure 1c supposing that M 1 -M 3 transit coherently (for a theoretical plot without E). We rsif.royalsocietypublishing.org J. R. Soc. Interface 12: 20141271 normalized U total by dividing by the respective U total at t ¼ -1 s ( -1 s was determined arbitrarily, noticing the small magnetic influence of E and I ). In this way, the difference in the number of magnets is cancelled out. We show the vertical axis in inverted form for an intuitive understanding corresponding to figure 1.

Conformation change and inhibition
Owing to the catalytic effect, the potential energy of E monotonically decreases, allowing the system to naturally proceed with reactions without an external energy input, and to reach a global stable state. By comparing the cases with E and the theoretically derived case without E, the reduction in the activation energy is clearly seen (we regard this decrease as the catalysis attained by E). Inhibition is also clearly shown in the global stable states (e.g. t ¼ 1 s), because I suppresses the decrease in the potential energy that E generates. The magnitude of the energy drop by I is a mere 6% of that obtained with E: As discussed, the previously defined reaction phases can be recognized as distinctive transitions.

Autocatalysis with multiple units combination
To test the designed system under more general conditions in a longer run, where multiple unit sets exist in space, we conducted experiments with five E -S sets and I -S sets, and show the representative trial results in figure 4. To prevent multiple S from gathering around the border of the container owing to their weak repulsion, we submerged iron plates 17 mm below the water surface level to weakly situate each S, while allowing them to orient themselves in random directions. Figure 4a shows a trial where Ss dock Es, whereas figure 4b shows Ss docking I s: In both cases, we initiated the reactions by manually placing five Es between the Ss, thus setting both initial conditions the same.
In figure 4a, after a brief interval of a quasi-stable state, the first conformation change (highlighted with a red circle) was triggered at 58.5 s, instantly followed by another conformation change at 60.1 s. The two new mobile Es released by the conformation changes traversed the field in the 12 o'clock direction  In general, the time of the first reaction is influenced by the randomly oriented S, which, nonetheless, have little influence on the later reaction speed once a reaction starts. After another conformation change was triggered by an E (5.9 s), the two released I in the centre of the field inhibited different Ss (at 9.2 and 19.6 s, highlighted with yellow circles). The final conformation change was invoked at 40.5 s. The most significant difference from the case in figure 4a, the docked E trial, is that an inhibited S did not create as strong a repulsive magnetic field as in the case of P, and hence it had less influence on the motion of the mobile units close by.
We conducted 30 trials for each of the two cases. The average durations for completing five reactions were 67.7 s (min ¼ 13.3, median ¼ 55. 7 During the experiments, a small number of trials were considered to be accidental errors, for example, because of magnets that jumped from S when changing conformation (this occurred five times before we reached 30 trials with Es), and that failed to conduct magnet flips (this occurred five times under the same conditions with Es). We also once terminated a trial with Es when no reaction occurred for more than 1 minute. Considering that conformation changes with a single S were reliable, the increase in the failure rate seems to indicate the magnetic influence of distant magnets. Within 41 potential inhibitions, I did not hold its position but slipped along the edge of S L and invoked a conformation change on five occasions (failure rate 12.2%).

Discussion
Unlike highly stochastic molecular reactions where thermal agitation is the driving force for transportation and massive rapid samplings of configurations for conformation change, our system rather exhibits a deterministic behaviour, whose dynamics could thoroughly be predicted by considering the positions of all involved magnets. Our emphasis is on presenting the possibility of catalytic behaviour carried out in the almost complete absence of environmental turbulence, thus, the units' morphology with respect to each reaction phase could be discussed. This aspect of the system, that it develops rather statically, at the same time, indicates that the mechanism shows a potential for smaller scales at which the influence of mass is more negligible. Incorporation of stochasticity through externally added kinetic turbulence or water agitation could nevertheless be feasible. By sufficiently shortening R2 in figure 1c, but still conserving the condition that R1 , R2, such environmental perturbation may still be able to invoke a sliding motion of M 2 , thus realizing conformation change of S: Regulating the agitation level and investigating the influence of catalytic enhancement would be of interest in future research.

Conclusion
With a special focus on the role of morphology, this study approaches the realization of a fully functional centimetresized, mechanical model of catalysis. We report on the construction and operation of the model, which contains both enzymes and inhibitors. To illustrate the analogous underlying processes of enzymatic behaviour, we first formulate the intermagnetic interactions attainable with permanent magnets. Then, we introduce physical units that instantiate the interaction and validate the desired behaviour where an enzyme triggers a kinematic reconfiguration of the target units, funnelling down the magnetic potential barrier (activation potential), whereas an inhibitor inhibits a reconfiguration by creating a barrier. As this phenomenon was attained at the pure physics level by combining morphology and magnetism, this study provides a platform at the intersection of classical mechanics (unit design), physics (magnetism) and chemistry (enzyme reaction). The obtained model extends the conventional definition of catalysis to systems of alternative scales, realizing 'mechanical' reactions with hands-on artefacts, which can expand the concept of manufacturing.
Funding statement. This work was partially supported by a Swiss National Science Foundation Fellowship number PA00P2_142208.
Endnote 1 The CAD data of all the designed units can be downloaded as electronic material in STL file format at http://www.shuhei.net/.