Using confocal microscopy and first passage time analysis, we measure and predict the rates of formation and breakage of polymer-depletion-induced bonds between lock-and-key colloidal particles and find that an indirect route to bond formation is accessed at a rate comparable to that of the direct formation of these bonds. In the indirect route, the pocket of the lock particle is accessed by nonspecific bonding of the key particle with the lock surface, followed by surface diffusion leading to specific binding in the pocket of the lock. The surprisingly high rate of indirect binding is facilitated by its high entropy relative to that of the pocket. Rate constants for forward and reverse transitions among free, nonspecific, and specific bonds are reported, compared to theoretical values, and used to determine the free energy difference between the nonspecific and specific binding states.

## I. INTRODUCTION

Colloidal particles mimic molecular systems in that they can form ordered crystalline phases,^{1} disordered glasses,^{2} and can even bind together tightly to form colloidal “molecules” that can themselves assemble into higher-order phases.^{3} Assembling colloids into higher order phases and structures frequently requires the colloids to be anisotropic or “patchy” such that specific (S) binding between particles leads to the formation of complex structures.^{4} The interaction specificity of these building blocks limits the kinetic pathways that systems can take to reach their ground state.^{5} The lock and key colloidal system,^{6} depicted in Fig. 1, forms colloidal molecules through anisotropic potential interactions; its behavior mimics the binding of drug or ligand in the binding pocket of a protein. In this system, the binding force is created by the presence of a polymer depletant, which creates an osmotic pressure that drives particles to reduce free (F) volume by binding either nonspecifically (NS) or specifically, as shown in Fig. 1. The synthesis and thermodynamics of specific binding of lock and key colloids was recently reported.^{6–9} Here, we report on the rich kinetics of bond formation in lock and key colloids. In particular, we find that two different pathways to a specific bond proceed at comparable rates. One occurs through a direct transition from free lock and key particles. The other, indirect pathway transitions through an intermediate nonspecifically bound lock and key pair, in which the key binds onto the spherical surface of the lock particle and then through a combination of rotations of the lock particle and sliding and rolling motions of the key, finds the dimple of the lock, and forms a specific bond. This finding is consistent with previous simulation results.^{8}

The importance of the role of nonspecific (NS) binding as a transitional step to specific binding has been demonstrated earlier for biological macromolecules,^{10,11} specifically for DNA interacting with proteins, as well as in models of bacteriophage tail attachment.^{12} The role of the equilibrium constant of different reaction pathways in determining clustering and self-assembly in anisotropic Janus colloids has also been shown.^{13} However, there is as yet no study that measures and models the kinetics of multiple reaction pathways in a colloidal system; establishing experimentally validated, first principles understanding of this complex chemical kinetics can improve the potential for self-assembly in a variety of systems, both natural and artificial.

Toward this aim, we use confocal microscopy to measure all of the transition rate constants shown in Fig. 1(a). We compare these results to the predictions of a diffusion-migration model of transition dynamics and find quantitative agreement for the constants that determine the relative magnitudes of the direct and indirect pathways to lock and key binding. The model is parameterized by direct measurement of the transition times from nonspecific lock-and-key pairs to free locks and keys; this determination of the net strength of the interaction potential is more accurate than what is possible from a simple theory for the depletion interaction. Model predictions include the success probability of nonspecific-to-specific binding as a function of interaction strength and the lifetimes of indirect binding events. We also predict the rates of formation of specific lock-and-key pairs and of the forward and reverse rates of formation of nonspecific bonds. The latter rates determine the overall rate of formation of specific lock-and-key bonds and, in a concentrated system, merit measurement and prediction because they control the kinetics of gel formation, which might inhibit the formation of macrocrystalline phases of lock-and-key bound pairs. From the measured rate constants, we also obtain the free energy difference between the nonspecific and specific lock-key bonds, which reveals interesting effects of entropy differences between the two states.

## II. MATERIALS AND METHODS

Lock and key particles were synthesized by polycondensation and free radical polymerization of 3-trimethoxysilyl-propylmethacrylate (TPM, ≥98%, Sigma-Aldrich), with a different fluorescent dye incorporated into each so they could be distinguished by confocal microscopy.^{7} The resulting lock particles had diameter *d*_{L} = 2.38 *μ*m ± 0.02 *μ*m and the keys had diameter *d*_{K} = 2.14 *μ*m ± 0.01 *μ*m as determined by analysis of *N*_{L} = 74 and *N*_{K} = 90 particles imaged by SEM (Fig. S1^{14}). TEM images of the lock particles showed their angle of aperture to be *α* = 0.58 ± 0.01 rad (for N = 3 measurements). The lock particles have a zeta potential of *ζ*_{L} = − 77.2 mV ± 2.22 mV; the keys have a zeta potential of *ζ*_{K} = − 88.0 mV ± 2.38 mV. The particular dimple shape of the lock particles^{14} is such that all six reactions shown in Fig. 1(a) occur at measurable rates, a condition that facilitates understanding the interrelationship among the six rate constants of this reaction system.

Stock solutions of 2 g/l polyethylene oxide (PEO, *M*_{v} = 600 000 g/mol, Sigma-Aldrich) at 1.5 mM NaCl (*κ*^{−1} = 7.9 nm) were prepared as the depletant. The radius of gyration of the PEO is estimated to be *R*_{g} = 50 nm.^{14,15} Dilute lock and key colloid suspensions of 2 ml volume were prepared by adding corresponding amounts of 1.5 mM NaCl solution, PEO solution, 25 *μ*l of a 1% wt. tetramethylammonium hydroxide solution in water (TMAH, Acros Organics) solution, and 25 *μ*l of a 5% wt. Pluronic F108 aqueous solution (Pluronic F108, Sigma-Aldrich) such that the final volume sample was 2 ml and pH = 9. PEO concentrations ranged from [PEO] = 0g/l to 1.4 g/l (*c*/*c*^{∗} from 0 to 0.74, where *c*^{∗} is the critical overlap concentration, $ c * =3 M w /4\pi R g 3 N A $). Finally, 300 *μ*L of the prepared suspensions were placed in an 8-well chamber (Lab Tek II), covered with silicone oil to prevent evaporation, and placed on the microscope.

To measure the kinetic rate coefficients of interaction, samples were imaged with an inverted confocal microscope (Leica TCS SP8). Two fluorescence channel imaging was performed to distinguish the two different particles. After 10 min, during which partial sedimentation of the particles to the coverslip occurred, time series of two dimensional images were acquired at 1.15 frames/s at a resolution of 512 × 512 pixels^{2} using a 63 × oil-immersion, *NA* = 1.40 Leica objective for 30 min, for a total capture of 2065 images. The pixel size was 117.3 × 117.3 nm^{2} and the total frame size was 60 × 60 *μ*m^{2}. A representative image series is shown in supplementary movie 1.^{14} Two-dimensional confocal microscopy images from each channel were separately analyzed by applying a Hough transform algorithm for circle detection^{16} implemented in MATLAB to find the particles’ centers to a resolution of ±14 nm. Histograms of bonded particles allowed resolution of specific, nonspecific, and free particles bonds at those with separation less than 2.2 *μ*m, between 2.2 and 2.6 *μ*m, and greater than 2.6 *μ*m, respectively (Fig. S2^{14}).

Representative image sequences of kinetic binding processes are shown in Figures 1(b) and 1(c), for both the direct formation of a lock-key specific bond, and the indirect formation, through a nonspecifically bound intermediate, respectively. Figure 1(b) shows an image sequence of a direct specific binding event: a key particle binds to the dimple of a lock particle upon finding the lock. Figure 1(c) shows a nonspecifically bound key particle diffusing on the surface of the lock particle until it binds to the dimple of the lock. For image series of these events, refer to supplementary movies 2 and 3.^{14}

The indirect binding mechanism shown in Figure 1(c) is not rare. For the depletant concentrations considered, the reaction rate for the indirect binding mechanisms was, on average, two times greater than the direct binding mechanism (Fig. S3^{14}). Moreover, we find that rate constants for the reverse reactions are all measurable as well. Therefore, although the thermodynamics of lock and key binding dictate the degree of binding at equilibrium—i.e., large times—the kinetic pathways by which equilibrium is achieved are complex and require that all six reactions be resolved. This kinetic complexity is a consequence of the anisotropy of the lock, and the fact that it offers two different binding states—specific and nonspecific—to the diffusing key.

The relative frequency of the kinetic events shown in Figure 1(a) varies with the concentration of depleting polymer and can be quantified by analysis of the time series of confocal microscopy images. These images are analyzed to yield the rate coefficients *k*_{F-S}, *k*_{F-NS}, *k*_{NS-S}, *k*_{NS-F}, *k*_{S-NS}, and *k*_{S-F} as defined below. Also measured were the lifetime distribution of specifically and nonspecifically bound pairs, the mean first passage time for nonspecific pairs to passage to specific pairs, and the success probability for a nonspecific pair to transition to a specific pair.

We measure the number density of locks and keys by counting the lock and key particles in each frame for the complete image time series and time-averaging their count. We then divide this number by the volume of the region, taken to be *V* = 2*r _{L}* ×

*A*, where

*A*is the area of the region,

*A*= (60

*μ*m),

^{2}and 2

*r*

_{L}= 2.4

*μ*m is the diameter of the lock particles. We also track the different events as they happen and calculate the rates at which the different events occur, normalized by volume. From these measurements, we calculate the event rate coefficients

*k*

_{F-S},

*k*

_{F-NS},

*k*

_{NS-S},

*k*

_{NS-F},

*k*

_{S-NS}, and

*k*

_{S-F}according to first and second order reaction kinetic processes. Specifically, the following equations for kinetic rate constants were used:

and

where *n*_{L}, *n*_{K}, *n*_{NS}, and *n*_{S} denote the number densities of locks, keys, nonspecifically bound, and specifically bound lock-key pairs; and *r*_{F-S}, *r*_{F-NS}, *r*_{NS-S}, *r*_{NS-F}, *r*_{S-NS}, and *r*_{S-F} denote the rate per unit volume at which free to specific, free to nonspecific, nonspecific to specific, nonspecific to free events, specific to nonspecific, and specific to free events, respectively, occur. Note the units of the rate coefficients vary depending on whether or not the binding process is uni-colloidal or bi-colloidal. The kinetic rate coefficients for the six equations are obtained by measuring the concentration of lock and key particles present (*n*_{L} and *n*_{K}, respectively, with units of *μ*m^{−3}) and the number of nonspecific and specific lock-key bonds formed and broken over the image acquisition time.

## III. THEORY

To model the kinetics of events starting from NS to other states, we use a diffusion-migration description of lock and key pair particle motion in terms of relative separation, *h*, defined as the surface to surface distance between particles, and relative orientation, *θ*, defined as the angle between the lock director and center-to-center vector between particles (Fig. S4^{14}). The governing equation for the evolution of probability density is the Smoluchowski equation,^{17} given by

where *ρ* is the probability density that the pair is at position (*h*, *θ*), *W* is the potential of mean force as a function of (*h*, *θ*), given by *W*(*h*, *θ*) = *Φ*(*h*) + *W _{g}*(

*h*,

*θ*), where

*Φ*(

*h*) is the interaction potential between particles given by a sum of repulsive electrostatic and depletion attractive potentials,

*W*(

_{g}*h*,

*θ*) is the configurational contribution to the potential of mean force associated with the choice of coordinates at a given position (

*h*,

*θ*), and

**D**is a diffusion tensor related to hydrodynamic interactions between lock and key in both the normal and tangential directions.

^{14}The Smoluchowski equation is a particular case of the more general master equation for stochastic process description. The main assumptions of this model are ergodicity and that the dispersion is dilute enough to prevent three or higher-body effects. The Smoluchowski equation can be written as

where **S** is the probability current. Transition between NS and F is taken in both experiments and theory as the separation *h* = *h _{a}*, a height at which a gap between the particles can be resolved in our system. For our system, h

_{a}= 372 nm; it is equal to the difference between the largest lock-key center-to-center separation distance used to define NS events, as defined above, and the sum of the lock and key average radii. A transition from NS to S is taken to occur at an orientation

*θ*=

*α*, with α = 0.58 rad, determined from TEM images, as above. The boundary conditions required to solve Eq. (7) are absorbing (probability sinks) at

*h*=

*h*and

_{a}*θ*=

*α*. The last absorbing boundary condition at

*θ*=

*α*states that we are assuming the binding is fast once a key finds its way to this position. Previous work

^{18}shows the existence of a depletion energy barrier due to sharp edges. This effect was found to be rather minor in our calculations.

^{14}The solution of Eq. (7) gives life times for different events as well as the relative occurrence of competing events, which can be estimated from lifetime distributions for events starting in NS and ending in state B (either F or S),

*w*

_{NS-B}(

*t*), as

where the integral is over the boundary 𝕊_{NS-B} between states NS and B, and d**n** is a unit vector normal to the boundary. The zeroth moment of *w*_{NS−B}(*t*),

is the success probability, which is the probability of crossing the boundary 𝕊_{NS-B} before crossing any other boundary. The success probability for a transition from NS to S is noted here as *P _{S}*. Normalization of

*w*

_{NS-B}(

*t*) by the success probability yields

which is the lifetime distribution conditional to events of the same type. The first moment of *p*_{NS-B}(*t*) gives us the mean first passage time, *t*_{MFP|NS-B}, of events of this kind, given by

Alternatively, the mean first passage time for all events leaving the NS state, *t*_{MFP|NS}, is given by

The kinetic rate constant between two states is then given by

## IV. RESULTS AND DISCUSSION

Because the theoretical form of the potential energy, *Φ*(*h*)—which combines both electrostatic and depletion contributions^{14}—is uncertain,^{21} due to the experimental uncertainty of the parameters that it depends on, we determine it from the measurements of the lifetime distributions for NS-F events. To accomplish this, we compare the lifetime distributions for NS-F events estimated from the diffusion-migration model for varying interaction strengths. In Figure 2(a), we show the experimental data for [PEO] = 1.0g/l and the NS-F lifetime distribution corresponding to the interaction strength that minimizes the weighed sum of squared errors between the experimental and modeled lifetime distributions. This was repeated for all the different [PEO], giving interaction strengths shown as the minimum of the potential energy, *Φ*_{min}, in Figure 2(b). Although the trend for the potential energy inferred from the lifetime of non-specific bonds on the colloid surface agrees with an *a priori* estimation from theory, within uncertainty limits, as shown in Fig. S5,^{14} there are quantitative differences. In this system, for the range of PEO concentrations studied, the minimum of the pair potential between lock and key varies from about 1 to 3 *k*_{B}*T*. Such analysis of the lifetime of non-specific bonds is a simple way to parameterize colloidal pair potential for comparison to theory and simulation.

The determination of the potential *Φ*(*h*) from the lifetime distribution for NS to F events fully specifies the diffusion-migration model. From it we compute the NS-S event lifetime distributions, *p*_{NS-S}(*t*), shown for the particular case of 1.0 g/l depletant on Figure 3(a), the mean first passage times for the NS to S and NS-F transitions (Figure 3(b)), and success probabilities, *P*_{S} (Figure 3(c)), for NS to S. These quantities are of particular interest because they probe how increased attraction between lock and key affects the dynamics of nonspecifically bound dynamics, leading in some cases to a transition from this bond to a specific bond.

The lifetime distribution of the NS-S event, *p*_{NS-S}(*t*), plotted in Figure 3(a), decays non-exponentially with time. Keys that successfully transition from NS to S spend a considerable time nonspecifically bound to the lock surface. This observation is consistent with previous findings showing the potential between lock and key has a secondary minimum in NS configuration.^{8} For example, 7.3% of the trajectories of keys bound nonspecifically to locks survive 6.1 s or more. By comparison, the time for a key colloid to diffuse a distance equal to its own radius is 5.7 s. These long trajectories endure because of the strength of the nonspecific depletion bond between the lock and key; unless Brownian fluctuations induce the nonspecific bond to break and for the key to become free, it continues surface diffusion on the lock until specific binding occurs.

The diffusion-migration model captures this physics with very good quantitative agreement, except at long times. The long-time disagreement between experimental and modeled lifetime distributions in Fig. 3(a) can also be seen in Figure 2(a) and is attributed to two possible factors. First, events with longer lifetimes have larger errors because they occur less frequently for an experiment of finite duration. However, this explanation is incomplete because deviation is systematically towards larger probabilities at long times for all concentrations and is outside the error bars of the experiments. Second, possible three-body and greater interactions in our system, especially at late times and high concentrations of PEO, are expected to slow down the kinetics of specific binding. These multiple-particle interactions occur, for example, when a nonspecifically bound lock-and-key pair encounters and binds with other particles, which amounts to minor qualitative differences. These multiparticle interactions are rare at the depletant concentrations studied here (cf. Fig. S7^{14}), but when they occur, they should produce longer NS-S and NS-F event lifetimes. For example, a key diffusing on the surface of a lock might be approached by another particle and interact with it, delaying its motion toward the dimple, or a key might interact with an occupied lock particle. Our diffusion-migration model assumes only pair particle interactions. Apart from this long-time deviation, the experimental and theoretical binding time distributions agree with each other for the range of [PEO] considered in our experiments.

As [PEO] is increased from 0.3 g/l to 1.4 g/l, the average event lifetime of NS-F events, *t*_{MFP|NS-F}, increases from 2.9 s to 6 s, and *t*_{MFP|NS-S} increases from 1.8 s to 2.9 s, as shown experimentally and confirmed by the model. From Fig. 3(b), the average transition times for NS-F events are longer than those for NS-S events. Because NS-F and NS-S processes compete, the average time for each process is dominated by the faster of the two, while the relative rates control the relative numbers of each transition type. The observed modest difference in average times between NS-F and NS-S is due to different shapes of the probability distributions for the transition times. While the NS-F process is essentially Poissonian, the NS-S process is non-Poissonian due to the finite distance to binding over which surface diffusion must occur, which creates an enhanced population of quickly binding states, relative to a Poissonian process. This trend is not as obvious in the experimental data, which show longer NS-S times than in model predictions. Discrepancies here are a consequence of the disagreement in the long-time tail in lifetime distributions, as explained above in the context of three-body interactions. This means there are a number of long-lasting events that shift average lifetimes, especially for NS-S events. Comparison of the lifetime distributions for NS-F (Fig. 2(a)) and NS-S (Fig. 3(a)) confirms that the former distribution generally falls above the latter, with this trend however becoming less clear at long times.

In Figure 3(c), we plot the experimental and model success probability *P*_{S}, defined as the probability that a key will specifically bind to the lock dimple given that it is initially nonspecifically bound to the lock surface versus polymer concentration. Our experiments show that as [PEO] is increased from 0 g/l to 1.2 g/l, P_{S} increases from 7.6% to 17.3%. Success probabilities extracted from our diffusion model are in good agreement with our experimental success probabilities for [PEO] below 1.4 g/l. As [PEO] increases, we expect that more nonspecifically bound lock and key pairs will specifically bind to each other, because keys will have more time to perform the successful search for the lock dimple. As the time that the key remains on the surface of the lock increases, as discussed above, so does the probability of NS-S binding between the lock and key particles. For this last polymer condition, we observe a large deviation in the experimental success probability from the other experimentally determined values and from the modeled result, which can be explained by the inapplicability of our diffusion model for *N*_{L} > 1, *N*_{K} > 1 multiparticle interactions (cf. Fig. S7 for evidence of multiparticle interactions at [PEO] = 1.4 g/l^{14}).

We now report the kinetic rate coefficients for the NS-S reaction, as well as the five others shown in Figure 1(a), as obtained by measuring the concentration of lock and key particles present (*n*_{L} and *n*_{K}, respectively, with units of *μ*m^{−3}) and the number of nonspecific and specific lock-key bonds formed and broken over the image acquisition time, as described above. We predict the NS-F and NS-S rate constants using the diffusion-migration model explained previously and computed using Eq. (14).

We find that as [PEO] is increased from 0 g/l to 1.4 g/l, the rate coefficient for nonspecific lock-key bond formation, *k*_{F-NS}, remains roughly constant at *k*_{F-NS} = 6.6 ± 0.5 *μ*m^{3}/s (Figure 4(a)). This is expected, since this rate is diffusion controlled and a lock-key collision counts as a binding event forming a nonspecific “bond,” which is ephemeral, unless depletion is present to slow breakage of this “bond.” The rate coefficient for direct lock-key bond formation, *k*_{F-S}, likewise remains constant for polymer concentrations ranging from 0 g/l to 1.2 g/l at a value of 0.67 ± 0.12 *μ*m^{3}/s. These rate constants do not change with polymer concentration because the rate of these events depends on the collisional dynamics of the lock and keys, which for a short range potential studied here is independent of potential strength. Free lock and key particles may collide with each other at the spherical surface of the lock or at the dimple of the lock particle, which has an angle of aperture *α* and thus occupies a fraction *f* = (1 − cos*α*)/2 of the lock surface area. The Smoluchowski collision rate between two spheres of unequal size *r*_{L} and *r*_{K} is given by $k=2 k B T 1 / r L + 1 / r K r L + r K /3\mu $.^{22} Here, *k*_{B} is the Boltzmann’s constant, *T* is the temperature, *μ* is the viscosity, and *r*_{L} and *r*_{K} are the lock and key radii, respectively. For our system, *T* = 293 K, *μ* = 1.002 × 10^{−3} Pa s, *r*_{L} = 1.19 *μ*m, and *r*_{K} = 1.07 *μ*m, yielding a collision rate of 10.8 *μ*m^{3}/s. On Figure 4(a), we compare the measured rate coefficients for F—NS and F—S lock-key binding to Smoluchowski collision rates, given by $ k F - NS = 1 \u2212 f k$ and *k*_{F−S} = *fk*. The agreement is satisfactory.

The observation of a finite rate coefficient of direct binding in the weak interaction limit is consistent with the study’s definition of direct binding—the lifetimes of these weakly interacting cases are short, and the time resolution of our measurements was constrained by our frame capture rate of 1.15 fps. Some disagreement between the theoretical collision rate and experimental values might be expected due to the quasi-2D nature of our system, because the lock and key particles are close to the coverslip. This implies that at any given time, there is a fraction of the surface area of these particles that cannot interact with other particles due to their proximity to the glass surface. Another factor to be considered is the effect of recently broken NS pairs that can bind faster than remote or far-away F pairs, although this effect would increase our theory predictions making agreement worse. However, the comparison of Figure 4(a) indicates that proximity and spatial correlation effects are not great.

In Figure 4(b), we show that the rate coefficient for unbinding of nonspecifically bound locks and keys, *k*_{NS-F}, decreases from 0.42 s^{−1} to 0.14 s^{−1} as [PEO] is increased from 0.3 g/l to 1.4 g/l, showing increased difficulty for keys to unbind from the lock at higher depletant concentration. We also observe that the rate coefficient for indirect specific binding, *k*_{NS-S}, is insensitive to changes in [PEO], as seen on Figure 4(b), remaining constant near a value of 0.064 ± 0.006 s^{−1}. The NS-S binding rate coefficient is independent of depleting polymer concentration due to the insensitivity of the surface diffusivity of keys on locks to small changes in surface-to-surface separation.^{23,24} The NS-S binding rate, on the other hand, depends on the concentration of NS-bound lock and key pairs, which is a function of polymer concentration.

In the particular lock and key system analyzed here, the strength of the specific bond is not so strong as to preclude unbinding during the long durations of the experiments. In Figure 4(c), we plot the measured S-NS and S-F event rate coefficients against polymer concentration. As depleting polymer concentration increases, the experimental rate coefficients for both specific unbinding processes decrease; the specific-to-nonspecific rate coefficients decrease from 0.4 s^{−1} to 0.07 s^{−1} for [PEO] equal to 0.3 g/l and 1.2 g/l, respectively, and specific-to-free rate coefficients decrease from 0.21 s^{−1} to 0.02 s^{−1} for [PEO] equal to 0.3 g/l and 1.2 g/l, respectively. The decrease in specific unbinding rate coefficients can be explained by an increase in the specific binding energy as [PEO] is increased. As binding energy increases, it becomes more difficult for key particles that are specifically bound to the lock dimple to unbind, and they remain in specific lock-key configuration for longer before unbinding.

While we can measure the rate constants for breaking a lock-key specific bond, shown in Figure 4(c), either to form a free pair or a nonspecifically bound one, a theoretical estimation of these rates would require intimate knowledge of both the 3-dimensional dependence of potential energy for the key in the pocket and hydrodynamic interactions for this geometry, which is not readily available. One thing to notice at this point is the fact that details of the shape of the lock pocket are not required to explain kinetic events in the direction of NS to S. This insensitivity is due to the nature of NS-S binding which is the result of diffusion of a key on the surface of the lock until the lip of the pocket is found, after which binding occurs faster than surface diffusion and the binding rate is not sensitive to the geometric details and complex hydrodynamics as the key enters into the pocket. The reverse process of escape from the pocket (i.e., from S to NS), however, involves more crucially the non-spherical shape of this pocket, the modeling of which is beyond the scope of this work.

Rate constants determined from transient experiments as explored here are related to the equilibrium behavior of the reactions involved. In particular, we can relate kinetic rate constants to equilibrium constants and obtain the equilibrium free energy difference between the nonspecific and specific lock-key bonds from the kinetics. At equilibrium, association and disassociation rates among free, nonspecific, and specific rates are equal. This yields the relations

and

which imply the following relationship between kinetic rates: *k*_{NS-S}/*k*_{S-NS} = *k*_{NS-F}*k*_{F - S}/*k*_{F-NS}*k*_{S-F}. At equilibrium, this ratio is equal to the concentration ratio of specific to nonspecific pairs, *n _{S}*/

*n*. Equilibrium also requires this ratio to be $ n S / n NS =exp \u2212 \Delta F NS - S / k B T $.

_{NS}^{14}Thus, we can obtain the free energy difference between the nonspecific and specific states, Δ

*F*

_{NS-S}, for all [PEO] by taking the ratio of rate constants as

or

where the difference between Eqs. (18) and (19) is simply the path one NS pair can take to get to S (cf. Fig. 1(a)).

A similar relation can be made to obtain the free energy difference between F and NS from Eq. (15) as *n _{NS}*/

*n*=

_{L}n_{K}*k*

_{F-NS}/

*k*

_{NS-F}. Equilibrium for this reaction requires $ n NS n 0 / n L n K =exp \u2212 \Delta F F - NS / k B T $,

^{14}where n

_{0}is a reference concentration. As a reference concentration we assume that one particle occupies the volume that defines the nonspecific state,

*V*= 4

_{NS}*π*(1 −

*f*)(

*R*′

^{3}−

*R*

^{3})/3, where

*f*is defined above,

*R*=

*r*

_{L}+

*r*

_{K}, and

*R*′ =

*R*+

*h*

_{a}; that is,

*n*

_{0}= 1/

*V*. The free energy difference between F and NS states is

_{NS}In Fig. 5, we plot the NS-to-S free energy difference obtained from experiment. We measure the free energy difference between NS and S two different ways: we can take the ratio of the NS-S and S-NS rate constants (direct measurement) using Eq. (18) or the product of the ratios of the four reactions (indirect measurement) using Eq. (19). The free energy difference between NS and S decreases from ∼2*k*_{B}*T* to 0 as [PEO] increases from 0 g/l to 1.4 g/l. Moreover, direct and indirect measurements of this difference are in agreement with each other, which serves as a thermodynamic consistency check for the measurements made.

Interestingly, the free energy measurements show that at depletant concentrations equal to 1 g/l or below, the nonspecific binding state is favored over the specific binding state. For [PEO] = 1.2ߙg/l and 1.4 g/l, the free energy difference between NS and S is close to zero, implying no preference between NS and S binding. Given the expected stronger potential of interaction for specifically bound pairs, the preference for nonspecific pairs seems counterintuitive. Nonetheless, the free energy estimated above includes contributions of both the potential energy of interaction and the associated entropy of each type of bond. Because the (energetically) weaker nonspecific bond is more prevalent, the entropy of this bond (as related to the nonspecific binding volume) must be much higher than that of the specific bond, as is indeed the case. In this colloidal system, the entropic penalty associated with the smaller number of particle configurations allowed for the specific binding state is large enough to favor nonspecific binding for [PEO] below 1.0 g/l.

This discussion shows the critical role of entropy on both the equilibrium and kinetic behavior of lock and key binding specifically, and for anisotropic colloids in general. For lock and key binding, the entropy is determined by the shape of the lock pocket relative to the key; the transition PEO concentration determining the point at which specific binding is favored relative to nonspecific binding would therefore certainly depend on complementarity between the key and pocket shapes. We can therefore envisage that certain pocket shapes—different than the one studied here (cf. Figure S1^{14})—might lead to potential energy and entropy changes upon binding that would combine to produce a more favorable free energy change for the specific bond, relative to the non-specific bond, than reported in Figure 5. Future work to generate a pocket-shape dependent free energy landscape—either by simulation or experiment—would be an interesting means to evaluate the role of shape complementarity on the energetic and entropic contributions to binding in this model lock-and-key system.

Figure 5 also shows the free energy difference between F and NS obtained from experiment and modeling. This free energy difference is obtained from the ratio of rate constants (see Eq. (20)) and the reference concentration. (Note that the reference concentration is independent of [PEO]. Therefore, relative changes along the curve are not a function of the choice of reference.) Our experiments indicate that the free energy difference between F and NS decreases with increasing [PEO] from a value of 0.7 *k*_{B}*T* at 0.3 g/l to −0.6 *k*_{B}*T* at 1.4 g/l. The corresponding free energy difference obtained from our modeling, using Eq. (20) and rate constants using the diffusion-migration model for *k*_{NS-F} and Smoluchowski collision rates for *k*_{F-NS}, is also shown in Figure 5 and is in very good agreement with our experiments. Absolute values of the NS-F free energy depend on the particular choice of reference concentration. Therefore, the preference of NS over F, as indicated by the sign of the free energy, will ultimately depend on the total particle concentration of the system.

## V. CONCLUSIONS

We have shown that specific lock-and-key binding occurs through two different kinetic pathways: a direct pathway from free particles to specifically bound ones and an indirect pathway involving a key particle nonspecifically bound to the spherical lock surface, which transitions by surface diffusion to a specific bond. We quantified lock-and-key binding kinetics and extracted event rate coefficients for the two binding pathways, demonstrating the importance of nonspecific binding pathways for the self-assembly of anisotropic particles. From a diffusion-migration analysis, we fit the lifetime event distributions from nonspecifically bound to free particles and thereby obtained experimentally derived estimates of the attractive interaction between particles as a function of depleting polymer concentration. Predicted rates of nonspecific-to-specific (NS-S) lock-key binding, as well as distributions of transition times for nonspecific to free particles (NS-F) and nonspecific-to-specific binding were found to agree with the experimentally measured quantities. From the measured rate constants, we computed free energy differences between different states of the system. We found that nonspecifically bound pairs are preferred to specifically bound pairs for depletant concentrations below 1.0 g/l, due to the large entropy penalty for formation of the specific bond relative to that for the nonspecific bond. Similar entropy penalties are well known to limit binding affinities in highly specific protein-ligand interactions.^{25} The measurements of lock-key binding kinetics are a starting point, not only for understanding more complex binding events in biology but also for determining rates of formation of complex colloidal materials involving particles with anisotropic interactions.

## Acknowledgments

Support from the US Army Research Office through the MURI program (Award No. W911NF10-1-0518) is acknowledged.