Force spectroscopy of polymer desorption: Theory and Molecular Dynamics simulation
Forced detachment of a single polymer chain, strongly-adsorbed on a solid substrate, is investigated by two complementary methods: a coarse-grained analytical dynamical model, based on the Onsager stochastic equation, and Molecular Dynamics (MD) simulations with Langevin thermostat. The suggested approach makes it possible to go beyond the limitations of the conventional Bell-Evans model. We observe a series of characteristic force spikes when the pulling force is measured against the cantilever displacement during detachment at constant velocity (displacement control mode) and find that the average magnitude of this force increases as grows. The probability distributions of the pulling force and the end-monomer distance from the surface at the moment of final detachment are investigated for different adsorption energy and pulling velocity . Our extensive MD-simulations validate and support the main theoretical findings. Moreover, the simulation reveals a novel behavior: for a strong-friction and massive cantilever the force spikes pattern is smeared out at large . As a challenging task for experimental bio-polymers sequencing in future we suggest the fabrication of stiff, super-light, nanometer-sized AFM probe.
In recent years single-molecule pulling techniques based on the use of laser optical tweezers (LOT) or atomic force microscope (AFM) have gained prominence as a versatile tool in the studies of non-covalent bonds and self-associating bio-molecular systems (1); (2); (3); (4); (5); (6); (7); (8); (9). The latter could be exemplified by the base-pair binding in DNA as well as by ligand-receptor interactions in proteins and has been studied recently by means of Brownian dynamic simulations and the master equation approach (10); (11). The LOT and AFM methods are commonly used to manipulate and exert mechanical forces on individual molecules. In LOT experiments, a micron-sized polystyrene or silica bead is trapped in the focus of the laser beam by exerting forces in the range . Typically, AFM (which covers forces interval in range) is ideal for investigations of relatively strong inter- or intramolecular interactions which are involved in pulling experiments in biopolymers such as polysaccharides, proteins and nucleic acids. On the other hand, due to the relatively small signal-to-noise ratio, the AFM experiments have limitations with regard to the mechanochemistry of weak interactions in the lower piconewton regime.
The method of dynamic force spectroscopy (DFS) is used to probe the force-extension relationship, rupture force distribution, and the force vs loading rate dependence for single-molecule bonds or for more complicated multiply-bonded attachments. Historically, the first theoretical interpretation of DFS has been suggested in the context of single cell adhesion by Bell (12) and developed by Evans (13); (14); (15). The consideration has been based on the semi-phenomenological Arrhenius relation which describes surface detachment under time-dependent pulling force, , with being the loading rate. It was also assumed that the effective activation energy, , may be approximated by a linear function of the force, i.e., . Here is the distance between the bonded state and the transition state where the activation barrier is located. The resulting Bell-Evans (BE) equation then gives the mean detachment force as a function of temperature and loading rate , i.e., , where is the desorption rate in the absence of applied pulling force.
As one can see from this BE-equation, the simple surmounting of BE-activation barrier results in a linear dependence of detachment force on the logarithm of loading rate, provided one uses the applied force as a governing parameter in the detachment process (i.e., working in an isotensional ensemble when is controlled and the distance from the substrate to the clamped end-monomer of the polymer chain fluctuates). For multiply-bonded attachments the interpretation problem based on this equation becomes more complicated since a non-linear relationship is observed (16). In this case chain detachment involves passages over a cascade of activation barriers. For example, Merkel et al. (16) suggested that the net rate of detachments can be approximated by a reciprocal sum of characteristic times, corresponding to jumps over the single barriers. In particular, regarding the detachment of biotin-streptavidin single bonds, it was suggested that two consecutive barriers might be responsible for the desorption process.
A simple example of multiply-bonded bio-assembly is presented by a singe-stranded DNA (ssDNA) macromolecule, strongly adsorbed on graphite substrate. The forced-induced desorption (or peeling) of this biopolymer has been studied analytically and by means of Brownian dynamics (BD) simulation by Jagota et al. (17); (18); (19); (20). In ref. (17) the equilibrium statistical thermodynamics of ssDNA forced-induced desorption under force control (FC) and displacement control (DC) has been investigated. In the latter case one works in an isometric ensemble where is controlled and fluctuates. It has been demonstrated that the force response under DC exhibits a series of spikes which carry information about the underlying base sequence of ssDNA. The Brownian dynamics (BD) simulations (18) confirmed the existence of such force spikes in the force-displacement curves under DC.
The nonequilibrium theory of forced desorption has been developed by Kreuzer et al. (21); (22); (23) on the basis of Master Equation approach for the cases of constant velocity and force-ramp modes in an AFM-experiment. The authors assumed that individual monomers detachments represent a fast process as compared to the removal of all monomers. This justifies a two-state model where all monomers either remain on the substrate or leave it abruptly. The corresponding transition rates (which constitute a necessary input in the Master Equation approach) must satisfy detailed balance. As a result of the Master Equation solution, the authors obtained a probability distribution of detachment heights (i.e., distances between the cantilever tip and the substrate) as well as an average detachment height as a function of the pulling velocity.
Irrespective of all these efforts, a detailed theoretical interpretation of the dynamic force spectroscopy experiments is still missing. For example, in terms of Kramers reaction-rate theory (24) the Arrhenius-like BE - model holds only when the effective activation energy . On the other hand, it is clear that for large forces (which we experience in AFM), the case when occurs fairly often. In this common case the general approach, based on the BE-model, becomes questionable. Besides, it can be shown (25), that the activation energy vs force dependence, , is itself a nonlinear function, so that the conventional BE - model, based on the linear approximation, , should be limited to small forces. Moreover, the Arrhenius - like relationship for the detachment rate, which was used in the BE - model, is a consequence of a saddle-point approximation for the stationary solution of Fokker- Planck equation (24). This contradicts the typical loading regimes, used in experiments, where applied force or distance grow linearly with time.
The present paper is devoted to the theoretical investigation of a single molecule desorption dynamics and aimed at interpretation of AFM - or LOT - based dynamic force spectroscopy in the DC constant-velocity mode. The organization of the paper is as follows: in Sec. II we give the equilibrium theory of detachment for the case of strong polymer adsorption. The mean force (measured at the cantilever tip) versus displacement diagram is discussed in detail. In particular, the characteristic force-“spikes” structure (which was first discussed in Ref. (17); (18)) can be clearly seen. In Sec. III we give a dynamical version of the detachment process. Our approach rests on construction of general free energy functions, depending on coarse-grained variables, which govern the non-linear response and structural bonding changes in presence of external forces. The corresponding free-energy-based stochastic equations (known as Onsager equations (26)) are derived and solved numerically. This solution makes it possible to provide not only force-displacement diagrams and the ensuing dependence on cantilever displacement velocity but also the detachment force probability distribution function (PDF). In Sec. IV the main theoretical results are then checked against extensive Molecular Dynamics (MD) simulation. A brief discussion of results is offered in Sec. V.
Ii Equilibrium theory at the strong adsorption case
Recently we suggested a theory of the force-induced polymer desorption (for relatively weak adsorption energy) in the isotensional (27); (28) and isometric (29) equilibrium ensembles supported by extensive Monte Carlo (MC) simulations. In the former case, the fraction of adsorbed monomers changes abruptly (undergoes a jump) when one varies the adsorption energy or the external pulling force. In the second case, the order parameter varies steadily with changing height of the AFM-tip, even though the phase transition is still of first order. The total phase diagram in terms of adsorption energy - pulling force, or, adsorption energy - end-monomer height, has been discussed theoretically and in terms of MC-simulations.
On the other hand, the AFM experiments deal with relatively strong forces ( (1)) so that in the case of a single molecule desorption experiment only a really strong adsorption energy is essential. This limit has been discussed in the recent papers by Jagota et al. (17); (18); (19); (20) and Kreuzer ey al. (21); (22); (23). Here we consider this problem in a slightly more general form. In so doing we distinguish between two different models: with frictionless- and strong-friction substrates, as indicated in Fig. 1.
ii.1 Frictionless substrate
This case has been considered in Refs. (17); (18); (21); (22); (23) and is based on the assumption that the force resisting sliding is sufficiently small, i.e., the cantilever tip and the contact point are both placed along the same -axis (see Fig. 1 (left panel)). The total partition function for a fixed cantilever distance , i.e., , is a product of partition functions of the adsorbed part , , of the desorbed portion (a stretched polymer portion), , and of the cantilever itself, , where is the number of desorbed polymer segments, and denotes the distance between the clamped end of this desorbed portion and the substrate. As a result,
where the integration interval, , and the step-function, , imply that restrictions, and , should be applied simultaneously. In this representation is the control variable (which is monitored by the corresponding AFM operating mode) whereas and are coarse-grained dynamic variables which should be integrated (in our case, an integral over , and summation over ) out. Moreover, if we introduce the function
then Eq. (1) can be rewritten as
In the strong adsorption regime, attains a simple form
where the dimensionless adsorption energy . The cantilever manifests itself as a harmonic spring with a spring constant , i.e., the corresponding partition function reads
Finally, we derive the partition function of the desorbed part of the polymer as function of the dynamic variables and , based of the Freely Jointed Bond Vector (FJBV) model (30); (31). The corresponding Gibbs free energy (i.e., the free energy in the isotensional-ensemble) is
where the dimensionless force . The corresponding distance .
where the so called Langevin function has been used. In the isometric-ensemble, the proper thermodynamic potential is the Helmholtz free energy, , which is related to by Legendre transformation,
as function of and .
In Eq. (11) the force should be expressed in terms of as follows: , where denotes the inverse Langevin function. The corresponding effective free energy function in terms of and reads
In the limit of a very stiff cantilever, , the cantilever partition function approaches a -function (21):
and Eq. (11) takes the form
For the isometric ensemble, i.e., in the -ensemble, the average force , measured by AFM-experiment, is given by
where is given by Eq. (11).
The numerical results, which follow from Eq. (15), are shown in Fig. 2. One can immediately see the ”sawtooth“-, or force-spikes structure on the force-displacement diagram as it was also found by Jagota et al. (17) in the limit of very stiff cantilever. Physically, spikes correspond to the reversible transitions , during which the release of polymer stretching energy is balanced by the adsorption energy. The corresponding thermodynamic condition reads . This condition also leads to the spikes amplitude law (17), i.e. the spikes amplitude gradually decreases in the process of chain detachment (i.e., with growing ).
This structure is more pronounced at larger adsorption energy and cantilever spring constant . Thus, while the force oscillates, its mean value remains nearly constant in a broad interval of distances , exhibiting a kind of plateau. Complementary information (for fixed at different values of ) is given on Fig. 3. One can verify that the plateau height is mainly determined by whereas the spikes amplitude is dictated by the cantilever spring constant .
ii.2 Strong polymer-substrate friction
In this limit one has to take into account the specific geometry of an AFM experiment, shown in Fig. 1 (right panel). For simplicity, an infinite friction of the polymer at the surface is assumed. The adsorbed polymer portion may be considered as a two-dimensional self-avoiding chain comprising segments. The last contact point (marked as in Fig. 1) can move due to adsorption or desorption elementary events. In Ref. (32) this was classified as the sticky case. In Fig. 1, is the distance from the cantilever base to to the substrate, is the height of the cantilever tip above the substrate, and is the distance between the cantilever tip and the contact point . Eventually, is the lateral distance between cantilever base and the contact point . One may assume that initially the desorbed portion of segments has occupied a distance of which, due to self-avoiding -configurations of an adsorbed chain, equals (where ).
The specific geometry of the AFM experiment in the case of strong polymer-substrate friction (shown in Fig. 1) brings about changes only in the cantilever partition function, i.e., instead of Eq. (5), one has
As a result, the total partition function in this case is given by
where again the variable should be excluded in favor of by means of the relation . In Eq. (17) the following constraints
have been taken into account.
The corresponding free energy functional in terms of dynamical variables and has the following form
The average force, which is measured in AFM-experiments, is given by
Iii Dynamics of desorption
In our recent paper (33) we have studied a single polymer force-induced desorption kinetics by making use of the notion of tensile blobs as well as by means of Monte Carlo and Molecular Dynamics simulations. It was clearly demonstrated that the total desorption time scales with polymer length as .
In order to treat a realistic AFM experiment in which the cantilever-substrate distance changes with constant velocity , i.e., , one has to consider the AFM tip dynamics. With this in mind, we will develop a coarse-grained stochastic model based on the free-energy functional Eq. (12). Before proceeding any further, we need to define the adsorption-desorption potential profile . This plays the role of the potential of mean force (PMF) which depends on .
iii.1 Stochastic Model
In the Helmholtz free-energy functional , given by Eq. (12) and Eq. (19), the free energy of the adsorbed portion is given by a simple contact potential, , where is an integer number in the range . Considering desorption dynamics (see below), we will treat as a continuous variable with a corresponding adsorption-desorption energy profile satisfying the following conditions:
For integer -values the energy profile has minima whereby we use the contact potential .
For half-integer values of the adsorption potential goes over maxima.
The activation barrier for monomer desorption, , and the corresponding adsorption activation barrier, , are proportional to the adsorption strength of the substrate whereby .
The adsorption-desorption energy profile satisfies the boundary conditions: (a fully adsorbed chain), and (an entirely detached chain).
One may show that the following energy profile, given as
meets the conditions (1) - (4).
The minima and maxima of Eq. (21) are located in the points defined by with denoting the continuous index of a monomer.
As a result,
In order to calculate the activation barriers, we determine first at the half-integer points, i.e.,
as well as at the integer points
Therefore, the activation barriers for the detachment, , and adsorption, , are given by
i.e., . Finally, one may readily see that and which is in line with condition (iv).
The total Helmholtz free energy for the frictionless substrate model is given by
whereas for the strong polymer-substrate friction model we have
These Helmholtz free energy functions govern the dissipative process which is described by the stochastic (Langevin) differential equations
where and are the Onsager coefficients. The random forces and describe Gaussian noise with means and correlators given by
The set of stochastic differential equations, Eq.(28) can be treated by a time integration scheme. Each realization of the solution provides a time evolution of and . In order to get mean values of the observables, these trajectories should be averaged over many independent runs . For example, in order to obtain the average force, Eq. (20), one should average over the runs
iii.2 Thermodynamic forces
where we have used (recall that )
On the other hand, a direct calculation shows that
Thus, for the force , given by Eq. (31), one has
In the strong friction case, Eq. (27) leads to a more complicated expression for the thermodynamic force:
For one obtains
where we have used
Taking into account Eq. (33), one finally derives
For the model, given by Eq. (27), the corresponding force reads
Finally, the variable should be expressed in terms of by making use of the relationship , where is the inverse Langevin function. A very good approximation for the inverse Langevin function, published in Ref. (34), is given by
iii.3 Quasistationary approximation
It could be shown that for a strongly stretched desorbed portion of the polymer chain, the variable rapidly relaxes to its quasi-stationary value (see Appendix A). In other words, can quickly adjust to the slow evolution of (governed by the Kramers process). In this quasi-stationary approximation , and from Eq. (37) one has , so that the following nonlinear equation for emerges
This could be represented as
The Onsager equation for the slow variable is given as
where is determined by Eq. (39).
Eventually, we get a system of so-called semi-explicit differential-algebraic equations (DAE) (35)
In this particular form of DAE one can distinguish between the differential variable and the algebraic variable . Eq. (44) can be solved numerically by making use of an appropriate Runge - Kutta (RK) algorithm, as shown in the Appendix B.
We have solved numerically our stochastic model, given by Eq. (43) and Eq. (41), for the case of frictionless substrate. To this end we used the second order Runge - Kutta (RK) algorithm for stochastic differential-algebraic equations (see Appendix B for more details). The advantage of the stochastic differential equations approach as compared to the Master Equation method (23) is that the former one gives a more detailed (not averaged) dynamic information corresponding to each individual force-displacement trajectory (as is often in an experiment). The result of averaging over runs is shown in Fig. 5 (left).
Fig. 5 (right panel) shows the resulting force - displacement diagram for and different detachment velocities. It it worth noting that the ”sawtooth” pattern can be seen for all investigated detachment velocities ranging between and . For larger velocities the plateau height of the force grows substantially. In other words, the mean detachment force increases as the AFM-tip velocity gets higher and the bonds stretching between successive monomers becomes stronger.
We have also studied the detachment force behavior as well as that of the cantilever tip distance from the substrate at the moment of a full detachment, (i.e. when ), by repeating the detachment procedure times and plotting the probability distribution functions (PDF) for different adsorption energies and detachment velocities - Fig. 6. As one can see from Fig. 6a, b, both the average and the dispersion of detachment force grow with which agrees with findings for reversible (i.e., when a broken bond can rebind) bond-breaking dynamics (36). In contrast, the mean cantilever tip distance variance decreases and its average value increases with growing (cf. Fig. 6c, d).
The average detachment force dependence on cantilever velocity is a widely covered subject in the literature in the context of biopolymers unfolding (37); (38); (39) or forced separation of two adhesive surfaces (40); (41); (36). Figure 7a, which shows the result of our calculations, has the characteristic features discussed also in ref. (40). One observes a well expressed crossover from a shallow-slope for relatively small detachment rates to a steep-slope region as detachment speed increases. One remarkable feature is that this crossover practically does not depend on the adsorption energy : the curve is merely shifted upwards upon increasing of . Therefore, the crossover is not related to a competition between the Kramers rate and the cantilever velocity but rather accounts for the highly nonlinear chain stretching as the velocity increases. The corresponding detachment distance of the cantilever tip (detachment height), Fig. 7b, reveals a specific sigmoidal shape in agreement with the results based on the Master Equation (23). At low velocities of pulling, , when the chain still largely succeeds in relaxing back to equilibrium during detachment, an interesting entropy effect is manifested in Fig. 7b: the (effectively) stiffer coil at leaves the substrate at lower values of than in the case of the colder system, . As the pulling velocity grows, however, this entropic effect vanishes and the departure from the substrate is largely governed by the stretching of the bonds rather than of the coil itself whereby the difference in behavior between and disappears.
Iv MD simulations
iv.1 The model
In our MD-simulations we use a coarse-grained model of a polymer chain of beads connected by finitely extendable elastic bonds. The bonded interactions in the chain is described by the frequently used Kremer-Grest potential, . The FENE (finitely extensible nonlinear elastic) potential is given by
with and .
In order to allow properly for excluded volume interactions between bonded monomers, the repulsion term is taken as Weeks-Chandler-Anderson (WCA) potential (i.e., the shifted and truncated repulsive branch of the Lennard-Jones potential,) given by
with or for , or , and , . The overall potential has a minimum at bond length . The nonbonded interaction between monomers are taken into account by means of the WCA potential, Eq. 46. Thus, the interactions in our model correspond to good solvent conditions.
The substrate in the present investigation is considered simply as a structureless adsorbing plane, with a Lennard-Jones potential acting with strength in the perpendicular –direction, . In our simulations we consider as a rule the case of strong adsorption , where is a temperature of Langevin thermal bath described below.
The dynamics of the chain is obtain by solving the Langevin equations of motion for the position of each bead in the chain,
which describes the Brownian motion of a set of bonded particles.
The influence of solvent is split into slowly evolving viscous force and rapidly fluctuating stochastic force. The random Gaussian force is related to friction coefficient by the fluctuation-dissipation theorem. The integration step is and time in measured in units of , where denotes the mass of the polymer beads, . In all our simulations the velocity-Verlet algorithm was used to integrate equations of motion (47).
The molecule is pulled by a cantilever at constant velocity . The cantilever is imitated by two beads connected by harmonic
spring and attached to one of the ends of the chain.
The mass of beads , forming the cantilever, was set either to or to . The equilibrium size of this harmonic spring was set to and the spring constant was varied in the range . The hydrodynamics radius of beads composing the cantilever was varied by changing the friction coefficient , taking into account the Stokes’ law, , where is the solvent viscosity.
Taking the value of the thermal energy J at K, the typical Kuhn length of nm and the mass of coarse-grained monomer as kg setups the unit of time in our simulations which is given in ps. The velocities used in simulations are in units of . Spring constants of our cantilever in real units are: .
Two typical snapshots of a polymer chain during slow detachment from an adsorbing substrate with different strengths of adsorption, and are shown in Fig. 9. Evidently, the chain is much more stretched for the strongly-attractive substrate where all adsorbed monomers stick firmly to the surface.
As we have already seen in Sec. III.4, the averaging of the force profile over many runs reveals the inherent sawtooth-structure of the force vs distance dependence (see Fig. 5) which is otherwise overshaded by thermal noise. Our MD-simulation result, depicted in Fig. 10, show the same tendency against the noisy background of a single detachment event. Therefore, for better clarity and physical insight, all our graphic results that are given below result from such averaging procedure.
Figure 11a shows how adsorption energy affects the force vs distance relationship. Apparently, with increasing the mean force (plateau height) is found to grow in agreement with our equilibrium theory results, given in Fig. 3. As suggested by our recent theory (27); (29), the plateau height goes up as , or as , for relatively small or large values, respectively. The amplitude of spikes increases with growing too, in line with the equilibrium findings (see Fig. 3). Moreover, as found by Jagota et al. (17), the amplitude of spikes follows an exponential law, , where is the number of desorbed polymer segments. On the other hand, the comparison of Fig. 11b and Fig.2 suggests that the stiffness of the cantilever spring constant affects mainly the spike amplitude especially at large .
Eventually, we demonstrate the impact of cantilever velocity, , as well as of its mass, , and friction coefficient, , on the force-distance profile. Apparently, these parameters affect differently strong the observed force - distance relationship. Similar to the results, obtained for our coarse-grained model in Sec. II, in the MD-simulation data the plateau height grows less than twice upon velocity increase of three orders of magnitude (see Fig. 12)! Only for a very massive, (), and strong-friction, (), cantilever, the plateau height grows significantly and gains a slight positive slope (see Fig.12d) whereby oscillations vanish. This occurs for the fastest detachment . Evidently, this effect is related to the combined role of the friction force in the case of rapid detachment along with the much larger inertial force () whereby the substrate-induced oscillations are overshadowed by the increased effort of pulling. In contrast, neither Fig. 12b, nor Fig. 12c indicate any major qualitative changes in the -vs--behavior when medium-friction, or mass cantilever alone are drastically changed.
The PDF of the detachment force and its velocity dependence are shown in Fig. 13. Similarly as in Sec. II, the average value and dispersion grow with increasing speed of pulling and this is weakly sensitive with regard to the adsorption strength of the substrate . Remarkably, the mean detachment force shows a similar nonlinear dependence on (cf. Fig. 7a). The crossover position does not change practically as the adhesion strength is varied, and the variation of the other parameters () towards a massive and strong-friction cantilever render this crossover considerably more pronounced.
The complementary PDF for the detachment height is given in Fig. 14a together with the corresponding average vs relationship. As predicted by our analytic model, cf. Section II, the height of final detachment of the chain from the substrate becomes larger for faster peeling and stronger adhesion , which is consistent with the MD data. One can see again the typical sigmoidal-shape in the vs dependence.
The two panels for different temperature, shown in Fig. 14b, indicate a smaller increase in at the higher temperature, provided the pulling velocity is sufficiently small too. This can be readily understood in terms entropic (rubber) elasticity of polymers and represents a case of delicate interplay between entropy and energy-dominated behavior. It is well known that a polymer coil becomes less elastic (i.e., it contracts) upon a temperature increase, cf. the lowest (grey) curve in Fig. 14b, (left panel) at , so that is smaller than in the corresponding lowest curve for in the right panel of Fig. 14b. This occurs at low values of . On the other hand, the softer chain (at ) stretches more easily and, therefore, goes up to for the highest speed instead of for . This entropic effect is well expressed at weak attraction to the surface, , which does not induce strong stretching of the bonds along the chain backbone. In contrast, at high , the bonds extend so strongly that the chain turns almost into a string and entropy effects become negligible. The energy cost of stretching then dominates and leads to higher values of at the higher temperature (cf. upper most green symbols in Fig. 14b) since it is now the elasticity of the bonds between neighboring segments which governs the physics of detachment. In this case the elastic constant of the bonds effectively decreases with an increase of so that the distance of detachment in the left panel of Fig. 14b for is higher than that for in the right panel.