Interactions between Protein Coated Particles and Polymer Surfaces

In this paper we investigate these nonspecific interactions by analyzing the response of protein coated magnetic particles to a rotating magnetic fiel...
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Interactions between Protein Coated Particles and Polymer Surfaces Studied with the Rotating Particles Probe M. Kemper,*,†,‡ D. Spridon,†,§ L. J. van IJzendoorn,† and M. W. J. Prins†,∥ †

Eindhoven University of Technology, The Netherlands Dutch Polymer Institute (DPI), The Netherlands § Alexandru Ioan Cuza University, Romania ∥ Philips Research Laboratories, The Netherlands ‡

S Supporting Information *

ABSTRACT: Nonspecific interactions between proteins and polymer surfaces have to be minimized in order to control the performance of biosensors based on immunoassays with particle labels. In this paper we investigate these nonspecific interactions by analyzing the response of protein coated magnetic particles to a rotating magnetic field while the particles are in nanometer vicinity to a polymer surface. We use the fraction of nonrotating (bound) particles as a probe for the interaction between the particles and the surface. As a model system, we study the interaction of myoglobin coated particles with oxidized polystyrene surfaces. We measure the interaction as a function of the ionic strength of the solution, varying the oxidation time of the polystyrene and the pH of the solution. To describe the data we propose a model in which particles bind to the polymer by crossing an energy barrier. The height of this barrier depends on the ionic strength of the solution and two interaction parameters. The fraction of nonrotating particles as a function of ionic strength shows a characteristic shape that can be explained with a normal distribution of energy barrier heights. This method to determine interaction parameters paves the way for further studies to quantify the roles of protein coated particles and polymers in their mutual nonspecific interactions in different matrixes.

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nonspecific interactions.4,5 Also oxidation of the substrate, increasing hydrophilicity, is used to minimize protein adsorption.6−8 Polymer materials are widely used as substrates for biosensing because polymers are easy to process and easy to modify. In this paper, we study the nonspecific interactions between protein-coated particles and polymer surfaces. As a model system for the sensor surface, we use polystyrene surfaces that were oxidized up to ∼23% using UV/ozone treatment.9 We investigate the interaction with myoglobin coated magnetic particles, since myoglobin is a biomarker for acute myocardial infarction.10 The pH of the suspending solution of the particles was varied from 3.3 to 10.2, and interactions were probed using the rotating particles probe. The rotating particles probe has been recently introduced as a probing technique for physicochemical interactions between particles and a substrate in a fluid.11 An ensemble of magnetic particles is brought close to a surface, and a rotating magnetic field is used to discriminate between bound and unbound particles. The fraction of unbound particles gives information about the interaction between the protein coated particles and the surface. Janssen et al. measured the interaction of

INTRODUCTION Point-of-care biosensors are used to rapidly measure low concentrations of analyte present in small amounts of a complex fluid like blood or saliva. Recent developments in biosensor design include the use of magnetic particles as labels for detection. The particles are coated with specific molecules, e.g., nucleic acids for hybridization assays or proteins for immunoassays.1,2 In a sandwich immunoassay, antibody coated particles are dispersed in a sample fluid in order to capture analyte molecules from solution, followed by magnetic actuation to bring the particles to a sensor surface, which is also coated with antibodies. Subsequently, unbound and nonspecifically bound particles are removed from the sensor surface and the sensor signal is measured. Detection of the particles can, for example, be performed by frustrated total internal reflection,2 surface plasmon resonance,3 or by magnetoresistivity.1 The sensitivity of biosensors is determined by the capture molecules, the detection principle, and the background signals caused by nonspecific interactions. Nonspecific interactions bind particles to the surface without an analyte molecule being present. This can be the binding of a coated particle on a coated substrate, for example, by antibody−antibody interactions. It can also be the binding of a coated particle on a bare part of the substrate. Often, sensor substrates are treated with blocking agents like bovine serum albumin or a detergent to minimize © 2012 American Chemical Society

Received: February 13, 2012 Revised: April 20, 2012 Published: May 9, 2012 8149

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streptavidin coated particles on a glass surface as a function of the ionic strength of the solution.11 The data reveal a clear increase of binding for increasing ionic strength. To explain the observations, the distance between particle and surface was calculated at which a minimum in interaction energy occurs, determined by electrostatic, van der Waals, gravitational, and magnetic energies. The experimental conditions at which particles were binding to the surface yielded a calculated distance of the energy minimum that was comparable to the surface roughness of the used magnetic particles. In this paper, we introduce a new model to interpret the experimental data of the rotating particles probe. The description is based on the energy barriers that the particles have to cross in order to bind to the surface. The key assumption of the model is that the particles experience a distribution of energy barrier heights, so that the particle binding data reflect an average barrier height as well as a variability of barrier heights. We validate the model by studying the binding of smooth protein coated magnetic particles to a polymer surface. We parametrize the barrier height and successfully reproduce the measured curves as a function of the ionic strength of the solution. The model is able to describe the experimental data for surfaces with different oxidation levels and for different pH values of the solution and allows the extraction of two characteristic barrier parameters for the nonspecific interactions between particles and surface.

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Figure 1. Experimental approach. (a) The particles are coated with myoglobin and the substrate is provided with a layer of polystyrene that is oxidized using a UV/ozone treatment. (b) Schematic of magnetic particles on a substrate. The particles rotate due to an applied rotating magnetic field. Optical observation of particle motion is used to distinguish between bound and unbound particles. The microscope picture shows an image of a magnetic particle: the particle (2.8 μm) is labeled with small tags (500 nm) so as to be able to record the rotation. (c) The measurement points reveal the fraction of unbound particles as a function of the ionic strength of the solution. The line represents a fit according to the distributed energy barrier model (eq 7). (d) Typical energy barrier between particle and surface, including electrostatic repulsion, van der Waals attraction, gravity, and magnetic energies.

MATERIALS AND METHODS

Figure 1 shows schematically how rotating particles probe experiments were performed: first, protein coated magnetic particles (with a small label to visualize rotation) were incubated on the substrate; then, a rotating magnetic field was applied to discriminate between bound and unbound particles and the fraction of unbound particles was calculated. For a range of ionic strengths, this fraction was recorded and curves were fitted using eq 7. The magnetic particles used in the experiments are Invitrogen Dynabeads M-270 Carboxylic Acid (diameter 2.8 μm), consisting of magnetite grains in a highly cross-linked polystyrene matrix with a hydrophilic glycidyl ether surface coating and carboxyl groups introduced on the surface.12 Contrary to the M-280 particles that were used before, the M-270 particles are smooth, minimizing the influence of surface roughness on the observed interactions. The smoothness makes it impossible to observe the rotation of naked particles. Therefore, small magnetic particles were used as optical tags. As tag particles, we used magnetic MagSense 500 nm polystyrene particles functionalized with streptavidin.13 The M-270 particles were functionalized using an EDC-NHS binding scheme; the carboxyl groups on the particles were activated using EDC and NHS in MES-buffer. Then, biotin EZ-link (Pierce) was incubated to introduce biotin on the surface. Unbound biotin EZ-link was washed away, and subsequently myoglobin (CalBiochem; Human, Recombinant E. coli) was incubated at 0.02 mg/mL. Finally, leftover activated carboxyl groups were quenched by incubation with ethanolamine (Sigma Aldrich). The small MagSense particle labels were bound to the M-270 particles by biotin-streptavidin coupling. Figure 1b shows a microscope image of a labeled particle. The particles were suspended in phosphate buffered saline (Sigma Aldrich, pH 7.4) that was diluted with deionized water to the desired ionic strength. For other pH values, hydrogen chloride or sodium hydroxide were added. The procedure of preparing and oxidizing the polystyrene surface has been described before.9 In short, the polystyrene surface was spincoated onto a glass coverslip on which first a polyimide attachment layer was applied. Oxidation was carried out by UV/ozone treatment for a specified time, thereby changing the surface oxygen content and hydrophilicity. A fluid cell of 9 mm diameter and 120 μm depth was created with a Secure-Seal imaging spacer that was stuck to the polystyrene substrate. Subsequently, the cell was filled with magnetic

particles suspended in solution with specified pH and ionic strength. The cell was closed with a glass coverslip and incubated for 5 min to be able to measure a steady-state situation. (as shown by Janssen et al.11) Details of the rotating particles probe setup have been described before.11 The rotating magnetic field is applied in the plane of the sample by a four coil setup with a soft iron yoke and slanted poles. The currents through the coils are 90° out of phase with respect to neighboring coils. A magnetic field of 20 mT and a rotation frequency of 1 Hz are used throughout all experiments. The particles on the bottom of the fluid cell are observed using a microscope (Leica DM6000M) with a 63× water immersion objective. Movies are recorded using a high speed camera (Redlake MotionPro HS-3) operated at 50 Hz and analyzed by counting the single particles, discriminating between “unbound” (rotating synchronously with the magnetic field) and “bound” (not following the rotating field). This procedure is based on the detection of particle rotation as a probe for binding and assumes that the application of torque does not change the fraction of bound particles. This assumption can be justified by several observations. At first, bound particles do not rotate, in spite of the applied torque. This implies that the bond between the particle and the surface is characterized by an energy barrier for rotation which exceeds the maximum energy that can be supplied by the application of the torque. This energy can be estimated as the product of the maximum torque and the angle of rotation required to overcome the energy barrier. Using the permanent moment of the M270 particles: 1.3 × 10−16 Am2 (ref 14) and an applied field of 20 mT, the maximum torque amounts to 650kT/rad. In order to estimate the required energy to break the bond along the rotation coordinate, we cannot use the potential energy curve for 8150

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dissociation perpendicular to the surface. Typical energy barriers for rotation along a single molecular bond are known for the C−C bond in hydrocarbons but cannot be applied here, because a biological bond is characterized by multiple noncovalent bonds spread out over the interacting surface. This is supported by the measurements of Janssen et al.14 that show the response of a protein pair upon applying a torque. The data for the protein G−IgG system clearly shows that dissociation of the protein G−IgG bond does not occur and instead, the protein complex behaves as a torsional spring. In the rotating particles probe, the nonspecific interaction between the particles and the surface is characterized by an interaction surface which is likely to be larger than the surface involved in the specific bond between proteins (viz., the protein G−IgG bond). A quantitative value for the energy required to dissociate the particle from the surface by applying an in-plane torque is difficult to give but might be estimated by calculating the surface energies involved. An estimate of the contact area can be obtained by considering the surface area of a sphere with a diameter of 2.8 μm (the size of the magnetic particle) that is in proximity of less than 0.1 nm from the surface and amounts to approximately 880 nm2. Assuming a surface free energy of 42.5 mJ/ m2 (for polystyrene15) results in an interaction energy of 9.4 × 103kT, which is much larger than the energy that can be supplied by the particles in the rotating magnetic field. Consequently, the applied torque is unlikely to be able to break the bond between the particles and the surface, and the rotating particles act as a valid probe that does not influence the fraction of bound particles.

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RESULTS First, the measurements using the rotating particles probe are shown and then we present the theoretical derivation of a model that can be used to fit all measured data. Experiments. We measured the interaction of myoglobin coated particles with a range of polystyrene surfaces of which the properties were systematically varied by UV/ozone oxidation. The oxidation times were varied from 0 to 120 s, leading to oxidation up to ∼23%, as determined with XPS measurements.9 The pH of the suspending solution was varied from 3.3 to 10.2. The fraction of unbound particles was measured as function of ionic strength. Ionic strength influences the height of the energy barrier that the particles have to cross to bind to the surface, as is explained in detail in the next section. We perform measurements up to an ionic strength of 300 mM. Higher concentrations are considered not relevant for applications in biosensors, since the physiological salt concentration is about 150 mM. Data points are averages over typically six fields of view, recorded on two or three samples, of typically tens of particles each. Error bars represent the standard deviation. The sometimes large deviations are caused by occasional variations in sample homogeneity and by sample-to-sample variations. However, the curves corresponding to differently oxidized samples can be clearly distinguished. Since all particles are on the bottom of the fluid cell, they stay in focus during the measurements. Figure 2 shows the fraction of unbound particles for differently oxidized surfaces in suspending solutions with a pH of 3.3, 7.4, and 10.2. All curves have a characteristic shape. With increasing oxidation time, a higher ionic strength was needed to bind the same fraction of particles. For example, at pH 7.4 (shown in Figure 2b) the concentration at which 50% of the particles was bound shifted from ∼10 mM to ∼100 mM for a change in oxidation time from 0 to 72 s. With increasing pH, a clear shift of the curves to higher ionic strengths was observed. For example, the concentration at which 50% of the

Figure 2. Fraction of rotating myoglobin coated magnetic particles on differently oxidized polystyrene surfaces (0−120 s) for various ionic strengths at pH values of 3.3 (a), 7.4 (b), and 10.2 (c). Data was fitted with eq 7, and the parameters are shown in Figure 4. Note the different ionic strength scales.

particles was bound for 60 s, oxidized polystyrene was ∼2 mM for pH 3.3, ∼ 25 mM for pH 7.4, and ∼200 mM for pH 10.2. Modeling of Particle−Surface Interactions. The interaction energy between a particle and the surface determines whether binding will occur. In all our experiments, we observe that particles which bind to a surface do not unbind again. This argues against a statistical model description that uses a dynamic equilibrium of binding and unbinding. Apparently, a model is required which explains that only a fraction of the particles is able to overcome an energy barrier and bind during incubation. At first, we discuss the energies that determine the barrier height and we show that the energy barrier can be parametrized as a function of the ionic strength of the solution. In practice, not all particles experience the exact same energy barrier due to small variations in particle or surface treatment. Therefore, subsequently, a model is introduced that describes the binding of particles by including a normal distribution of energy barrier values. Energy Barrier Height. Protein coated particles approaching the polystyrene surface experience a number of different interactions. These interactions have been described extensively (e.g., by Leckband and Israelachvili16) and include the wellknown electrostatic and van der Waals forces as well as 8151

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hydrophobic interactions. Also hydrogen bridge formation, ion correlation forces, and steric forces may play a role on the molecular scale. In order to parametrize the barrier height in our model, we use the electrostatic and van der Waals forces and include gravitational and magnetic forces that are specific for our experimental setup. Since the origin of hydrophobic interactions is still debated and no generally accepted model is available, we do not add a term corresponding to these forces to the expression for the energy barrier. Note that parameters obtained from fitting the data will also include the other interactions. For example, the hydrophobic interaction will increase the attractive interactions and thereby lower the energy barrier. We calculate the electrostatic and van der Waals interactions based on a macroscopic description of matter (continuum equations). Electrostatic interactions are calculated by adapting the Poisson−Boltzmann equation for a sphere close to a flat surface, and the interaction energy is described in terms of the surface and particle potentials as well as the ionic strength of the solution.16 The ionic strength determines the Debye screening length, which is the effective range of electrostatic interactions. The electrostatic interaction is repulsive for particles and surfaces that have the same sign of charge. The van der Waals energy is calculated for a sphere at a distance to a flat surface using the Hamaker constant.17 For interactions of nonconducting solids or proteins in water or salt solutions, the Hamaker constant is typically 10−20 J, causing an attractive interaction.16 Gravitational energy for the M-270 particles is Eg = 6.8 × 10−14h J with h the height in meters of the center of mass of the particle above the surface. The magnetic energy is Em = 1.3 × 10−16h J, caused by a small vertical magnetic gradient in the setup.18 Taken all together, the (electrostatic, van der Waals, gravitational, and magnetic) energies form a curve of potential energy with a deep minimum close to the surface, which is caused by the van der Waals attraction. If the electrostatic repulsive energy is larger than the van der Waals attraction, an energy maximum is present and a secondary minimum further away from the surface is formed. The difference between the maximum and secondary minimum determines the energy barrier that particles have to cross to bind to the surface, as illustrated in Figure 1d. At high ionic strengths, the electrostatic interactions are shielded at a very short distance, so van der Waals interactions take over and the barrier disappears. The influence of gravity and magnetic energy on the shape of the energy landscape at short distances is negligible, but they cause the particles to sediment onto the surface. The energy expressions do not allow an analytical solution for the energy barrier height. Therefore, we have calculated the barrier height numerically for the range of ionic strengths used in the experiments. The surface potential was kept constant at −30 mV, while particle surface potentials were chosen in the range of polystyrene surface potentials measured at various pH values.19 The Hamaker constant was varied from 0.5 × 10−20 to 1.5 × 10−20 J, and this did not have any influence on the general shape of the curves. Calculated values are shown in Figure 3. To allow for an analytical derivation of the rate of the binding of particles to the surface, we need an analytical expression for the barrier height. The expression should cover the conditions on which particles can cross the barrier and bind to the surface. On the time scale of the experiment, energy barriers in excess of 25kT are not likely to be crossed.16 Therefore, we parametrize

Figure 3. Symbols show calculated energy barrier values (the difference between the maximum and the secondary minimum of the potential energy) for a surface potential of −30 mV and particle potentials of −20, −25, and −30 mV with A = 10−20 J. The lines show the parametrization of the barrier height as a function of ionic strength as given in eq 1.

the function for barrier heights up to a few hundred kT. The energy barrier height can be conveniently described using eq 1:

Eb =

C′ + D′ Ic

(1)

The calculated energy barrier values and the corresponding parametrized curves are shown as a function of ionic strength in Figure 3. Distributed Energy Barrier Model. The height of the energy barrier that particles have to cross to bind to the surface is very sensitive to variations in the exact particle, surface, and environmental properties. A difference in potential of only 1 mV (for either particle or surface) already causes a change in barrier height of 30kT in physiological conditions (150 mM). Once the particles crossed the energy barrier of height Eb to adsorb to the surface, they will not unbind anymore, since the primary minimum of van der Waals attraction in theory is infinitely deep. Indeed, in experiments, unbinding of particles was very rarely observed, which means the barrier for unbinding was larger than 25kT.16 The rate of change of the number of unbound particles (F) with a given energy barrier can be expressed using the Arrhenius equation: ⎛ −E ⎞ dF = −k 0 exp⎜ b ⎟F ⎝ kT ⎠ dt

(2)

with k0 the Arrhenius preexponential (attempt frequency), k Boltzmann’s constant, and T temperature. The number of unbound particles with barrier Eb after an incubation time t is given by ⎛ ⎛ − E ⎞⎞ F = F0 exp⎜ −tk 0 exp⎜ b ⎟⎟ ⎝ kT ⎠⎠ ⎝

(3)

with F0 the initial number of unbound particles with unique barrier Eb. In order to model the inhomogeneity of the particle and surface, a distribution of energy barrier values, f(Eb), is introduced. The total number of particles is represented by S0. We assume that particles are initially randomly distributed over all possible states, so F0(Eb) = S0 f(Eb) dEb. The number of unbound particles S can be found by integrating F (eq 3) over all energy barrier values: 8152

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⎛

∞

S = S0

⎛ − E ⎞⎞

∫−∞ f (E b)exp⎜⎝−tk 0 exp⎝ kTb ⎠⎟⎠ dE b ⎜

⎟

(4)

If we assume for f(Eb) a normal energy distribution with mean energy barrier height Eb,0 and width σ, we get 1 S = σ 2π S0

∞

⎛ −(E − E )2 ⎞ b b,0 ⎟⎟ · 2 2σ ⎝ ⎠

∫−∞ exp⎜⎜

⎛ ⎛ − E ⎞⎞ exp⎜ −tk 0 exp⎜ b ⎟⎟ dE b ⎝ kT ⎠⎠ ⎝

(5)

The second exponent in the integral can be approximated with a stepfunction at Ec = kT ln(2tk0). This assumption is valid if the spread in energy barrier values is larger than the thermal energy spread ({(√2σ)/(kT)} > 1, as shown by Suuberg20), which is reasonable since a 1 mV variation in surface potential causes already a 30 kT change in barrier height. Thus, the integral can be expressed as an error function: ⎛ Ec − E b,0 ⎞⎤ S 1⎡ = ⎢1 − erf⎜ ⎟⎥ ⎝ S0 2⎣ 2 σ ⎠⎦

Figure 4. Interaction parameters C and D as a function of polystyrene oxidation time, extracted from the comprehensive set of measurements using eq 7 and restricting D > 1.4. Panel a shows parameter C, panel b shows parameter D. Dashed lines are guides to the eye.

(6)

By filling in the parametrized energy barrier height from eq 1, an expression is obtained that describes the fraction of unbound particles as a function of ionic strength. ⎛ −C ⎞⎤ S 1⎡ = ⎢1 − erf⎜ + D ⎟⎥ S0 2 ⎢⎣ ⎝ Ic ⎠⎥⎦ C=

C′ 2σ

and

D=

an increasing energy barrier height for increasing oxidation time and pH. C quantifies the influence of the ionic strength on the barrier height. The ionic strength directly influences electrostatic interactions by changing the shielding. Van der Waals interactions are not directly influenced by ionic strength, but because shielding changes the position of the energy barrier, the contribution of the van der Waals energy to the barrier height is influenced as well. Therefore, C includes contributions of electrostatic as well as van der Waals energies. Also other interactions, like hydrophobic interactions, will contribute to C. Electrostatic interactions are not only determined by the ionic strenght but also by the particle and surface potentials: higher potentials cause more repulsion and thereby a high energy barrier. Van der Waals interactions are determined by the Hamaker constant: a higher constant indicates more attraction, thus a lower energy barrier. The first trend observed in the measurements is increasing C, thus increasing energy barrier height for increasing oxidation time, which was observed at all pH values. This can be explained by the fact that oxidized polystyrene has a more negative surface potential than nonoxidized polystyrene, and therefore, the electrostatic repulsion will be increased, raising the energy barrier.19 Also the decrease of the Hamaker constant that has been reported for oxidation of polystyrene increases the energy barrier height.21 On top of that, oxidation lowers the water contact angle on polystyrene, reducing its hydrophobicity and thereby the hydrophobic interactions.9 The second trend is that C increases for increasing pH values. For (oxidized) polystyrene surfaces, an increasing pH gives rise to increasing negative surface potential.19 Since both the surface and the particles were negatively charged, a high pH causes high ζ potentials and thereby high electrostatic repulsion, which directly increases the barrier height. At low pH, the opposite holds: both surface and particle will have a low (absolute) charge, so there is weak electrostatic repulsion, leading to a lower energy barrier. To characterize (unknown) biosensor substrates, it is possible to use rotating particles probe experiments and extract parameter C. A high C value means that little nonspecific

with

−D′ + kT ln(2tk 0) 2σ

(7)

This formula can be used for fitting the measured data. Basically, C (mM) expresses the position of the curve while D determines the value for infinite ionic strength as well as the slope of the curve. Two different regimes can be distinguished: first, if not all particles bind at infinite ionic strength (S/S0 > 0.025), D < 1.4; second, if all particles do bind at infinite ionic strength (S/S0 < 0.025), D > 1.4. In view of the data, we are in the second regime.

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DISCUSSION All measurements were fitted with eq 7 and D > 1.4, corresponding to the regime where all particles bind at infinite ionic strength. The fitted curves in Figure 2 show that the measured data was reproduced as a function of ionic strength and a clear distinction can be made between all measurement series. Interaction parameters C and D were extracted for each measurement series and are shown in Figure 4. Interaction parameter D was found to vary only between 1.4 and 3 without a trend correlated to oxidation time or pH. Interaction parameter C varied in the range ∼2−1000 and showed a clear trend of increasing values for increasing surface oxidation time as well as for increasing pH. This reflects the shift of the curves to higher ionic strengths as shown in Figure 2. Variations in C and D directly give information about the change in energy barrier height if Ec (defined after eq 5, dependent on incubation time and attempt frequency) and σ are assumed to be constant. This corresponds to a fixed incubation time as used in our experiments. Therefore, an increasing value of C as observed in the experiments describes 8153

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varies with oxidation time of the surface and pH of the solution. Increasing oxidation and increasing pH both lead to increasing C values, which corresponds to a shift of the curve and binding of the particles at higher ionic strengths. The shift is caused by the increase of barrier height by the increase of the electrostatic interaction, by the decrease of the Hamaker constant, or by change of the hydrophobic interactions. Thus, a high pH and long surface oxidation are favorable in order to minimize nonspecific binding of myoglobin coated particles to polystyrene surfaces. For future work, more experimental parameters should be varied to independently determine barrier height and spread and to unravel the different contributions to the barrier. For example, use of nonionic surfactants might allow one to independently study the influence of the hydrophobic interactions. In conclusion, we have presented a novel method to analyze interactions between protein coated particles and a surface, using the rotating particles probe with a distributed energy barrier model. Our measurements on myoglobin coated particles and polystyrene surfaces reveal a consistent picture for a wide range of fluid conditions (ionic strength, pH) and surface conditions (oxidation state). The method paves the way for further studies to quantify the mechanisms underlying nonspecific interactions between protein coated particles and polymer surfaces in different matrixes.

interaction will occur, caused by high electrostatic repulsion, a low Hamaker constant or other interactions. Using eq 7, we can relate the found parameters to actual energy barrier heights (eq 1), only taking into account electrostatic, van der Waals, gravitational, and magnetic energies. The used incubation time of 5 min gives Ec = kT ln(2tk0) ≈ (6.4 + ln k0)kT. Because of this logarithmic dependence, Ec is very insensitive to variations in k0 and a range of 10 < k0 < 108 only gives rise to 8.7 < Ec < 25, expressed in units of kT. Further, to make an estimate of C′ and D′, a value of the energy barrier spread σ needs to be chosen. The difference in surface potential between oxidized and nonoxidized polystyrene is about 20 mV.19 The accuracy of particle surface potential measurements typically is a few millivolts. We estimate the variation over the surface to be small compared to these values. Since, according to our calculations, 1 mV change in surface potential corresponds to 30 kT change in energy barrier, we estimate σ ≈ 10 kT. A lower limit for σ can be found from the observation that at high ionic strength all particles bind. By setting Eb,0 = D′ = σ, together with the found D and calculated Ec-range, the minimum value of σ is between 1.7 and 8 kT. This conforms to our choice of σ ≈ 10 kT. Using the values found for C and D (2 < C < 1000 and D ≈ 2), the following ranges are found for C′ and D′, expressed in units of kT: 28 < C′ < 14 000 −19 < D′ < −3

(8)

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

S Supporting Information *

Theoretically, at infinite ionic strength, Eb = D′. The negative value of D′ corresponds with the observation that at high ionic strength all particles bind. We can compare the experimentally derived values in eq 8 with the energy barrier calculations. The calculated values found for C′ are typically 20 000 to 30 000kT, a few times larger than the experimentally estimated maximum value of 14 000kT. The values found for D′ are in the range −60 to 160kT, so they deviate somewhat more. This could be explained by the fact that our calculations included only electrostatic and van der Waals interactions, while in the measurements, for example, hydrophobic interactions will play a role as well. Hydrophobic interactions will tend to lower the energy barrier, compared to only electrostatic, van der Waals, gravitational, and magnetic interactions, which is what we observe when comparing calculated and measured energy barrier heights. Our model, however, just used a parametrized energy barrier (as function of ionic strength of the solution) without details about the different contributions, so its use is not limited by this.

Typical snapshot of the field of view of a movie taken with the rotating particles probe. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research forms part of the research programme of the Dutch Polymer Institute (DPI, Project No. 677). D. Spridon acknowledges financial support from the European Social Fund in Romania, under the responsibility of the Managing Authority for the Sectoral Operational Programme for Human Resources Development 2007-2013 (Grant PSDRU/88/1.5/S/47646).

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REFERENCES

(1) Martins, V. C.; Cardoso, F. A.; Germano, J.; Cardoso, S.; Sousa, L.; Piedade, M.; Freitas, P. P.; Fonseca, L. P. Femtomolar limit of detection with a magnetoresistive biochip. Biosens. Bioelectron. 2009, 24, 2690−2695. (2) Bruls, D. M.; Evers, T. H.; Kahlman, J. A. H.; van Lankvelt, P. J. W.; Ovsyanko, M.; Pelssers, E. G. M.; Schleipen, J. J. H. B.; de Theije, F. K.; Verschuren, C. A.; van der Wijk, T.; van Zon, J. B. A.; Dittmer, W. U.; Immink, A. H. J.; Nieuwenhuis, J. H.; Prins, M. W. J. Rapid integrated biosensor for multiplexed immunoassays based on actuated magnetic nanoparticles. Lab Chip 2009, 9, 3504−3510. (3) Wang, Y.; Dostalek, J.; Knoll, W. Magnetic nanoparticleenhanced biosensor based on grating-coupled surface plasmon resonance. Anal. Chem. 2011, 83, 6202−6207. (4) Darain, F.; Yager, P.; Gan, K. L.; Tjin, S. C. On-chip detection of myoglobin based on fluorescence. Biosens. Bioelectron. 2009, 24, 1744− 1750.

CONCLUSIONS We have measured the interaction of myoglobin coated particles on oxidized polystyrene surfaces using the rotating particles probe, which quantifies interactions by recording the fraction of rotating particles. Since no particles are observed to unbind during the measurements, we have developed a distributed energy barrier model to explain the shape of the measured curves as a function of the ionic strength of the solution. The interaction barrier is characterized by two parameters, one describing the ionic strength dependence of the barrier (C) and one describing the barrier height at very high ionic strength (D). The high ionic strength term is found to vary in a small range (1.4−3), independent of oxidation time and pH. In contrast, the ionic strength dependence term clearly 8154

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dx.doi.org/10.1021/la300630n | Langmuir 2012, 28, 8149−8155