Faster Convergence of Diffusion Anisotropy Detection by Three-Step

Mar 16, 2016 - We focus on the issue of limited number of samples in the single particle tracking (SPT) when trying to extract the diffusion anisotrop...
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Faster Convergence of Diffusion Anisotropy Detection by Three-Step Relation of Single-Particle Trajectory Yu Matsuda,*,† Itsuo Hanasaki,*,‡ Ryo Iwao,§ Hiroki Yamaguchi,§ and Tomohide Niimi§ †

Institute of Materials and Systems for Sustainability, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8603, Japan Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Naka-cho 2-24-16, Koganei, Tokyo 184-8588, Japan § Department of Micro-Nano Systems Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8603, Japan ‡

ABSTRACT: We focus on the issue of limited number of samples in the single particle tracking (SPT) when trying to extract the diffusion anisotropy that originates from the particle asymmetry. We propose a novel evaluation technique of SPT making use of the relation of the consecutive three steps. More specifically, the trend of the angle comprised of the three positions and the displacements are plotted on a scatter diagram. The particle anisotropy dependence of the shape of the scatter diagram is examined through the data from the standard numerical model of anisotropic two-dimensional Brownian motion. Comparison with the existing method reveals the faster convergence in the evaluation. In particular, our proposed method realizes the detection of diffusion anisotropy under the conditions of not only less number of data but also larger time steps. This is of practical importance not only when the abundant data is hard to achieve but also when the rotational diffusion is fast compared to the frame rate of the camera equipment, which tends to be more common for smaller particles or molecules of interest.

S

of practical importance in the fabrication of metamaterials from particle dispersion.17 Prolate particles such as rods and ellipsoids have two kinds of translational diffusion coefficients D∥ and D⊥ and the rotational one Dr based on their shape (cf. Figure 1a). In general, D∥ and D⊥ are different. It has been shown that this diffusion anisotropy can be evaluated from the trajectory data.18−21 The essence of diffusion anisotropy is described by the time correlation function for the translationalrotational coupling18 or second and fourth moments of the displacement of the particle,19,20 which enabled the quantitative evaluation. Another approach of characterization is based on the large deviation principle.21 It is the fundamental principle of statistical mathematics that includes the law of large numbers and the central limit theorem. Therefore, it is rather an application of universal principle to a specific problem. Whereas the MSD is a kind of variance characteristics as a function of time, the application of the large deviation principle to the random trajectories broadens the scope of characterization. Indeed, the approach has been extended to the characterization of bacterial motility,22 while the qualitative difference of selfpropulsion from the Brownian motion leads to the difference in the useful measure. There are more and more situations where

ingle-particle/molecule tracking (SPT/SMT) is an effective diagnostic method to extract nonbulk information, and hence it has been applied to various fields of microscopy studies such as biochemistry1−4 and polymer chemistry.5−7 In particular, there are quite a few two-dimensional systems of particle/molecule diffusion such as those in cellular membranes,2,8,9 Langmuir films or monolayers,10 and solid−liquid interfaces.11,12 The motions of individual particles are affected by the shape of the particle as well as the surrounding environment. The trajectories are usually characterized by calculating the mean square displacement (MSD). The longterm behavior of the MSD indicates modes of the particle motion: normal, anomalous, and confined diffusion.2,13 For example, when a MSD is a linear function of time, the motion of particles are regarded as normal diffusion, and the diffusion coefficient can be calculated from the MSD. The transient behavior of the MSD reflects more information, but it is not regarded as convenient for practical use. In fact, it is not necessary to solely rely on the MSD to characterize the Brownian motion of particles, and more characteristics are readily evaluated from analysis techniques of the SPT/SMT trajectory data from different schemes. There is more to Brownian motion than MSD even in simple systems.14−16 When the particles of interest are not spherical, or do not have sufficient symmetry, the Brownian motion exhibits anisotropy. The characteristics of such Brownian motions are © XXXX American Chemical Society

Received: January 29, 2016 Accepted: March 16, 2016

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DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry

why the above-mentioned focus on the data analysis technique is much less explored. Furthermore, the time resolution of the trajectory also blurs relation between the apparent result and possible mechanism.29 It is also possible to consider the analysis techniques when the sufficient amount of data for the abovementioned techniques is unavailable, although there is obviously a fundamental limit. Characterization beyond the overall diffusion coefficient provides much more information in experimental measurements also when only moderate amount of data is available.30 In this study, we propose a novel technique for the detection of diffusion anisotropy for limited amount and time-resolution of SPT/SMT trajectory. Evaluation of the relation among consecutive three time steps of positions realizes faster convergence for diffusion anisotropy detection compared to the existing method.

Figure 1. (a) Schematic of the axes for the translational and rotational diffusion coefficients. The point “C” is the particle center of diffusion. (b) Schematic of the displacements and the relative angle γ defined from three consecutive trajectory steps. (c) Definitions of the axes α and β that indicate the displacement characteristics in terms of the norm of displacements per time step and their orientation relative to the previous steps (cf. eq 11).

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MODEL OF TWO-DIMENSIONAL ANISOTROPIC BROWNIAN MOTION We consider a trajectory of a particle having anisotropic translational diffusion coefficients D∥ and D⊥ in the longitudinal and the perpendicular direction of the particle, respectively (D∥ ≥ D⊥ > 0), as well as the rotational diffusion coefficient Dr as shown in Figure 1a. The particle is assumed as a single point indicating the positions of the maker and the center of diffusion that are indistinguishable. This assumption is reasonable in most experimental conditions, because the position of maker and that of center of diffusion cannot be spatially resolved in a microscope image due to the diffraction limit of a microscope.1−3 The trajectory of the particle is generated by the twodimensional Wiener process described by the overdamped Langevin equations as follows:18,21,31−33

the larger amount of the number of samples drastically helps the characterization of the stochastic process including Brownian motion.23 Since the advantages of SPT/SMT originate from the larger amount of information before being averaged out in the macroscopic measurements, it is a reasonable and promising direction to pursue. On the other hand, it is not always feasible to collect plenty of data. In spite of the drastic advances in the measurement techniques of SPT/SMT, there remains a situation that limited number of samples are available.24−28 That is part of the reason ⎤

⎡

d x(t ) ⎥ ⎢ dt ⎣ y(t )⎦ dθ(t ) dt

⎡ 2 2 ⎢ 2D cos θ(t ) + 2D⊥ sin θ(t ) =⎢ ⎢⎣ 2D − 2D⊥ cos θ(t )sin θ(t )

(

=

(

)

⎛ D + D⊥ ⎞1/2 L=⎜ ⎟ ⎝ 2Dr ⎠

η=

1 Dr

(2)

d θ (t ) = dt

2 ξθ(t )

(7)

where x(t) = (x(t),y(t))T, Ξ(t) = (ξx(t), ξy(t))T, û(t) = (cos θ(t), sin θ(t))T, and I is the 2 × 2 unit matrix. We fix τ = Dr = 1 to simplify the discussion21 in this study; thus, the anisotropy parameter η is the only tunable parameter. Equations 6 and 7 are numerically solved by the Heun’s method18,21 expressed by

(3)

(4)

θ(t + Δt ) = θ(t ) +

2Δt ξθ(t )

x(t + Δt ) = x(t ) +

Δt [M(t + Δt ) + M(t )]·Ξ(t ) 2

(8)

(9)

D − D⊥

where M(t) is described as

D + D⊥

(5)

M (t ) =

the Langevin eqs 1 and 2 are nondimensionalized18,21 as

2(1 − η) (I − u(̂ t )u(̂ t ))]·Ξ(t )

2(1 + η) u(̂ t )u(̂ t ) +

2(1 − η) (I − u(̂ t )u(̂ t )) (10)

The discretized time Δt is the time step in the numerical calculation. The time step Δt/Dr(= Δt in this study) corresponds to the time interval between each frame in SPT/ SMT experiment. Since the rotational motion of a particle is

dx(t ) = [ 2(1 + η) u(̂ t )u(̂ t ) dt +

)

2Dr ξθ(t )

where (x(t), y(t)) is the position of the particle (the center of diffusion) and θ (t) is the orientation of the particle at the time t. The variables ξx, ξy, and ξθ are the Gaussian random variables that satisfy ⟨ξx⟩ = ⟨ξy⟩ = ⟨ξθ⟩ = 0 and ⟨ξx(t)ξx(t′)⟩ = ⟨ξy(t)ξy(t′)⟩ = ⟨ξθ(t)ξθ(t′)⟩ = δ(t − t′), where ⟨•⟩ stands for the ensemble average of •. By introducing the characteristic length L, time τ, and anisotropy parameter η as follows:

τ=

⎤ 2D⊥ cos θ(t )sin θ(t )⎥ ⎡ ξx(t )⎤ ⎥ (1) ⎥⎢ 2D sin 2θ(t ) + 2D⊥ cos2 θ(t ) ⎥⎦ ⎢⎣ ξy(t )⎥⎦ 2D −

(6) B

DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX

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Figure 2. Sample trajectories under the conditions of Δt = 1.0 × 10−2 and N = 104 for (a) η = 0.5 and (b) η = 0.9. Scatter diagrams for diffusion anisotropy that originates from particle asymmetry for (c) η = 0.5 and (d) η = 0.9. The definitions of α and β are illustrated in Figure 1. The solid lines of ellipses are obtained from eq 13. The dashed lines show the circle with the radius equal to the major axis for comparison.

overlooked with large Δt, the detection of anisotropic diffusive motion will be difficult at large Δt. For example, the diffusion anisotropy of the trajectories of Δt < 1.0 × 10−2 have been examined in the previous studies.18,21 The range of anisotropy parameter is 0 ≤ η ≤ 1. When η = 0, D∥ = D⊥, corresponding to the isotropic particle. The diffusion coefficient D⊥ = 0 if η = 1; however, prolate particles with finite lengths yield D⊥ > 0.33,34 Therefore, the anisotropy parameter with the range 0 ≤ η < 1 (0 ≤ η ≤ 0.99 for numerical reason) is investigated in this study.

cos γn =

Δxn·Δxn − 1 |Δxn||Δxn − 1|

(12)

It should be noted that |Δxn|/Δt does not represent the ordinary speed as the displacement is indifferentiable in the Wiener process. In other words, it makes sense only for finite Δt, and the scale of Δt affects the (α, β) through the rotation of particles with respect to the fixed experimental frame of Cartesian coordinate. The shape of the scatter diagram depends on the anisotropy parameter η, because the particle tends to travel longer distance toward its longitudinal direction during a sufficiently small yet finite time interval Δt. Finally, to characterize the shape of the scatter diagram, the radius of gyration tensor Rg is defined by35

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OUR PROPOSED METHOD OF DIFFUSION ANISOTROPY DETECTION Ratio of Squared Radii of Gyrations in the Three-Step Relation. We consider a trajectory of a particle with the number N of time steps besides the arbitrary initial position. The position of the particle at the time nΔt (n: integer, 0 ≤ n ≤ N) is expressed as x(nΔt). Throughout this paper, x(nΔt) is briefly denoted as xn. The relative angle and the translational distance of the particle motion are derived from three consecutive time series of trajectory data. First, we consider the relative displacements Δxn = xn+1 − xn and Δxn−1 = xn − xn−1 for n ≥ 1 as illustrated in Figure 1b. Second, the relative angle γn is defined as the angle between Δxn−1 and Δxn. Third, as shown in Figure 1c, a scatter diagram for n ≥ 1 in the new α−β space is introduced by ⎛ αn ⎞ |Δxn| ⎛ cos γn ⎞ ⎜⎜ ⎟⎟ ⎜ ⎟= Δt ⎝ sin γn ⎠ ⎝ βn ⎠ (11)

Rg2

⎡ ⎢ ⎢ =⎢ ⎢1 ⎢ ⎢⎣ N

1 N

N

∑ (αi − ⟨α⟩)2 i=1

N

∑ (αi − ⟨α⟩)(βi − ⟨β⟩) i=1

1 N

⎤

N

∑ (αi − ⟨α⟩)(βi − ⟨β⟩)⎥ i=1

1 N

N

∑ (βi − ⟨β⟩)2 i=1

⎥ ⎥ ⎥ ⎥ ⎥⎦

(13)

We define the ratio of squared radii of gyration Rr2 =

1 N 1 N

Rr2

as

N

∑i = 1 (βi − ⟨β⟩)2 N

∑i = 1 (αi − ⟨α⟩)2

(14)

For large N, the statistical quantities19,36 in eq 13 can be calculated as

where cos γn is defined as C

DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry ⎡⟨α 2⟩ 0 ⎤ ⎥ lim R g 2 = ⎢ ⎢⎣ 0 ⟨β 2⟩⎥⎦ N →∞

(15)

because ⟨α⟩ = ⟨β⟩ = 0. Equation 15 is a diagonal tensor whose eigenvalues are ⟨α2⟩ and ⟨β2⟩. This indicates that the principal axes of the tensor coincides with the α- and β-axes. The root of eigenvalues are radii of gyration and show the extent of the scatter diagram in the direction of the α- and β-axes. Equation 14 is also reduced to ⟨R r 2⟩ = lim R r 2 = N →∞

⟨β 2⟩ ⟨α 2⟩

⟨Rr2⟩

(16)

⟨Rr2⟩

This means that lies in the range of 0 < ≤ 1. When ⟨α2⟩ = ⟨β2⟩ (⟨Rr2⟩ = 1) corresponding to the trajectory of η = 0, the shape of the scatter diagram is a circle. On the other hand, ⟨Rr2⟩ < 1 for η ≠ 0 indicates an ellipsoidal shape of the scatter diagram. We consider the anisotropy parameter η ≠ 1 in this study; thus, ⟨Rr2⟩ ≠ 0. Figure 2 shows examples of trajectories and scatter diagrams under the conditions of η = 0.5 and 0.9, Δt = 1.0 × 10−2, and N = 104. The shape of the scatter diagram is

Figure 3. (a) The ratio ⟨Rr2⟩ of squared radii of gyration obtained as a function of the anisotropic parameter η and the time step Δt. The examined data points (filled points) are fitted with a curved plane (mesh representation). (b) ⟨Rr2⟩ as a function of η when Δt = 0.06. (c) ⟨Rr2⟩ as a function of Δt when η = 0.5. The horizontal axis of the inset is shown in logarithmic scale. The cross sections corresponding to parts b and c are illustrated in part a.The solid lines of parts b and c correspond to the fitting by eq 17.

not circular but ellipsoidal with the ratio ⟨R r 2⟩ of major-tominor axes being 0.87 for η = 0.5 and 0.54 for η = 0.9. The diffusion anisotropy can be better recognized from Figure 2c,d than from the trajectory. One can extend this framework to three-dimensional case by considering an ellipsoid particle having three translational and two rotational diffusion coefficients. In this case, we can obtain two relative angles from three consecutive steps: one of the angles can be determined by considering orthogonal projection of xn+1 on the space spanned by xn−1 and xn. The other angle is made by the planes spanned by xn−1−xn and xn−xn+1. The three-dimensional shape of a scatter diagram has to be considered by calculating the three-dimensional form of the radius of the gyration tensor. Relation between ⟨Rr2⟩ and η. The dependence of the ratio of squared radii of gyration ⟨Rr2⟩ on the anisotropy parameter η and the time step Δt is numerically studied for large N. The trajectories are generated by eqs 8 and 9 under the conditions of 0 ≤ η ≤ 0.99 and 1.0 × 10−4 ≤ Δt ≤ 1.0 × 10−1. The total number of time steps is set at N = 106. It is considered that N = 106 is enough large to evaluate ⟨Rr2⟩, because the off-diagonal elements of the radius of gyration tensor are negligibly small under this condition (the ratio of the diagonal and off-diagonal elements is O(10−5)). We calculate ⟨Rr2⟩ for 16 independent trajectories and averaged them for each condition (i.e., total 1.6 × 107 data points for each condition). The averaged values are treated as best estimate values of ⟨Rr2⟩ for each condition. Indeed, the standard deviations of O(10−5) are small compared with ⟨Rr2⟩ of O(10−1). Figure 3 shows the calculated ⟨Rr2⟩ as a function of η and Δt. The obtained ⟨Rr2⟩ data is well fitted by the following simple polynomial equation by the least-squares method, ⟨R r 2⟩(η , Δt ) = 1 + p10 η + p20 η2 + p11 ηΔt

where the degree of freedom adjusted R-square is 0.998. Since ⟨α2⟩ = ⟨β2⟩ at η = 0, the constant term of eq 17 is fixed to be 1 to satisfy ⟨Rr2⟩(0, Δt) = 1. The errors show the 95% confidence bounds obtained by considering the Student’s t cumulative distribution function. In our proposed detection method, Rr2 is calculated from the trajectory by eq 14. Therefore, η can be deduced from eq 17 by substituting the calculated Rr2 for ⟨Rr2⟩ under specified conditions of Δt. When we calculate η by solving eq 17 for a given Rr2, the error δη/η can be estimated as O(10−1) from the error propagation37 due to the errors shown in eq 18.

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EVALUATION OF η FROM FINITE NUMBER OF SAMPLES AND FRAME INTERVALS Our proposed method shares the notion in common with the pioneering work of ref 18 in a sense that both methods focus on the consecutive three time steps of trajectory data. As also mentioned in ref 21, techniques based on different schemes that evaluate the same property is worth existing because different origins can lead to the same value of an employed technique. The test of different techniques based on different principles on the same data improves the certainty in the mechanism, which is different from the statistical certainty of a convergence of the quantity. The latter can vary depending on the schemes, and it is important in practice as the amount of trajectory data are usually limited in SPT/SMT experiments. In addition to the amount of data, the frame interval is also limited by the camera specification. We show that our proposed method is also advantageous in the evaluation under the smaller amount of data N and larger frame interval Δt, by comparison with ref 18. From this point of view, we compare our method with that of ref 18 because convergence of higher order statistical quantities19,20 are poor in general.

(17)

p10 = ( −1.41 ± 2.28) × 10−2 p20 = −0.897 ± 0.028

(18)

p11 = −1.44 ± 0.16 D

DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry The trajectories are generated by eqs 8 and 9 with the known anisotropy parameter ηth. The anisotropy parameters ηes are estimated from the trajectories, and they approach ηth with an increasing amount of the sampled data. The dependence of the error in ηes on the total number of data N is examined. We consider a single trajectory for each condition to simplify the discussion. In ref 18, a three-step correlation function is defined as follows: C(0, t , t ′) = 2⟨{[x(t ) − x(0)]·[x(t ′) − x(t )]}2 ⟩ − ⟨|x(t ) − x(0)|2 |x(t ′) − x(t )|2 ⟩

(19)

where t′ > t and t′ = 2t in this discussion in common with ref 18, and it can be reduced to the analytical expression as follows: C(0, t , 2t ) = η2(1 − e−4t )2

(20)

Thus, C(0, t, 2t) is obtained from the trajectory data using eq 19 and η is extracted from eq 20. The accuracy of C(0, t, 2t) becomes poor beyond t > 0.25,18 and too-short time range faces the difficulty in the fitting convergence. Therefore, we calculate C(0, t, 2t) in the range of 0 ≤ t ≤ 0.25 and average NΔt/0.5 data in this discussion (note 0 ≤ 2t ≤ 0.5). Figure 4 shows the histogram of ηes for 256 independent trials. One trial consists of the evaluation of ηes by processing a trajectory generated under the conditions of ηth = 0.5, Δt = 1.0 × 10−2, and N = 104. Average of 256 trials yields ηes = 0.50 ± 0.03 for our method and ηes = 0.46 ± 0.19 for the method of ref 18 where the errors are the values containing the standard deviations and the fitting errors shown in eq 18. As shown in the figure, the distribution of ηes based on our method is narrower than that of ref 18. The broad peak around η = 0.5 and small peak at η = 0 are observed in the distribution of ref 18. The latter at η = 0 is caused by the data with C(0, t, 2t) < 0 due to small amount of the data. Figure 4 suggests the higher reliability of our proposed method in the evaluation of ηes under the same conditions of N and Δt. Comparison of ηes between our proposed method and ref 18 as a function of the number of samples N is shown in Figure 5a for trajectories of ηth = 0.5, Δt = 1.0 × 10−2. The data shown in the figure are the average of 256 trials, and error bars are the standard deviations and the fitting errors. It can be observed from the figure that the estimation errors in our proposed method is substantially smaller than those of ref 18. The plotted points for N = 104 corresponds to Figure 4. This advantage becomes more prominent when N is smaller. The estimation result for the trajectories of Δt = 1.0 × 10−1 by our method is

Figure 5. Accuracy of estimated diffusion anisotropy η as a function of the number of samples N for the cases of (a) ηth = 0.5 and (b) ηth = 0.1. The error bars indicate the standard deviations of 256 trials. The longest and shortest caps of error bars indicate the cases of our method and those of ref 18 with Δt = 10−2, respectively. The case with Δt = 10−1 is also shown for our proposed method with intermediate size of caps for the error bars in part a.

also shown in this figure. Even when Δt = 1.0 × 10−1, our method provides higher accuracy of ηes for N ≥ 103. Besides the substantial difference in the random errors in terms of standard deviations, it appears that the method of ref 18 accompanies a systematic deviation, i.e., the underestimation of η especially when N is smaller at least when η = 0.5. On the other hand, our proposed method does not seem to have systematic error at the comparable scale with the random one. Next, we confirm the estimation accuracy for weaker anisotropy. Figure 5b shows ηes for the trajectories of ηth = 0.1 with Δt = 1.0 × 10−2. As expected, the errors are larger for the estimation of smaller η for both methods. Our proposed method requires N ≥ 104 for η = 0.1, because there are many trials that lead to Rr2 > 1 for small N (the probability for Rr2 > 1 is larger than 0.2 when N < 103) and because ∂⟨Rr2⟩/∂η is small when 0 ≤ η ≤ 0.3 as shown in Figure 3b. Nevertheless, our proposed method gives smaller errors than those of ref 18 for N ≥ 104. We have discussed the performance of our proposed method on condition that the data of N steps are consecutive. There are many situations that the continuous tracking duration of each identical particle is short, e.g., less than 10. This is typically caused by the short focal depth and/or large diffusion coefficient in general and the blinking or bleaching of

Figure 4. Distributions of ηes obtained from 256 trials of evaluation by our proposed method and ref 18 where each trial uses the independent set of data consisting of a trajectory with N = 104 under the condition of ηth = 0.5 and Δt = 1.0 × 10−2. E

DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry fluorescent labels. However, the method of ref 18 requires at least 20 consecutive time series of data to obtain a single curve of C(0, t, 2t), and shorter time range of fitting increases the error and causes the difficulty in the convergence of the fitting iteration procedure. In contrast, our proposed method requires only three consecutive steps to calculate Rr2 by construction. Now, we consider actual experiment conditions. The rotational diffusion coefficient of rhodamine 6G in o-terphenyl is Dr = 0.2 1/s at 253 K.38 The frame rate of SPT/SMT measurement by a recent EM-CCD camera is usually in the range of O(10−2) s to O(10−1) s. The nondimensional time step of Δt = 10−2 corresponds to the dimensional time step of Δt/Dr = 5 × 10−2 s. This estimation shows that our proposed method is applicable in this case.

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CONCLUSIONS In this article, we have proposed the detection method of the diffusion anisotropy from particle trajectories. The translational distance and the relative angle comprised of three consecutive data of particle positions are statistically characterized. The ratio Rr2 of squared radii of gyration (cf. eq 14) defined from the two-dimensional parameter space consisting of these norm and angle are evaluated, and the one-to-one correspondence to the diffusion anisotropy η (cf. eq 5) is exploited. The estimation of η shows smaller errors compared to the existing method and its advantage is more prominent when the number of samples are smaller and the time step is larger. Furthermore, the requirement of only three consecutive steps for a unit of statistical evaluation makes it helpful for the experimental situations where tracking of an identical particle for a long duration is hard to achieve.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported in part by the Grant for Scientific Research from the JGC-S Scholarship Foundation and the Daiko Foundation.

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REFERENCES

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DOI: 10.1021/acs.analchem.6b00390 Anal. Chem. XXXX, XXX, XXX−XXX