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From a Protein’s Perspective: Elution at the Single-Molecule Level Logan D. C. Bishop† and Christy F. Landes*,†,‡ Department of Chemistry, ‡Department of Electrical and Computer Engineering, and Smalley-Curl Institute, Rice University, Houston, Texas 77251, United States
Acc. Chem. Res. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 08/22/18. For personal use only.
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CONSPECTUS: Column chromatography is a widely used analytical technique capable of identifying and isolating a desired chemical species from a more complicated mixture. Despite the method’s prevalence, theoretical descriptions have not advanced to accommodate today’s common analyte, proteins. Proteins are increasingly used as biologics, a term that refers to biological pharmaceuticals, and present new complexities for chromatographic separation. Large variations in surface charge, chemistry, and structure among protein analytes expose the limits in the current theoretical framework’s ability to predict the efficiency of a column without empirical data. The bottleneck created by empirical optimization is a strong motivation for a renewed effort to achieve an indepth understanding of the range of interactions that occur between a protein analyte and the stationary phase that together enable its selective separation from other constituents of a mixture. The physical and chemical processes that dictate the amount of time an analyte spends in the column are often abstracted by theory and treated as statistical distributions. Until recently, these distributions could not be mapped experimentally as traditional experimental techniques could not reveal underlying heterogeneity in structure, charge, and dynamics. Aligning the latest experimental and theoretical advances is thus a hurdle to be overcome so that significant progress can be made toward a predictive chromatographic theory. In this Account, we detail the work of the Landes Lab in developing single-molecule techniques that refine the stochastic theory of chromatography as a first step toward predictive chromatographic column design. We provide a brief review of the development of stochastic theory and establish a mathematical framework to put the discussed physical chemistry in context. We describe our investigations of three pertinent phenomena: mobile/stationary phase exchange, adsorption/desorption kinetics, and hindered diffusion. We highlight experimental evidence that points to nonuniform behavior. Then, we describe our work in developing single-molecule techniques that can evaluate these effects on a protein-by-protein basis. We highlight two developments: fast imaging via super temporal-resolved microscopy (STReM) and visualizing diffusion within pores via a combination of fluorescence correlation spectroscopy and super-resolution optical fluctuation imaging (fcsSOFI). Both methods offer new ways to study chromatographic elution at the single-protein level. Such methods can identify the rare heterogeneities that prevent efficient separations and advance the field closer to predictively optimized protein separations.
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INTRODUCTION Bringing a new protein based therapeutic, termed biologic, to market averages approximately $2.6 billion, of which 50% can be dedicated to isolating and purifying the active compound.1,2 A hidden contributor to this cost is the iterative optimization of each stationary phase/mobile phase interaction, for example, packing density, porosity, ligand density, scale-up, etc.3−5 Ionexchange chromatography uses electrostatic interactions between analyte and stationary phase to improve the separation. Similarly, affinity chromatography retains the analyte via complexation with antibodies bound to the stationary phase surface.6 Improving chromatographic isolation of biologics requires a better understanding of the underlying analyte−stationary phase interactions and how these processes can be predictively selected to optimize macroscale elutions. One major problem in chromatographic separations is that subpopulations of slow (or fast) eluting analytes cause a tail (or front) on an elution peak.7 This analyte population, made up of a small fraction of © XXXX American Chemical Society
the overall sample, causes overlap with other constituent peaks reducing the separation efficiency and the amount of product recovered. It is clear from the simple example of elution peak tailing that better experimental methods are necessary to probe the chemistry and physics that drive subpopulation behavior and to relate mechanistic details to macroscale observables using the standard dictates of statistical mechanics. Doing so will enable predictive design of new separation media, thereby minimizing time intensive iterative optimization. The structure of this Account mirrors this goal by introducing a theoretical framework to direct the discussion on three phenomenological processes where tailing can arise: Exchange between mobile and stationary phase, adsorption/desorption events, and the hindrance of diffusion in pores. Increasing the predictive power of theory to describe these effects will improve the Received: May 15, 2018
A
DOI: 10.1021/acs.accounts.8b00211 Acc. Chem. Res. XXXX, XXX, XXX−XXX
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Accounts of Chemical Research fundamental understanding of protein chromatography and provide a much-needed speed up in the development of biologics.
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THEORY An explicit description of chromatographic elution must account for heterogeneity in the mechanics of flow for each protein and its interaction with the stationary phase. Three central lines of theory have been utilized to describe chromatography: the plate model of Martin and Synge,8 the differential model of Lapidus and Amundson,9 and the stochastic theories authored by Giddings and Eyring.10 The first two models describe elution as the transfer of ensemble analyte mass between volumetric “plates”, serving as the underlying theory for the van Deemter equation.11 Modeling elution from the ensemble view fails to capture the rare effects that lead to tailing in elution. Revealing the driving forces that create the tailing population and eliminating them from the column requires examination on a molecule-by-molecule basis. Giddings and Eyring’s model describes the motion of a single molecule down the length of a column as a random walk with periods of motion and periods of static adsorption.12 The elution time (tr) is the sum of time spent in the mobile phase (tm) and time spent in the stationary phase (ts) (eq 1).10 tr = tm + ts
(1)
A statistical expansion of ts models a Poisson process of adsorption times (τ) sampled from an exponential distribution. Time spent in the stationary phase is then ts = ∑mi τi. Early theories treated tm as constant across all molecules (tm = t0) within the length of the column (d) at mobile phase flow speed (μ). While Giddings posited the existence of separate site types, others expanded the analytical solution to account for more than one site chemistry.13,14 The stationary phase time of any protein is then the sum of all time segments spent adsorbed to the surface for n types of adsorption sites across m adsorption events per site type. Given constant mobile phase time (t0), elution time is then described by eq 2. n
tr = t0 +
m
∑ ∑ τi ,j i
j
(2) Figure 1. (A) An elution profile (top) can be viewed as the sum of many independent molecular trajectories (bottom). (B) Elution viewed as a molecular path between stationary phase beads, characterized by intermittent moments of adsorption on a stationary phase bead. (C) Each adsorption site (green) will have a characteristic adsorption rate constant (ki) and desorption rate constant (ki−1). (D) Interaction with the porous stationary phase complicates the motion of a molecule as the position of the molecule in reference to a surface can introduce hindrance to diffusive motion. Protein structure data obtained from PDB 1A4V.23
A full elution profile (Figure 1A, top), results from combining all random walks of the analyte molecules (Figure 1A, bottom) as they travel down the column. Using this model, McQuarrie introduced the use of characteristic functions (CFs) to analytically derive the elution profile, a development brought to maturity by Dondi et al.13,15 Current ensemble statistics treat elution as a Lèvy process, a more detailed extension of the CF theory that provides a stable analytical solution.15,16 The development of single-molecule methods has reinvigorated interest in the stochastic theory of chromatography as there are now experimental methods to track statistical distributions of analyte molecules.17−21 For example, renewal theory, a more formal description of the previously cited random walk, has been used to model movement of an analyte down a column on a particle-by-particle basis.19,22 Recognizing that the amount of time in the mobile phase is dependent on a nonuniform flow speed (μk) across varying distances (dk) between γ = ∑ni mi + 1 moments of motion, we amend eq 2 to produce eq 3:
γ
tr =
∑ k
dk + μk
n
m
∑ ∑ τi ,j i
j
(3)
Expanding tm acknowledges the possibility of tailing effects due to nonuniform motion caused by the porosity of the stationary phase. We propose eq 3 as the guiding mathematical framework for our current and future work as it expresses the single-molecule nature of elution and is not analyte specific. B
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Accounts of Chemical Research Equation 3 provides explicit variables for probing elution via experimental single-molecule techniques. In this Account, we explore our lab’s ongoing research probing the importance of these variables to chromatographic separations. We first discuss how to overcome the experimental challenges to quantifying the tm.24,25 Then, we describe widefield kinetics studies that inform the probabilistic distribution of τ as well as elucidate the values for n and m.26,27 Finally, correlation-based imaging is introduced to understand intrapore diffusion, capturing dk.28 The compilation of these factors provides tunable parameters for column design and offers a direct way to investigate the mechanics of elution.
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MOBILE/STATIONARY PHASE INTERCHANGE Analyte transport in the mobile phase occurs at faster speeds than can be tracked by the frame rate of detectors.29 Singleparticle tracking (SPT) would be ideal for directly observing a protein as it exchanges between the mobile and stationary phases, and methods have been previously described in detail by this lab and others.18,30 Here, we focus on our work to understand protein motion on a surface and emphasize the distributions of distance and flow velocity that compose tm. We close with a discussion of our ongoing development of the imaging technique super temporal-resolved microscopy (STReM), which we hope will enable both spatial and temporal tracking of fluorescently labeled proteins as they exchange between mobile and stationary phases. Recent single-protein studies show that protein−surface interactions are more complicated than can be described by simple adsorption−desorption between the mobile and stationary phases27,31−33 and instead are governed by hydrophilic character, surface roughness, and oxygen content. Moringo et al. performed an exploratory study into the effect of increasing oxygen-containing functional groups on polystyrene surfaces in tuning the surface-hopping of lysozyme, a model protein.24 Superlocalization techniques in combination with a SPT algorithm, Troika,34 allowed us to characterize individual proteins as they explored the surface (Figure 2A).24 Increasing oxygenated functional group density reduced the mobility of lysozyme measured via single frame displacement. This effect was attributed to a reduction of available hydrophobic adsorption sites on the polystyrene surface. Competitive adsorption of water at the surface increased the average distance (dk) traveled by lysozyme between adsorption events by reducing the likelihood of reabsorption to the surface immediately after desorption. Modeling a distribution for the random distances traveled between adsorptions requires knowledge of the statistical connectedness of adsorption events. Mabry et al. performed autocorrelation studies of BODIPY C12 fatty acid over trimethylsilyl silica to quantify the uniformity of the surface in relation to adsorption probability.31 The study, originally designed to identify the roles of specific vs nonspecific adsorption sites, included a brief aside on the correlation between adsorption events over a distance. Autocorrelation analysis of adsorption events found significant correlation at distances under 300 nm (Figure 2B) indicating that molecular adsorptions occurred in clusters with 300 nm radii. Two major conclusions were drawn from this data. First, correlation between adsorption events implies that the surface is nonhomogenous in terms of binding probabilities. This conclusion supports the idea that many kinds of site chemistries can be present (n > 1 in eq 2) on what is
Figure 2. (A) Single-particle tracking of lysozyme on an oxygen plasma treated polystyrene surface. The area explored by the protein is tuned by the amount of time the surface is treated.24 (B) Autocorrelation of the distance traveled by BODIPY C12 fatty acid on trimethylsilyl silica in different concentrations of methanol. Higher values of autocorrelation indicate that the quantities are statistically related.31 (C, left) Standard image using a stationary double-helix phase mask.40,41 (C, right) Image taken using a rotating DHPM.25 Time of adsorption is encoded into the angle separating the two lobes of PSF.25 Panel A adapted with permission from ref 24. Copyright 2017 American Chemical Society. Panel B adapted with permission from ref 31. Copyright 2014 American Chemical Society. Panel C adapted with permission from ref 25. Copyright 2016 American Chemical Society.
otherwise considered a homogeneous surface. Second, the data also suggests that the chance of one adsorption increases the likelihood of subsequent adsorptions simply because the protein is in closer proximity to the surface, a phenomenon posited by Scott and Fritz much earlier.22 These studies highlight that the distances between events can be quantified, albeit at an unsuitable resolution for complete predictability. Tuning tm requires development of techniques that can accurately map the distribution of distances between adsorption events despite the high speed at which these events occur. Capturing protein dynamics faster than the frame rate of common CCD detectors is thus a current objective of the Landes Lab and is being approached through inclusion of a rotating double-helix (DH) phase mask in the Fourier plane of the detector path in a method called STReM. The DH phase mask was originally developed to compress 3D sample information into a 2D image and turns an C
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A common route for tuning adsorption chemistry in ion exchange chromatography is through functionalization of the stationary phase surface with charged ligands. Kisley et al. investigated the adsorption of fluorescently labeled αlactalbumin to agarose stationary phases functionalized with counter-charged ligands of two different forms: engineered pentargininamide ligands vs stochastic clusters of monoargininamide ligands (Figure 3A).26 Clustered monoargininamide sites only approached the adsorption kinetics of the pentaargininamide sites when loaded with 10000 times as many ligands and yielded a broader distribution of adsorption kinetics. Engineered pentaargininamide clusters induced a narrower distribution of adsorption kinetics than stochastic clusters despite an identical interfacial charge, analogous to fewer site chemistries (n in eq 3) with a larger number of adsorption events at each site (m), yielding a narrower elution curve. Statistics describing analyte−stationary phase interactions were collected via observation of protein residence times on the surface revealing heterogeneity in desorption/adsorption dynamics between sites. Desorption times (td) and waiting times between adsorptions (ta) were tracked at each type of ligand site. Rate constants for desorption/adsorption were calculated and found to be linear, analogous to the linear isotherm regime (Figure 3B).10 Interestingly, kinetics among different ligand sites were nonuniform, strongly supporting a description in which several different types of adsorption site chemistries are present (n > 1 in eq 2). Combining these statistics with the mathematical construction of Lèvy processes allowed for simulated elution (Figure 3C). Despite having identical interfacial charges, the variation in local ligand conditions contributed to vastly different elution profiles. Rare site chemistries, recognizable only on a site-by-site basis, clearly led to asymmetries in the final elution curves and reduced the expected separation efficiency. Adsorption kinetics are also strongly affected by the interactions of other chemical components in the mobile phase that share chemical or physical properties with the target protein. In a follow up study, an agarose-supported surface was functionalized with argininamide ligands and introduced to a laminar flow of fluorescently labeled α-lactalbumin.27 Ionic content of the analyte mixture was controlled by varying the concentration of NaCl. It was found that the salt concentration increased the number of adsorptions at any one site (m) as well as affected the availability of sites across the surface (Figure 3D). Lower concentrations of salt lead to many rarebinding sites, while increased salt content lead to fewer sites with higher binding counts, possibly a result of the adsorption mechanism changing. This result implies that tuning adsorption effects requires knowledge of contributions from the functionalization of the stationary phase as well as the chemical content of the mobile phase, analogous to tuning both m and n in eq 2. Theories proposed to describe the reasoning behind the types of site chemistries (n) and adsorption populations (m) rely on the introduction of new steric effects and ionic shielding. Varying agarose concentration tunes the pore sizes on the surface. It was suggested that a decrease in pore size enabled steric screening of charged ligand sites reducing the likelihood of adsorption and shortening the time adsorbed by reducing interaction energy.47 This idea is expanded in the next section as we note that pore structure can drastically change the amount of accessible polymer surface.
otherwise radially symmetric point-spread function (PSF) into a pair of lobes rotating along a shared axis.18,30,35 STReM encodes temporal information in the angle of the DH PSF by relating lobe orientation to the rotational position of the DH phase mask. The unique angle of the PSF relative to the angular frequency of the phase mask and frame rate is then used to correlate the subframe arrival time of an adsorbing fluorescently labeled protein (Figure 2C), thereby improving the ability to track mobile phase−stationary phase exchange by 20 times faster than the camera frame rate. STReM offers promise in tracking a protein’s transition from mobile to stationary phases, one key to understanding and eventually tuning the construction of chromatographic columns. For example, STReM recently was used to resolve an additional desorption step for α-lactalbumin on nylon 6,6 that was undetectable at standard temporal resolutions.36 Calculating flow speeds through a column, via experiment and simulation, is a current topic of interest,37,38 but simulating flow through porous media is computationally expensive and is not easily generalizable across the many possible mesoporous structures.39 As such, we identify future advancements of STReM in improving time resolution as particularly crucial to the goal of achieving a predictive chromatographic theory. Optimizing chromatography will depend on tuning the macrostructure to enable controlled flow through the column and a narrow distribution of tm.42 However, capturing the motion of the protein is only one facet to fully modeling elution. The other half of the equation, ts, requires a mechanistic understanding of the adsorption/desorption kinetics.
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ADSORPTION AND DESORPTION PROCESSES The standard mechanism for protein ion-exchange chromatographic separation relies on the adsorption of the protein onto counter-charged ligands on the stationary phase.43 The probability of adsorption and the length of time spent adsorbed should be a function of unique charge and chemical or physical properties of the selected analyte compared to those of the contaminants and the eluent. As described by eq 3, a summation of all adsorption events forms the full contribution of the time in the stationary phase (ts) to the total elution time. Studying these events requires capturing the statistical moments of three variables: the number of adsorption events (m), the number of adsorption site chemistries (n), and the distribution of adsorption times (τ). Increasing separation efficiency requires reducing the variance of m and τ, as well as isolating only one site chemistry (n = 1), thereby removing one major contributor to anomalous tailing or fronting effects (Figure 1A).7 Discovering the statistical moments of these underlying distributions requires investigation of each independent protein rather than elucidation from the ensemble system. To this end, our lab has focused on examining the dynamics of a model protein (α-lactalbumin) on standard stationary phase systems (agarose and silica) in the single-molecule regime.26,27,44,45 The methods46 described attempt to visualize the dynamics of adsorption, quantify the number of events (m) and number of individual sites, and then construct the statistics that describe ts. This approach hints at predictive tuning of adsorption chemistry to fit the characteristics of the analyte− protein, reducing the cost of separations for biologics by removing repetitive optimization of the column. D
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plays in adsorption is key to elucidating the chemical effects that dictate ts. Similarly, a fully descriptive model of chromatography also requires an understanding of how stationary phase features affect protein motion.
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HINDERED DIFFUSION IN PORES The previous sections have, by necessity, not made assignments about analyte motion to make data analysis tractable. A full description of the standard elution process requires knowledge of the diffusive motion of a protein through the varied pore geometry of the stationary phase.48−50 Variation in microstructure such as pore shape, size distribution, and their effects on flow and accessible surface area need to be understood analytically to tune the efficiency of chromatographic separations. Current analytical models for diffusion within pores lack the predictive power to analytically evaluate the diffusion within the stationary phase. Hlushkou et al. performed a theoretical investigation into diffusion when analyte size is comparable to pore throat diameter.51 Commonly used theoretical models assume that each analyte has an effective diameter of zero, reducing both complexity of the expression and predictive power. This assumption does not reflect reality and even small variations in relative sizes of analyte and pores have a marked effect on the diffusive motion of the protein through the stationary phase (Figure 4A). Here, we note that a particle having an infinitesimally small molecular diameter has a larger range of spaces that it can explore (Figure 4A, left). Increasing d the particles’ diameter to 94 nm (λ = tracer = 0.237 ) to d pore
accurately reflect elution greatly decreases pore accessibility by limiting the number of access points to the mesoporous structure. (Figure 4A, right). Bottle-necking at smaller pore openings reduces the total explorable volume of the stationary phase not only altering the diffusivity of each protein but also affecting the local concentration on the macroscale (Figure 4B). Doing so effectively decreases the number of possible adsorption sites encountered by an analyte during its journey down the column. Conversely, a smaller λ increases protein flow through the porous volume and offers more possible adsorption sites. Availability of sites increases due to more accessible surface area leading to more adsorption events (m) and a change in the competition at the stationary phase surface. Analytical models posited first by Renkin52 and later refined by Dechdilok and Deen53 do not accurately account for steric hindrance as they were posed in reference to a single, uniform cylindrical pore isolated from the mesoporous structure. This assumption is insufficient given the stochastic construction of the porous media used in chromatography. Pore shape is nonuniform and flow effects can vary greatly at the entrance and exit between pores. A central part of the investigation by Hlushkou et al. highlighted significant deviation from these theorems when the analyte’s molecular diameter approached 20% of the pore throat diameter (λ = 0.2, Figure 4C). Approaching this limit led to a steep drop in diffusivity while the presented theories predicted a gentler decrease. In total, the study highlights the shortcomings of current theories in modeling diffusive motion through porous media. One avenue for solving this computationally intractable problem is to measure diffusive motion directly via superresolution microscopy. The Landes Lab has pioneered a technique combining super-resolution optical fluctuation
Figure 3. (A) Super-resolution images of α-lactalbumin adsorption events on (left) stochastic and (right) engineered ligand clusters of argininamide on functionalized agarose.26 (B) Four selected pentaargininamide binding sites (left) and the distribution of desorption/adsorption times at those sites (right).26 (C) Projection of rate constants from panels A and B using stochastic theory simulates elution curves for columns comprised of stochastic vs engineered ligand clusters.26 (D) Super-resolution images show that variations in the ionic strength of the mobile phase both influence the number of protein adsorptions at any single site and also tune the available sites (m and n in eq 2).27 Panels A, B, and C adapted with permission from ref 26. Copyright 2014 PNAS. Panel D adapted with permission from ref 27. Copyright 2014 Elsevier.
Studies of α-lactalbumin interacting with porous agarose have shown that the traditional belief that surface sites share a homogeneous chemistry is fundamentally incorrect.26,27 Microscale heterogeneities, whether it be ligand clustering or shifts in ionic strength, can have macroscale implications for the final elution profile. Understanding the role that surface engineering E
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the diffusion within. fcsSOFI was extended to another system of interest, nanochannels. These, like traditional pores, represent an avenue of interest in the study of stationary phase morphology.56−58 Understanding diffusion dynamics within porous structures is critical to achieving high-resolution separations as it plays a crucial role in varying tm (via diffusion) and tuning ts (via pore accessibility). A proper description of these factors requires knowledge of the microscopic contributions to the macroscopic elution, thereby requiring single-molecule methods to observe the motion of proteins in the column.19,39 Though the methods described operate on simplified separation environments, their continued development to measure motion within the nano- and mesoporous stationary phase structure will be key to directed design of the stationary phase.
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CONCLUSION The continuous rise in demand and price within the global pharmaceutical market creates an implicit imperative that the process of developing new drugs be streamlined. It is the view of the authors that this necessary speed up can be achieved, in part, through the creation of a unified theory of chromatography. The paragraphs preceding describe what we believe are the most important factors for developing such a theory. We described the ongoing work in our lab to visualize elution mechanisms at the experimental level and to marry them with new theoretical results in the field. Knowledge of the interchange between the mobile and stationary phase, adsorption dynamics occurring on the surface, and diffusive motion of proteins as they enter the porous stationary phase is the key to building the next generation of chromatographic columns. Adequate understanding of these phenomena should help in reducing the cost and time of producing the necessary pharmaceuticals for a changing medical landscape. Looking forward, the goal of Landes Lab is to continue developing high through-put imaging techniques capable of distinguishing dynamics occurring at the single protein level. To this end, we have begun to integrate machine learning techniques into the image analysis to capture salient patterns undetectable by human observation or current algorithms. These efforts, in concert with the theoretical framework we have proposed, are necessary steps in developing the next generation of single-molecule methods to evaluate chromatography. We stress the need for increased attention with modern computational tools. An updated theoretical framework will enable anticipatory design of future stationary phases to avoid structures that cause elution defects before column construction. Transitioning to numerical simulations will avoid the simplifications of previous analytical theories and allow parallels between the theoretically modeled variables and experimental observables seen under the microscope. This marriage of theory and experiment not only speeds the development of functionalized surfaces but also points to specifically engineered stationary phase morphologies. With these new methodologies, we will continue to study elution at a granularity far smaller than in the past, building an ensemble view from a protein’s perspective.
Figure 4. (A) Simulated images depicting dependence of stationary phase accessibility on ratio of protein diameter to pore diameter (λ).51 (B) The differences in explorable volume dependent on the value of λ.51 (C) A theoretical mapping of the reduction of mean diffusion based on the ratio of the diameter of the protein to the pore diameter.51 (D) Simulated experiments for simultaneous protein tracking within a pore and imaging pore size distributions. (left) Examination of particle motion within a 1D pore using SPT. (right) The same data analyzed by fcsSOFI for diffusion characterization as well as subdiffraction pore imaging.28 Panels A, B, and C adapted with permission from ref 51. Copyright 2017 American Chemical Society. Panel D adapted with permission from ref 28. Copyright 2015 American Chemical Society.
imaging (SOFI)54 with fluorescence correlation spectroscopy (fcs).55 Combining these two methodologies allows for tracking protein diffusion within pores (via fcs) while imaging the pore itself (via SOFI). The method, termed fcsSOFI, has enabled observation of diffusion within a single pore and has rendered significant improvements over using SPT alone (Figure 4D).28 Use of fcsSOFI allowed for the detection of two identifiable pores and quantification of the 1D diffusion of simulated particles inside these pores. In comparison, SPT methods identified the edges of the pore but failed to quantify
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AUTHOR INFORMATION
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*E-mail: cfl
[email protected]. F
DOI: 10.1021/acs.accounts.8b00211 Acc. Chem. Res. XXXX, XXX, XXX−XXX
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(9) Lapidus, L.; Amundson, N. R. Mathematics of Adsorption in Beds 0.6. The Effect of Longitudinal Diffusion in Ion Exchange and Chromatographic Columns. J. Phys. Chem. 1952, 56, 984−988. (10) Giddings, J. C.; Eyring, H. A Molecular Dynamic Theory of Chromatography. J. Phys. Chem. 1955, 59, 416−421. (11) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography. Chem. Eng. Sci. 1956, 5, 271−289. (12) Giddings, J. C. The random downstream migration of molecules in chromatography. J. Chem. Educ. 1958, 35, 588. (13) Mcquarrie, D. A. On Stochastic Theory of Chromatography. J. Chem. Phys. 1963, 38, 437−445. (14) Fornstedt, T.; Zhong, G. M.; Guiochon, G. Peak tailing and mass transfer kinetics in linear chromatography. J. Chromatogr. A 1996, 741, 1−12. (15) Dondi, F.; Remelli, M. The characteristic function method in the stochastic theory of chromatography. J. Phys. Chem. 1986, 90, 1885−1891. (16) Dondi, F.; Cavazzini, A.; Pasti, L. Chromatography as Lévy Stochastic process. J. Chromatogr. A 2006, 1126, 257−267. (17) Pasti, L.; Cavazzini, A.; Felinger, A.; Martin, M.; Dondi, F. Single-molecule observation and chromatography unified by levy process representation. Anal. Chem. 2005, 77, 2524−2535. (18) Shen, H.; Tauzin, L. J.; Baiyasi, R.; Wang, W. X.; Moringo, N.; Shuang, B.; Landes, C. F. Single Particle Tracking: From Theory to Biophysical Applications. Chem. Rev. 2017, 117, 7331−7376. (19) Hlushkou, D.; Gritti, F.; Daneyko, A.; Guiochon, G.; Tallarek, U. How Microscopic Characteristics of the Adsorption Kinetics Impact Macroscale Transport in Chromatographic Beds. J. Phys. Chem. C 2013, 117, 22974−22985. (20) McCain, K. S.; Hanley, D. C.; Harris, J. M. Single-molecule fluorescence trajectories for investigating molecular transport in thin silica sol-gel films. Anal. Chem. 2003, 75, 4351−4359. (21) Zhong, Z. M.; Lowry, M.; Wang, G. F.; Geng, L. Probing strong adsorption of solute onto C18-silica gel by fluorescence correlation imaging and single-molecule spectroscopy under RPLC conditions. Anal. Chem. 2005, 77, 2303−2310. (22) Scott, D. M.; Fritz, J. S. Model for chromatographic separations based on renewal theory. Anal. Chem. 1984, 56, 1561−1566. (23) Chandra, N.; Brew, K.; Acharya, K. R. Structural Evidence for the Presence of a Secondary Calcium Binding Site in Human αLactalbumin. Biochemistry 1998, 37, 4767−4772. (24) Moringo, N. A.; Shen, H.; Tauzin, L. J.; Wang, W.; Bishop, L. D. C.; Landes, C. F. Variable lysozyme transport dynamics on oxidatively functionalized polystyrene films. Langmuir 2017, 33, 10818. (25) Wang, W.; Shen, H.; Shuang, B.; Hoener, B. S.; Tauzin, L. J.; Moringo, N. A.; Kelly, K. F.; Landes, C. F. Super Temporal-Resolved Microscopy (STReM). J. Phys. Chem. Lett. 2016, 7, 4524−4529. (26) Kisley, L.; Chen, J. X.; Mansur, A. P.; Shuang, B.; Kourentzi, K.; Poongavanam, M. V.; Chen, W. H.; Dhamane, S.; Willson, R. C.; Landes, C. F. Unified superresolution experiments and stochastic theory provide mechanistic insight into protein ion-exchange adsorptive separations. Proc. Natl. Acad. Sci. U. S. A. 2014, 111, 2075−2080. (27) Kisley, L.; Chen, J. X.; Mansur, A. P.; Dominguez-Medina, S.; Kulla, E.; Kang, M. K.; Shuang, B.; Kourentzi, K.; Poongavanam, M. V.; Dhamane, S.; Willson, R. C.; Landes, C. F. High ionic strength narrows the population of sites participating in protein ion-exchange adsorption: A single-molecule study. J. Chromatogr. A 2014, 1343, 135−142. (28) Kisley, L.; Brunetti, R.; Tauzin, L. J.; Shuang, B.; Yi, X. Y.; Kirkeminde, A. W.; Higgins, D. A.; Weiss, S.; Landes, C. F. Characterization of Porous Materials by Fluorescence Correlation Spectroscopy Super-resolution Optical Fluctuation Imaging. ACS Nano 2015, 9, 9158−9166. (29) Michalet, X.; Siegmund, O. H. W.; Vallerga, J. V.; Jelinsky, P.; Millaud, J. E.; Weiss, S. Detectors for single-molecule fluorescence imaging and spectroscopy. J. Mod. Opt. 2007, 54, 239−239.
Logan D. C. Bishop: 0000-0003-2265-4374 Christy F. Landes: 0000-0003-4163-6497 Notes
The authors declare no competing financial interest. Biographies Logan D. C. Bishop is an aspiring computational chemist pursuing his interests in applying advanced machine learning methods to statistical mechanics. He graduated from the University of Texas at Austin with a double major in Chemistry and Computer Science then joined the Landes Lab at Rice in Fall of 2016. His current project involves studying the probabilistic distributions that dictate chromatographic elution. Christy F. Landes is a Professor in the Department of Chemistry and the Department of Electrical and Computer Engineering at Rice University in Houston, Texas. After graduating from George Mason University in 1998, she received a Ph.D. in Physical Chemistry from the Georgia Institute of Technology in 2003. She was a postdoctoral researcher at the University of Oregon and a NIH postdoctoral fellow at the University of Texas at Austin. She moved to Rice in 2009, earning a NSF CAREER award for her tenure-track work in 2011 and an ACS Early Career Award in experimental physical chemistry in 2016.
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ACKNOWLEDGMENTS Christy Landes thanks the Welch Foundation (Grant C-1787), and Logan D. C. Bishop acknowledges that this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program (Grant 1450681).
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GLOSSARY tr: total elution time tm: time in mobile phase ts: time on stationary phase τx: time of a single adsorption event dk: distance traveled between adsorption events μk: flow speed between adsorption events
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REFERENCES
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