Spatially-dependent model for rods and cones in the retina.

We develop a mathematical model for photoreceptors in the retina. We focus on rod and cone outer segment dynamics and interactions with a nutrient source associated with the retinal pigment epithelium cells. Rod and cone densities (number per unit area of retinal surface) are known to have significant spatial dependence in the retina with cones located primarily near the fovea and the rods located primarily away from the fovea. Our model accounts for this spatial dependence of the rod and cone photoreceptor density as well as for the possibility of nutrient diffusion. We present equilibrium and dynamic solutions, discuss their relation to existing models, and estimate model parameters through comparisons with available experimental measurements of both spatial and temporal photoreceptor characteristics. Our model compares well with existing data on spatially-dependent regrowth of photoreceptor outer segments in the macular region of Rhesus Monkeys. Our predictions are also consistent with existing data on the spatial dependence of photoreceptor outer segment length near the fovea in healthy human subjects. We focus primarily on the healthy eye but our model could be the basis for future efforts designed to explore various retinal pathologies, eye-related injuries, and treatments of these conditions.

Binding, brightness, or noise? Extracting temperature-dependent properties of dye bound to DNA.

We present a method for extracting temperature-dependent thermodynamic and photophysical properties of SYTO-13 dye bound to DNA from fluorescence measurements. Together, mathematical modeling, control experiments, and numerical optimization enable dye binding strength, dye brightness, and experimental noise (or error) to be discriminated from one another. By focusing on the low-dye-coverage regime, the model avoids bias and can simplify quantification. Utilizing the temperature-cycling capabilities and multi-reaction chambers of a real-time PCR machine increases throughput. Significant well-to-well and plate-to-plate variation is quantified by using total least squares to account for error in both fluorescence and nominal dye concentration. Properties computed independently for single-stranded DNA and double-stranded DNA by numerical optimization are consistent with intuition and explain the advantageous performance of SYTO-13 in high-resolution melting and real-time PCR assays. Distinguishing between binding, brightness, and noise also clarifies the mechanism for the increased fluorescence of dye in a solution of double-stranded DNA compared to single-stranded DNA; in fact, the explanation changes with temperature.

Inferring the Nominal Molecular Mass of an Analyte from Its Electron Ionization Mass Spectrum.

The performance of three algorithms for predicting nominal molecular mass from an analyte's electron ionization mass spectrum is presented. The Peak Interpretation Method (PIM) attempts to quantify the likelihood that a molecular ion peak is contained in the mass spectrum, whereas the Simple Search Hitlist Method (SS-HM) and iterative Hybrid Search Hitlist Method (iHS-HM) leverage results from mass spectral library searching. These predictions can be employed in combination (recommended) or independently. The methods were tested on two sets of query mass spectra searched against libraries that did not contain the reference mass spectra of the same compounds: 19,074 spectra of various organic molecules searched against the NIST17 mass spectral library and 162 spectra of small molecule drugs searched against SWGDRUG version 3.3. Individually, each molecular mass prediction method had computed precisions (the fraction of positive predictions that were correct) of 91, 89, and 74%, respectively. The methods become more valuable when predictions are taken together. When all three predictions were identical, which occurred in 33% of the test cases, the predicted molecular mass was almost always correct (>99%).

Prevalence estimation and optimal classification methods to account for time dependence in antibody levels.

Serology testing can identify past infection by quantifying the immune response of an infected individual providing important public health guidance. Individual immune responses are time-dependent, which is reflected in antibody measurements. Moreover, the probability of obtaining a particular measurement from a random sample changes due to changing prevalence (i.e., seroprevalence, or fraction of individuals exhibiting an immune response) of the disease in the population. Taking into account these personal and population-level effects, we develop a mathematical model that suggests a natural adaptive scheme for estimating prevalence as a function of time. We then combine the estimated prevalence with optimal decision theory to develop a time-dependent probabilistic classification scheme that minimizes the error associated with classifying a value as positive (history of infection) or negative (no such history) on a given day since the start of the pandemic. We validate this analysis by using a combination of real-world and synthetic SARS-CoV-2 data and discuss the type of longitudinal studies needed to execute this scheme in real-world settings.

Reducing Bias and Quantifying Uncertainty in Fluorescence Produced by PCR.

We present a new approach for relating nucleic-acid content to fluorescence in a real-time Polymerase Chain Reaction (PCR) assay. By coupling a two-type branching process for PCR with a fluorescence analog of Beer's Law, the approach reduces bias and quantifies uncertainty in fluorescence. As the two-type branching process distinguishes between complementary strands of DNA, it allows for a stoichiometric description of reactions between fluorescent probes and DNA and can capture the initial conditions encountered in assays targeting RNA. Analysis of the expected copy-number identifies additional dynamics that occur at short times (or, equivalently, low cycle numbers), while investigation of the variance reveals the contributions from liquid volume transfer, imperfect amplification, and strand-specific amplification (i.e., if one strand is synthesized more efficiently than its complement). Linking the branching process to fluorescence by the Beer's Law analog allows for an a priori description of background fluorescence. It also enables uncertainty quantification (UQ) in fluorescence which, in turn, leads to analytical relationships between amplification efficiency (probability) and limit of detection. This work sets the stage for UQ-PCR, where both the input copy-number and its uncertainty are quantified from fluorescence kinetics.

Monoclonal Antibodies as SARS-CoV-2 Serology Standards: Experimental Validation and Broader Implications for Correlates of Protection.

COVID-19 has highlighted challenges in the measurement quality and comparability of serological binding and neutralization assays. Due to many different assay formats and reagents, these measurements are known to be highly variable with large uncertainties. The development of the WHO international standard (WHO IS) and other pool standards have facilitated assay comparability through normalization to a common material but does not provide assay harmonization nor uncertainty quantification. In this paper, we present the results from an interlaboratory study that led to the development of (1) a novel hierarchy of data analyses based on the thermodynamics of antibody binding and (2) a modeling framework that quantifies the probability of neutralization potential for a given binding measurement. Importantly, we introduced a precise, mathematical definition of harmonization that separates the sources of quantitative uncertainties, some of which can be corrected to enable, for the first time, assay comparability. Both the theory and experimental data confirmed that mAbs and WHO IS performed identically as a primary standard for establishing traceability and bridging across different assay platforms. The metrological anchoring of complex serological binding and neuralization assays and fast turn-around production of an mAb reference control can enable the unprecedented comparability and traceability of serological binding assay results for new variants of SARS-CoV-2 and immune responses to other viruses.

Revealing thermodynamics of DNA origami folding via affine transformations.

Structural DNA nanotechnology, as exemplified by DNA origami, has enabled the design and construction of molecularly-precise objects for a myriad of applications. However, limitations in imaging, and other characterization approaches, make a quantitative understanding of the folding process challenging. Such an understanding is necessary to determine the origins of structural defects, which constrain the practical use of these nanostructures. Here, we combine careful fluorescent reporter design with a novel affine transformation technique that, together, permit the rigorous measurement of folding thermodynamics. This method removes sources of systematic uncertainty and resolves problems with typical background-correction schemes. This in turn allows us to examine entropic corrections associated with folding and potential secondary and tertiary structure of the scaffold. Our approach also highlights the importance of heat-capacity changes during DNA melting. In addition to yielding insight into DNA origami folding, it is well-suited to probing fundamental processes in related self-assembling systems.

Towards Quantitative and Standardized Serological and Neutralization Assays for COVID-19.

Quantitative and robust serology assays are critical measurements underpinning global COVID-19 response to diagnostic, surveillance, and vaccine development. Here, we report a proof-of-concept approach for the development of quantitative, multiplexed flow cytometry-based serological and neutralization assays. The serology assays test the IgG and IgM against both the full-length spike antigens and the receptor binding domain (RBD) of the spike antigen. Benchmarking against an RBD-specific SARS-CoV IgG reference standard, the anti-SARS-CoV-2 RBD antibody titer was quantified in the range of 37.6 µg/mL to 31.0 ng/mL. The quantitative assays are highly specific with no correlative cross-reactivity with the spike proteins of MERS, SARS1, OC43 and HKU1 viruses. We further demonstrated good correlation between anti-RBD antibody titers and neutralizing antibody titers. The suite of serology and neutralization assays help to improve measurement confidence and are complementary and foundational for clinical and epidemiologic studies.

Analysis and uncertainty quantification of DNA fluorescence melt data: Applications of affine transformations.

Fluorescence-based measurements are a standard tool for characterizing the thermodynamic properties of DNA systems. Nonetheless, experimental melt data obtained from polymerase chain-reaction (PCR) machines (for example) often leads to signals that vary significantly between datasets. In many cases, this lack of reproducibility has led to difficulties in analyzing results and computing reasonable uncertainty estimates. To address this problem, we propose a data analysis procedure based on constrained, convex optimization of affine transformations, which can determine when and how melt curves collapse onto one another. A key aspect of this approach is its ability to provide a reproducible and more objective measure of whether a collection of datasets yields a consistent "universal" signal according to an appropriate model of the raw signals. Importantly, integrating this validation step into the analysis hardens the measurement protocol by allowing one to identify experimental conditions and/or modeling assumptions that may corrupt a measurement. Moreover, this robustness facilitates extraction of thermodynamic information at no additional cost in experimental time. We illustrate and test our approach on experiments of Förster resonance energy transfer (FRET) pairs used study the thermodynamics of DNA loops.

Affine analysis for quantitative PCR measurements.

Motivated by the current COVID-19 health crisis, we consider data analysis for quantitative polymerase chain-reaction (qPCR) measurements. We derive a theoretical result specifying the conditions under which all qPCR amplification curves (including their plateau phases) are identical up to an affine transformation, i.e. a multiplicative factor and horizontal shift. We use this result to develop a data analysis procedure for determining when an amplification curve exhibits characteristics of a true signal. The main idea behind this approach is to invoke a criterion based on constrained optimization that assesses when a measurement signal can be mapped to a master reference curve. We demonstrate that this approach: (i) can decrease the fluorescence detection threshold by up to a decade; and (ii) simultaneously improve confidence in interpretations of late-cycle amplification curves. Moreover, we demonstrate that the master curve is transferable reference data that can harmonize analyses between different labs and across several years. Application to reverse-transcriptase qPCR measurements of a SARS-CoV-2 RNA construct points to the usefulness of this approach for improving confidence and reducing limits of detection in diagnostic testing of emerging diseases. Graphical Abstract Left: a collection of qPCR amplification curves. Right: Example of data collapse after affine transformation.

Foundations of modeling in cryobiology-III: Inward solidification of a ternary solution towards a permeable spherical cell in the dilute limit.

In the previous two manuscripts we outlined the general theory of heat and mass transport in a cell-liquid-ice system with general boundaries and nonideal and nondilute assumptions. Here we simplify the models considerably by presenting a reduction to a spherically symmetric system-a spherical cell with an encroaching spherical ice front. We also reduce to linear approximations of the nonideal nondilute models, essentially assuming dilute and ideal conditions. We derive the resulting nondimensional combined heat and mass transport model for a ternary solution and present numerical solutions. We include an analysis of the effects of varying some nondimensional parameters on rates of ice growth with comments on the necessity of models that account for spatially varying quantities in cryobiology.

Dynamic Measurement of Nanoflows: Realization of an Optofluidic Flow Meter to the Nanoliter-per-Minute Scale.

The ultimate performance of flow-based measurements in microfluidic systems is currently limited by their accuracy at the nanoliter-per-minute scale. Improving such measurements (especially in contexts that require continuous monitoring) is challenging because of constraints associated with shrinking system geometries and limitations imposed by making precise measurements of smaller quantities in real time. A particularly interesting limit is the relative uncertainty as flow approaches zero, which diverges for most measurement methods. To address these problems, we have developed an optofluidic measurement system that can deliver and record light in a precise interrogation region of a microfluidic channel. The system utilizes photobleaching of fluorophore dyes in the bulk flow and can identify zero flow to better than 1 nL/min absolute accuracy. The technique also provides an independent method for determining nonzero flow rates based on a robust scaling relationship between the fluorescence emission and flow. Together, these two independent approaches enable precise measurement of flow to within 5% accuracy down to 10 nL/min and validation of flow control to within 5% uncertainty down to 2 nL/min. We also demonstrate that our technique can be used to extend a calibrated flow meter well below its specified range (e.g., 500 nL/min) and to make dynamic measurements of similar relative uncertainties to the calibrated meter, which would have otherwise expanded significantly in this regime.

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces.

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts-namely, transport in the "bulk phases" away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.

Combining Fragment-Ion and Neutral-Loss Matching during Mass Spectral Library Searching: A New General Purpose Algorithm Applicable to Illicit Drug Identification.

A mass spectral library search algorithm that identifies compounds that differ from library compounds by a single "inert" structural component is described. This algorithm, the Hybrid Similarity Search, generates a similarity score based on matching both fragment ions and neutral losses. It employs the parameter DeltaMass, defined as the mass difference between query and library compounds, to shift neutral loss peaks in the library spectrum to match corresponding neutral loss peaks in the query spectrum. When the spectra being compared differ by a single structural feature, these matching neutral loss peaks should contain that structural feature. This method extends the scope of the library to include spectra of "nearest-neighbor" compounds that differ from library compounds by a single chemical moiety. Additionally, determination of the structural origin of the shifted peaks can aid in the determination of the chemical structure and fragmentation mechanism of the query compound. A variety of examples are presented, including the identification of designer drugs and chemical derivatives not present in the library.

Foundations of modeling in cryobiology-I: concentration, Gibbs energy, and chemical potential relationships.

Mathematical modeling plays an enormously important role in understanding the behavior of cells, tissues, and organs undergoing cryopreservation. Uses of these models range from explanation of phenomena, exploration of potential theories of damage or success, development of equipment, and refinement of optimal cryopreservation/cryoablation strategies. Over the last half century there has been a considerable amount of work in bio-heat and mass-transport, and these models and theories have been readily and repeatedly applied to cryobiology with much success. However, there are significant gaps between experimental and theoretical results that suggest missing links in models. One source for these potential gaps is that cryobiology is at the intersection of several very challenging aspects of transport theory: it couples multi-component, moving boundary, multiphase solutions that interact through a semipermeable elastic membrane with multicomponent solutions in a second time-varying domain, during a two-hundred Kelvin temperature change with multi-molar concentration gradients and multi-atmosphere pressure changes. In order to better identify potential sources of error, and to point to future directions in modeling and experimental research, we present a three part series to build from first principles a theory of coupled heat and mass transport in cryobiological systems accounting for all of these effects. The hope of this series is that by presenting and justifying all steps, conclusions may be made about the importance of key assumptions, perhaps pointing to areas of future research or model development, but importantly, lending weight to standard simplification arguments that are often made in heat and mass transport. In this first part, we review concentration variable relationships, their impact on choices for Gibbs energy models, and their impact on chemical potentials.

Minding the margins: Evaluating the impact of COVID-19 among Latinx and Black communities with optimal qualitative serological assessment tools.

COVID-19 disproportionately affected minorities, while research barriers to engage underserved communities persist. Serological studies reveal infection and vaccination histories within these communities, however lack of consensus on downstream evaluation methods impede meta-analyses and dampen the broader public health impact. To reveal the impact of COVID-19 and vaccine uptake among diverse communities and to develop rigorous serological downstream evaluation methods, we engaged racial and ethnic minorities in Massachusetts in a cross-sectional study (April - July 2022), screened blood and saliva for SARS-CoV-2 and human endemic coronavirus (hCoV) antibodies by bead-based multiplex assay and point-of-care (POC) test and developed across-plate normalization and classification boundary methods for optimal qualitative serological assessments. Among 290 participants, 91.4 % reported receiving at least one dose of a COVID-19 vaccine, while 41.7 % reported past SARS-CoV-2 infections, which was confirmed by POC- and multiplex-based saliva and blood IgG seroprevalences. We found significant differences in antigen-specific IgA and IgG antibody outcomes and indication of cross-reactivity with hCoV OC43. Finally, 26.5 % of participants reported lingering COVID-19 symptoms, mostly middle-aged Latinas. Hence, prolonged COVID-19 symptoms were common among our underserved population and require public health attention, despite high COVID-19 vaccine uptake. Saliva served as a less-invasive sample-type for IgG-based serosurveys and hCoV cross-reactivity needed to be evaluated for reliable SARS-CoV-2 serosurvey results. Using the developed rigorous downstream qualitative serological assessment methods will help standardize serosurvey outcomes and meta-analyses for future serosurveys beyond SARS-CoV-2.

Optimal classification and generalized prevalence estimates for diagnostic settings with more than two classes.

An accurate multiclass classification strategy is crucial to interpreting antibody tests. However, traditional methods based on confidence intervals or receiver operating characteristics lack clear extensions to settings with more than two classes. We address this problem by developing a multiclass classification based on probabilistic modeling and optimal decision theory that minimizes the convex combination of false classification rates. The classification process is challenging when the relative fraction of the population in each class, or generalized prevalence, is unknown. Thus, we also develop a method for estimating the generalized prevalence of test data that is independent of classification of the test data. We validate our approach on serological data with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) naïve, previously infected, and vaccinated classes. Synthetic data are used to demonstrate that (i) prevalence estimates are unbiased and converge to true values and (ii) our procedure applies to arbitrary measurement dimensions. In contrast to the binary problem, the multiclass setting offers wide-reaching utility as the most general framework and provides new insight into prevalence estimation best practices.

On the challenge of unambiguous identification of fentanyl analogs: Exploring measurement diversity using standard reference mass spectral libraries.

Fentanyl analogs are a class of designer drugs that are particularly challenging to unambiguously identify due to the mass spectral and retention time similarities of unique compounds. In this paper, we use agglomerative hierarchical clustering to explore the measurement diversity of fentanyl analogs and better understand the challenge of unambiguous identifications using analytical techniques traditionally available to drug chemists. We consider four measurements in particular: gas chromatography retention indices, electron ionization mass spectra, electrospray ionization tandem mass spectra, and direct analysis in real time mass spectra. Our analysis demonstrates how simultaneously considering data from multiple measurement techniques increases the observable measurement diversity of fentanyl analogs, which can reduce identification ambiguity. This paper further supports the use of multiple analytical techniques to identify fentanyl analogs (among other substances), as is recommended by the Scientific Working Group for the Analysis of Seized Drugs (SWGDRUG).

Reproducibility in cytometry: Signals analysis and its connection to uncertainty quantification.

Signals analysis for cytometry remains a challenging task that has a significant impact on uncertainty. Conventional cytometers assume that individual measurements are well characterized by simple properties such as the signal area, width, and height. However, these approaches have difficulty distinguishing inherent biological variability from instrument artifacts and operating conditions. As a result, it is challenging to quantify uncertainty in the properties of individual cells and perform tasks such as doublet deconvolution. We address these problems via signals analysis techniques that use scale transformations to: (I) separate variation in biomarker expression from effects due to flow conditions and particle size; (II) quantify reproducibility associated with a given laser interrogation region; (III) estimate uncertainty in measurement values on a per-event basis; and (IV) extract the singlets that make up a multiplet. The key idea behind this approach is to model how variable operating conditions deform the signal shape and then use constrained optimization to "undo" these deformations for measured signals; residuals to this process characterize reproducibility. Using a recently developed microfluidic cytometer, we demonstrate that these techniques can account for instrument and measurand induced variability with a residual uncertainty of less than 2.5% in the signal shape and less than 1% in integrated area.

Modeling in higher dimensions to improve diagnostic testing accuracy: Theory and examples for multiplex saliva-based SARS-CoV-2 antibody assays.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic has emphasized the importance and challenges of correctly interpreting antibody test results. Identification of positive and negative samples requires a classification strategy with low error rates, which is hard to achieve when the corresponding measurement values overlap. Additional uncertainty arises when classification schemes fail to account for complicated structure in data. We address these problems through a mathematical framework that combines high dimensional data modeling and optimal decision theory. Specifically, we show that appropriately increasing the dimension of data better separates positive and negative populations and reveals nuanced structure that can be described in terms of mathematical models. We combine these models with optimal decision theory to yield a classification scheme that better separates positive and negative samples relative to traditional methods such as confidence intervals (CIs) and receiver operating characteristics. We validate the usefulness of this approach in the context of a multiplex salivary SARS-CoV-2 immunoglobulin G assay dataset. This example illustrates how our analysis: (i) improves the assay accuracy, (e.g. lowers classification errors by up to 42% compared to CI methods); (ii) reduces the number of indeterminate samples when an inconclusive class is permissible, (e.g. by 40% compared to the original analysis of the example multiplex dataset) and (iii) decreases the number of antigens needed to classify samples. Our work showcases the power of mathematical modeling in diagnostic classification and highlights a method that can be adopted broadly in public health and clinical settings.