Decoding Complex Chemical Mixtures with a Physical Model of a Sensor Array
[['Mammalian and insect olfactory systems are capable of recognizing tens of thousands of odors – mostly organic compounds with diverse chemical structures and properties [1]–[4] .', 'The olfactory tasks commonly faced by such systems include detecting odors , estimating their strength , identifying their source , and recognizing one specific odor in the background of another [5] .', 'The sense of smell exhibits amazing sensitivity and discriminatory power , distinguishing between closely related compounds and detecting vanishingly small odorant concentrations [6] .', 'Olfactory signaling is mediated by a superfamily of several hundred G-protein-coupled receptors ( GPCRs ) – a significant fraction of the total number of genes in many higher eukaryotes [7]–[11] .', 'In mammals , GPCRs are located on the surfaces of the cilia projected from olfactory sensory neurons; typically receptors of only one type are expressed in a given neuron [12] .', 'Odor recognition is combinatorial , with one odorant activating multiple receptors and one receptor responding to multiple odorants [12]–[16] .', 'The resulting complex patterns of receptor activation enable robust identification of many more odors than would have been possible with “lock and key” receptors reacting to only one analyte .', 'Moreover , several studies provide evidence for widespread inhibitory responses in which receptors are antagonized by odorants [14] , [15] , [17]–[20] .', 'The idea of combinatorial recognition has been adapted to artificial arrays in which multiple sensors with partially overlapping selectivities respond to a given analyte [21]–[24] .', 'While the output of these cross-specific arrays in response to single compounds can generally be interpreted through pattern recognition algorithms [24]–[27] , computational analysis becomes more difficult when the array is presented with a mixture of compounds .', 'Indeed , the non-linear nature of sensor responses to multiple ligands makes it hard to train discriminatory algorithms on a “typical” subset of patterns .', "The non-linear dependence of sensor output on ligand concentrations is generic in reporter systems and may be compounded by potential binding interference of the two ligands , saturation of the sensor output [28] and , of particular concern , potential antagonistic action of one ligand on another's activity [19] .", 'As a result , responses to complex mixtures have primarily been used to “fingerprint” specific mixtures rather than identify their constituents quantitatively [29]–[32] .', 'There are relatively few studies which focus on the quantitative analysis of mixtures: for example , Heilig et al . used a single sensor and Fourier transformation techniques to analyze a binary mixture of and [33] , White et al . trained artificial neural networks to identify relative concentrations in binary mixtures [34] , and Woodka et al . used a non-negative least squares method to quantify the composition of analyte mixtures with up to five components [35] .', 'Here we describe a physical model of receptor-ligand recognition that explicitly relates observed response patterns to component concentrations and receptor properties , making it easier to quantify mixture constituents .', 'We use Bayesian inference to predict absolute concentrations of each ligand in arbitrary mixtures of uridine diphosphate ( UDP ) sugar nucleotides applied to a combinatorial array of four GPCRs .', 'Furthermore , we develop a universal metric of receptor array performance , and use it to study the fundamental limits imposed on the accuracy of ligand recognition by the physics and biology of receptor-ligand interactions .', 'Finally , we provide design guidelines for constructing cross-specific arrays optimized for mixture recognition , and demonstrate that inhibitory responses are essential for simultaneous detection of all components in a complex mixture .'], ['Our sensor array is comprised of four engineered receptors ( L-3 , H-20 , K-3 and 2211 ) with distinct but overlapping specificities for four types of nucleotide sugars: UDP-glucose ( UDP-Glc ) , UDP-galactose ( UDP-Gal ) , UDP-glucosamine ( UDP-GlcNAc ) and UDP .', 'The receptors were evolved in vitro from the human UDP-glucose receptor using directed mutagenesis of the residues involved in ligand binding ( see Materials and Methods ) [36] .', 'Nucleotide sugars and their derivatives are key constituents in polysaccharide synthesis and other cellular processes .', 'Their structural similarity makes them a challenging target for array-based discriminatory analysis .', 'To assess receptor-ligand interactions quantitatively in our sensor array , we functionally expressed the receptors in S . cerevisiae .', 'To do so , we replaced the yeast pheromone receptor with one of the sensor GPCRs in strains in which the pheromone response pathway was modified to respond to the heterologous receptor by inducing transcription of the E . coli lacZ gene [37] .', 'In this fashion , the extent of GPCR activation following ligand addition could be directly monitored as the level of produced in the cell , which we measured using a fluorescence-based assay ( Materials and Methods ) .', 'Applying a mixture of nucleotide sugars to the receptor array yields a complex pattern of responses of the four receptor-bearing strains .', 'The response of each receptor depends on the concentration of all components in the mixture , on the receptor-ligand binding affinities , and on the efficacy with which each ligand activates the receptor .', 'Nonetheless , the contents of arbitrary nucleotide sugar mixtures can be deciphered using array readout as input to a physical model of receptor-ligand interactions .'], ['We have developed a Bayesian algorithm that allows determination of all the constituents in an unknown mixture from the output of a cross-specific sensor array .', 'Our algorithm employs a physical picture of sensor-analyte interactions to model the non-linear relationship between ligand concentrations and the reporter response .', "After appropriate calibration of each sensor's response to each analyte of interest , the algorithm interprets the integrated output of the entire array and , with a sufficient number of variably tuned sensors , reliably returns the amount of each chemical in a complex mixture .", 'We also provide quantitative guidelines for designing optimal sets of sensors .', 'Three general principles emerged from our computational and theoretical studies of array design .', 'First , the optimal parameters of the sensors exhibit weak dependence on the relative amounts of compounds in a mixture .', 'Thus a given set of optimal sensors will remain near-optimal through a sizable range of ligand concentrations .', 'Nonetheless , analyzing a mixture where both compounds are present in roughly similar amounts is better accomplished with a set of sensors different from those fine-tuned to measure a small amount of one compound in the presence of a large excess of the other .', 'Second , the maximum number of ligands in a mixture whose levels can all be determined simultaneously is simply twice the number of sensors in the array .', 'This linear relationship is different from the exponential relationship between ligands and receptors in olfactory systems [4] , [12] .', 'The problem addressed by the olfactory system , to recognize a very large number of individual odors with a limited repertoire of receptors , is not the same as that solved by our algorithm , to determine all the constituents in a complex mixture .', 'In fact , even the most skilled human nose can simultaneously detect and distinguish no more than a handful of odorants .', 'Third , the optimum design of receptors for the array demands that one of the ligands function as a strong agonist of a receptor and a second ligand as a strong antagonist of that receptor .', 'Antagonists sharpen the discriminatory powers of the array by heightening the differences in the receptor response to individual compounds .', 'As a result , a mixture of chemicals produces an array readout which is not a superposition of responses to individual ligands , and whose intensity pattern may be fine-tuned for maximum recognition through receptor-ligand binding energies .', 'Accordingly , odors that function as antagonists to a subset of olfactory receptors could potentially increase the discriminatory power of the olfactory system , and in particular enable it to resolve mixtures that contain those odors .', 'Recent analysis of olfactory receptors suggests that some odorants do possess antagonist activity [14] , [17]–[20] .', 'Our theoretical framework provides a rationale for the existence of such antagonists and underscores their role in both olfactory systems and artificial receptor arrays .'], ['The L-3 mutant was isolated using a procedure similar to that previously employed with the H-20 and K-3 mutants [36]: oligonucleotides with randomized sequences corresponding to the codons to be mutagenized were utilized to generate overlapping PCR products .', 'The L-3 motif corresponds to amino acid residues LLxSA on TM7 .', 'Mutant libraries were generated by gap repair using overlapping PCR products and transformed to media selective for recombined plasmids .', 'To select for functional mutants , libraries were replica-plated to selective SC-His media [40] containing one of six ligands: UDP-Gal , UDP-Glc , UDP-galNAc , UDP-GlcNAc , UDP or dTDP-glucose ( 50 of 1 mM solution spread on 30 mL of SC “Leu-His agar medium in 8 . 5 cm Petri plates ) . Yeast growth media was supplemented by 1 mM 3AT , a competitive inhibitor of the HIS3 reporter gene product , which sets the threshold for reporter gene activation . Functional receptor mutants that showed qualitatively disparate responses to the panel of ligands were selected for further analysis . Among these , the H-20 and K-3 mutants , described earlier [36] , and the L-3 mutant , described here , were selected to be utilized alongside the 2211 “parent” in a four-receptor array for analysis of mixtures of UDP-Glc , UDP-Gal , UDP-GlcNAc and UDP .', 'Our assays were based on microtiter assays described previously [41] .', 'Yeast strains expressing each of the four mutant receptors were diluted to of in flasks .', 'Cultures were then grown overnight in 100 mL selective media to an of .', 'Serial dilutions of each ligand or mixture of ligands were prepared in yeast culture medium in -well culture blocks .', 'Ligands or mixtures of ligands were transferred in aliquots in quadruplicate to deep-well polypropylene plates using a BioMek robotic liquid handler .', 'of suspended yeast cells in medium ( undiluted from the overnight cultures ) were then aliquotted into each well and mixed .', 'The cultures were sealed with foil tape and incubated at on a plate shaker at 400–500 rpm for hours ( H-20 , K-3 and L-3 receptors ) or overnight ( 2211 receptor ) .', 'After incubation , substrate [41] ( FDG solution; 0 . 5 mM fluorescein di-beta-D-galactopyranoside , 2 . 3% Triton X-100 , and 0 . 127 M Pipes , pH 7 . 2 ) was mixed with an equal volume of Pierce Y-PER solution ( Thermo Scientific ) and distributed in aliquots to black plates .', 'aliquots of the yeast/ligand cultures were then transferred into the black plates and mixed gently but thoroughly by pipetting , taking care to avoid generating bubbles .', 'A single layer of paper towel was placed on top of each plate and the plates were then individually wrapped in aluminum foil and incubated without shaking at for approximately one hour before reading on an automated fluorescent plate reader ( Perkin Elmer EnVision ) .', 'Microtiter plate-based assays are often subject to edge- or plate-bias due to uneven heating or discrepancies in timing across a single plate or among plates .', 'While no obvious plate effects were seen , it is very difficult to control for all possible variations in a single experiment .', 'Due to the number of samples and the need to make efficient use of materials , each of the mixture experiments was split across two plates per receptor .', 'In mixtures of equal proportions , samples containing UDP , UDP-Gal and UDP-GlcNAc but lacking UDP-Glc were on Plate 1 , while all mixtures containing UDP-Glc were on Plate 2 .', 'In the UDP-Gal/UDP-Glc binary mixtures of unequal proportions , samples containing 90% , 80% or 60% UDP-Glc were on Plate 1 , while samples containing 40% , 20% or 10% UDP-Glc were on Plate 2 .', 'For each single ligand or combination of ligands , a series of measurements was performed at several values of the total concentration ( M ) : for H-20 , K-3 , L-3 and for 2211 .', 'The total chemical potential is then given by , where is the number of measurements in the series , are known chemical potential differences between two consecutive measurements , and is the chemical potential at the mM reference point .', 'Note that in order to reconstruct the total chemical potential for all points in the series , only needs to be predicted .', 'Each series of measurements was replicated four times; fluorescence counts were normalized to separately for each plate ( Dataset S1 ) .', 'For a single receptor interacting with a single ligand , we model the normalized reporter fluorescent intensity as: ( 1 ) where is the receptor efficacy , is the background intensity ( a small amount of background fluorescence observed in the absence of ligand binding ) , is the free energy of receptor-ligand binding , ( is the Boltzmann constant , and is the temperature ) , and is the chemical potential .', 'We compute the log-likelihood of the data by assuming that fluorescence measurements are Gaussian-distributed around values from Eq .', '( 1 ) : ( 2 ) where are measured intensities and is the noise parameter .', "The log-likelihood is used to estimate the posterior probability of all model parameters according to the Bayes' formula [38]: ( 3 ) where on the right-hand side the likelihood from Eq .", '( 2 ) is multiplied by the product of priors for each model parameter and divided by evidence .', 'combines data from all experimental replicates .', "We use uniform priors ( invariant with respect to translations , ) : ( 4 ) for , and , and Jeffrey's priors ( invariant with respect to rescaling , ) for : ( 5 ) We have used , , , in our calculations .", 'The reporter response to a mixture of ligands is given by ( 6 ) where is the probability that receptor is bound by ligand and is the partition function .', 'is the binding free energy between receptor and ligand ( , ) , is the efficacy , and is the background intensity .', 'The background intensity for receptor is the average from all calibration experiments involving that receptor .', 'is the chemical potential of ligand , which can be expressed through the total chemical potential and the relative concentrations ( , ) : ( 7 ) where .', 'Note that an arbitrary choice of the ligand in the denominator leads to several equivalent representations of the relative concentrations .', 'The log-likelihood of the observed pattern of fluorescence intensities from multiple receptors interrogated by a mixture of ligands is given by ( 8 ) Here is defined in Eq .', '( 6 ) ( in the interests of brevity , we suppress its dependence on for each receptor-ligand combination ) .', 'denotes fluorescence measured for receptor at the total chemical potential , is the total number of measurements , and is the noise parameter .', 'Similarly to Eq .', '( 3 ) , the log-likelihood is used to estimate the posterior probability .', "We employ a uniform prior for with and a Jeffrey's prior for 's with .", 'We estimate all posterior probabilities by nested sampling [38] – a Bayesian Monte-Carlo ( MC ) technique that yields an ensemble of models from which the average value of each parameter and its standard deviation are computed .', 'Unlike other methods such as MC sampling of the product of likelihood and priors , nested sampling allows us to keep track of the evidence , yielding absolute values of the posterior probability .', 'The Hessian matrix in the low-noise limit can be written as ( Text S1 ) : ( 9 ) where , is the log-likelihood , is the total number of measurements for receptor , and is the noise parameter .', 'Uncertainties for each predicted parameter are given by the diagonal elements of the inverse Hessian matrix: ( 10 ) Software used in this study , called RANSA ( Receptor Array Nested Sampling Algorithm ) , is available at http://olfaction . rutgers . edu .']]
['Introduction', 'Results', 'Discussion', 'Materials and Methods']
['Combinatorial sensor arrays , such as the olfactory system , can detect a large number of analytes using a relatively small number of receptors .', 'However , the complex pattern of receptor responses to even a single analyte , coupled with the non-linearity of responses to mixtures of analytes , makes quantitative prediction of compound concentrations in a mixture a challenging task .', 'Here we develop a physical model that explicitly takes receptor-ligand interactions into account , and apply it to infer concentrations of highly related sugar nucleotides from the output of four engineered G-protein-coupled receptors .', 'We also derive design principles that enable accurate mixture discrimination with cross-specific sensor arrays .', 'The optimal sensor parameters exhibit relatively weak dependence on component concentrations , making a single designed array useful for analyzing a sizable range of mixtures .', 'The maximum number of mixture components that can be successfully discriminated is twice the number of sensors in the array .', 'Finally , antagonistic receptor responses , well-known to play an important role in natural olfactory systems , prove to be essential for the accurate prediction of component concentrations .']
['Mammalian and insect olfactory systems are combinatorial in nature - instead of activating a single specialized receptor , each analyte invokes a complex pattern of responses across the receptor array .', 'The advantage of such systems lies in their ability to detect a large number of analytes with a relatively small number of receptors .', 'However , the complexity of array responses to mixtures of analytes makes quantitative prediction of component concentrations a challenging task .', 'Here we show that combinatorial output from an array of four engineered G-protein-coupled receptors can be used to predict the concentration of each component in mixtures of highly related sugar nucleotides .', 'We employ a physical model of ligand-receptor interactions and carry out Bayesian analysis of the array output .', 'Furthermore , our in silico designs of receptor arrays reveal that antagonistic responses , in which the receptor is bound by the ligand but there is no downstream reporter activity , are necessary for precise recognition of mixture components .', 'This conclusion provides a rationale for the widespread inhibitory responses observed in olfactory systems .', 'Our methodology can be employed with both biological systems and artificial receptor arrays ( “electronic noses” ) designed for various industrial needs .']
['physics', 'analytical chemistry', 'chemistry', 'biology', 'computational biology', 'biophysics']