IOP PUBLISHING PHYSICAL BIOLOGY doi:10.1088/1478-3975/7/4/046002 Phys. Biol. 7 (2010) 046002 (12pp) Quantifying enzymatic lysis: estimating the combined effects of chemistry, physiology and physics Gabriel J Mitchell1 , Daniel C Nelson2 and Joshua S Weitz1,3 1 School of Biology, Georgia Institute of Technology, Atlanta, GA, USA Institute for Bioscience and Biotechnology Research and Department of Veterinary Medicine, University of Maryland, Rockville, MD, USA 3 School of Physics, Georgia Institute of Technology, Atlanta, GA, USA 2 E-mail: [email protected] Received 29 June 2010 Accepted for publication 8 September 2010 Published 4 October 2010 Online at stacks.iop.org/PhysBio/7/046002 Abstract The number of microbial pathogens resistant to antibiotics continues to increase even as the rate of discovery and approval of new antibiotic therapeutics steadily decreases. Many researchers have begun to investigate the therapeutic potential of naturally occurring lytic enzymes as an alternative to traditional antibiotics. However, direct characterization of lytic enzymes using techniques based on synthetic substrates is often difficult because lytic enzymes bind to the complex superstructure of intact cell walls. Here we present a new standard for the analysis of lytic enzymes based on turbidity assays which allow us to probe the dynamics of lysis without preparing a synthetic substrate. The challenge in the analysis of these assays is to infer the microscopic details of lysis from macroscopic turbidity data. We propose a model of enzymatic lysis that integrates the chemistry responsible for bond cleavage with the physical mechanisms leading to cell wall failure. We then present a solution to an inverse problem in which we estimate reaction rate constants and the heterogeneous susceptibility to lysis among target cells. We validate our model given simulated and experimental turbidity assays. The ability to estimate reaction rate constants for lytic enzymes will facilitate their biochemical characterization and development as antimicrobial therapeutics. of their own cell wall during growth, repair and division [4]. Unregulated production of these enzymes, called autolysins, can result in bacterial autolysis [5]. Other lytic enzymes called exolysins are excreted by a variety of organisms to protect against bacterial infection or to accelerate degradation of potentially toxic cell wall fragments, e.g. human lysozyme [6, 7]. As another example, bacteriophage (or phage) lytic enzymes, also known as endolysins, are utilized by phages to burst out of their bacterial hosts late in the infection cycle [8–10]. Once a possible bacteriolytic agent has been identified, the quantitative details of enzymatic lysis are essential to characterizing its potential as a therapeutic agent. However, in the case of many lytic enzymes, such quantitative characterization of enzymatic lysis has proved problematic. Regardless of origin, many lytic enzymes often bind to the 1. Introduction An increasing number of pathogenic bacteria are currently resistant to what were previously effective antibiotics, e.g. methicillin-resistant Staphylococcus aureus. The rise of single- and multi-drug-resistant bacteria has spurred efforts to develop alternatives to antibiotics. Leading candidates for alternatives to antibiotics are metabolites, peptides and enzymes produced by organisms, including viruses and bacteria, that eliminate bacterial cells in natural conditions [1, 2]. Lytic enzymes are a prime example within this class of alternative antibiotics. Lytic enzymes are bacteriolytic agents that can cause bacterial lysis by cleaving bonds in the cell wall’s peptidoglycan network responsible for cell rigidity and containment of the cytoplasmic membrane [3]. For example, many bacteria make lytic enzymes to selectively cleave parts 1478-3975/10/046002+12$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK Phys. Biol. 7 (2010) 046002 G J Mitchell et al three-dimensional superstructure of the peptidoglycan, with some enzymes requiring secondary binding sites such as cell wall-associated carbohydrates or teichoic acids [10]. As such, these enzymes typically do not hydrolyze small molecular weight substrates that mimic only the bonds to be hydrolyzed, which are often used for determination of kinetic constants for other classes of hydrolytic enzymes. Instead, turbidity assays of solutions of bacteria are used to explore the action of lytic enzymes on intact bacterial cells [11]. In turbidity assays, the turbidity of the solution (as measured through forward scattering) decreases in time as the cells lyse due to the action of the enzymes. The central challenge in the analysis of turbidity assays is to infer the microscopic details of lysis from the macroscopic turbidity data. Thus far, this has not been possible. Instead, current methods define the specific ‘activity’ of a lytic enzyme in a turbidity assay as inversely proportional to the time it takes for a known titer of enzyme to reduce the turbidity of a solution of cells by half [12, 13]. Such a definition, though quantitative, has no obvious relationship with the underlying kinetics of the enzyme reaction or the mechanics of lysis. Current definitions of ‘activity’ cannot be used to determine the binding rate, de-binding rate or catalytic rate of an enzyme. Further, qualitatively distinct turbidity curves can have the same ‘activity’ (see figure 1(a)). In this paper, we propose a new standard for the analysis of turbidity assays based on the solution of an inverse problem [15, 16]. The key insight of our approach is that the entire turbidity time series can and should be used to quantitatively assess the properties of lytic enzymes. In doing so we integrate simple models of the physics of light scattering, the chemistry of lytic enzyme kinetics and the cellular physiology involved in lysis. The inputs to the estimation procedure are turbidity time series measurements of mixtures of bacterial cells and lytic enzymes. This input specifies the data for an inverse problem, the solution of which provides estimates for parameters describing the chemical kinetics of lytic enzymes acting on cell surfaces as well as a measure of susceptibility to lysis of the physiologically heterogeneous bacterial population. As support for our approach we provide a demonstration of its application on synthetic data and on experimental turbidity time series measurements obtained from the action of eggwhite lysozyme on Micrococcus lysodeikticus cells. M. lysodeikticus is a Gram-positive bacterium whose cell wall is comprised of pure peptidoglycan to which lysozyme binds; as such, it is a model organism for studying lysozyme kinetics [6, 17, 18]. For the synthetic data, we achieve good convergence to specified rate constants and widely varying susceptibility distributions. For the lysozyme data, we obtain an estimate for kinetic rate constants of egg-white lysozyme consistent with previous values obtained through independent experiments [19–21] based on spectroscopic methods. The quantitative details of enzymatic lysis are relevant to a number of biotechnological applications including protein extraction [22], the engineering of transgenic livestock resistant to microbial infection [23] and in the design and assessment of therapeutic or antimicrobial treatments based on these enzymes [3, 24]. Of particular interest to us are treatments based on phage-derived endolysins which are capable of lysing susceptible Gram-positive bacteria when added exogenously [10, 25]. Several animal models of infection carried out over the past decade support the therapeutic efficacy of these enzymes against group A streptococci [9], group B streptococci [26], pneumococci [13], enterococci [27], S. aureus [28] and Bacillus spp. [14]. In all these cases the complexities of the interaction between the cell wall, binding domains and the lytic enzymes make it difficult to isolate a small molecular weight, homogeneous substrate to conduct direct measurements of enzyme activity. Our approach to the quantitative analysis of the relevant chemistry using simple turbidity assays paves the way for studies of the design and engineering of these endolysins and other lytic enzymes in a variety of applications. 2. Materials and methods 2.1. Physics, chemistry and physiology of lysis The turbidity of cell suspensions declines when lytic enzymes are added to susceptible bacterial cells (see figure 1(a)). The precise value of turbidity at a given time reflects a combination of physical, chemical and physiological processes. For a large range of cell densities (∼106 –108 cells ml−1 ) the turbidity of a solution of bacteria is linear in the concentration of both intact cells and lysed cells. The pre-factors of such linear relationships can be established via calibration experiments. A decline in turbidity is the result of a shift in the balance from intact cells (which are relatively opaque) to lysed cells (whose contents contribute less to turbidity, as in figure 1(b)). When lytic enzymes are mixed with bacterial cells in the solution they will adsorb to cell surfaces and cleave bonds, eventually causing lysis. With knowledge of the precise chemical rate constants associated with the reaction of the enzyme and substrate as well as the initial amount of substrate and enzyme molecules in a solution, it is possible to compute a reaction time series. This computed reaction time series describes the theoretically expected distribution of the number of intact bonds, complexes and cleaved bonds on a percell basis (see figure 1(c)). Importantly, not all cells are equally susceptible to lysis and so any quantitative analysis of turbidity assays that improves upon conventional measures requires a means for taking into account population level heterogeneity in lysis susceptibility. Lysis occurs when a pore of a critical size forms in the cell wall and the cytoplasm pushes the cell membrane out into the environment due to the pressure difference across the membrane [29] (see figure 1(b) for an example in Bacillus spp.). When such a pore will form for any particular cell is a potentially complicated (and as yet unknown) function of the cell’s internal state which includes the number of bonds cleaved, cell size, internal pressure, cell wall thickness, growth phase and other variables [30, 31]. As such, our model proposes the use of a ‘lysis susceptibility function’ that describes a population-averaged susceptibility to lysis as a function of the number of bonds cleaved on the cell wall. The lysis susceptibility function can be thought of as representing the distribution of susceptibilities in a 2 Phys. Biol. 7 (2010) 046002 G J Mitchell et al (c) t3 intact fraction t2 lysed fraction all intact 85% bonds cleaved 80% cells lysed 50% bonds cleaved 40% cells lysed 15% bonds cleaved 20% cells lysed few intact all lysed t3 t2 t1 0 t1 pores form time max turbidity t1/2 fraction of bonds cleaved (b) enzyme 1 enzyme 2 enzyme 3 fraction of cells lysed (a) time heterogeneity in susceptibility to lysis 0 turbidity max Figure 1. Turbidity assays as a result of the physics, chemistry and physiology of lysis. (a) Qualitatively distinct turbidity curves can have the same half lysis time, t1/2 , i.e. the same ‘activity’. (b) Lytic enzymes catalyze reactions which form pores in bacterial cell walls. Light scattering is determined by the shape and abundance of these cells, which eventually lyse due to the action of the enzymes (reprinted with permission from Schuch et al [14]). (c) The fraction of lysed cells at any given time depends on the number of bonds cleaved as determined by the reaction time series (blue curve) as well as the population susceptibility to lysis (black curve which contains the shaded areas ‘lysed fraction’ and ‘intact fraction’). Susceptibility to lysis of a population is defined as the fraction of cells that will lyse given that the x fraction of bonds have been cleaved, where x ranges from 0 to 1. For example, in this illustration 40% of cells will lyse if 50% of bonds have been cleaved, which occurs at time t2 . As the number of bonds cleaved increases with time, a larger fraction of cells will lyse. For stochastic reactions the contributions to lysis from all paths must be summed to compute the fraction of lysed cells. The turbidity of a solution of bacteria undergoing lysis is determined by the fraction of intact and lysed cells, which is only indirectly related to the reaction time series. chemical rate constants and the lysis susceptibility function given a set of experimentally measured turbidity time series [16]. The solution of this inverse problem involves technical challenges distinct from those that arise in the computation of the forward problem [32]. The mathematical details of the formulation of both the forward and inverse problems are described in the section that follows. physiologically heterogeneous population. It is a nonparametric representation of the population’s susceptibility to lysis that requires no special knowledge of microscopic variables, other than the number of bonds cleaved. Together, the decline in turbidity can be viewed as a combination of physics, chemistry and physiology as follows. When enzymes are added to bacterial cells they begin to cleave bonds on cell surfaces. The fraction of bonds cleaved per cell with time can be predicted using a reaction model such as Michaelis–Menten kinetics (see figure 1(c), left panel). At a given point in time, intact cells will have a fraction of their bonds broken (see figure 1(c), middle panel). Because of heterogeneity in susceptibility to lysis, only a subset of cells will lyse (the light shading in the left panel of figure 1(c)) whereas other cells will remain intact even as their cell walls are being digested by the action of enzymes (the dark shading in the left panel of figure 1(c)). Once the fraction of lysed and intact cells are known, data from light-scattering calibration experiments can be used to predict the optical density as a function of time (figure 1(c), right panel). Hence, given knowledge of an enzyme reaction mechanism, knowledge of a lysis susceptibility function and independent light-scattering experiments with different initial concentrations of cells and enzymes, it is possible to solve the forward problem to predict the decline of turbidity in all wells. Our main interest, however, is in solving the associated inverse problem (see figure 2), in which we attempt to infer 2.2. Turbidity, lysis and enzyme reaction models Our analysis begins with a description of a reaction mechanism with N species parameterized by k ∈ RK . A typical form for this reaction is k+ kf S + E C→ H + E, (1) k− where, in this case, N = 4 and k = (k+ , k− , kf ) ∈ R3 and S, E, C and H denote substrate, enzyme, complex and product respectively. Associated with this reaction mechanism is a dynamical system, also parameterized by k, which may be deterministic or stochastic (see appendix A). The reaction coordinate z denotes the state, for example, z = (S, E, C, H ) for the reaction model in equation (1). Given an initial distribution of chemical species wm (z, 0) ∈ W (W is a space of C2 functions) and a time evolution operator φkt : W → W we have wm (z, t|k) = φkt (wm (z, 0)) 3 (2) Phys. Biol. 7 (2010) 046002 G J Mitchell et al Figure 2. Forward and inverse problems. In the forward or direct problem, we can compute the model turbidity time series given the susceptibility distribution P (x) and a reaction model time series, qm (x, t|k), with parameters in the vector k. The associated inverse problem is to determine {k, P (x)} which best fits a given turbidity time series, ρ̃m (t). a probability density function for the reaction state at time t for the mth well (a well is a self-contained reaction chamber). This notation holds for both deterministic and stochastic models. Due to conservation of mass wm (z, t|k) will satisfy dzwm (z, t|k) = 1 (3) where nm is the concentration of cells and the constants ρintact and ρlysed can be determined experimentally. From this we form a model for the actual signal from which we will sample our data, ρ̃m (t|k, P ) = ρm (t|k, P ) − ρintact nm + ξm (t), where each ξm is a random signal satisfying for all t. Implicit in this formulation is the assumption that the reaction is well mixed, so that mean-field descriptions of the state are applicable. Our model for susceptibility to lysis is based on the idea that the vulnerability of a cell is determined by the fraction of cell wall bonds which have been cleaved. We obtain a distribution for the fraction of bonds cleaved qm (x, ti |k) ∈ Q (Q is a space of C2 functions over domain = [0, 1]) in the mth well through qm (x, t|k) = X(wm (z, t|k)), (8) ξm (t) = 0 ξm (t)ξm (t ) = σ 2 δ(t − t )δmm . and (9) By defining cm = (ρlysed − ρintact )nm (10) we obtain an expression for the (adjusted) turbidity function ρ̃m as dxqm (x, t|k)P (x) + ξm (t) (11) ρ̃m (t|k, P ) = cm (4) where X is a kind of normalized projection operator, which maps the marginal distribution in the projected subspace to the rescaled domain . For example, in the case of deterministic reactions a delta function in N dimensions centered at some point z is mapped to a one-dimensional delta function centered around the fraction of bonds cleaved x, the rescaled coordinate. Once again qm (x, t|k) obeys dxqm (x, t|k) = 1 (5) which forms a set of M integral equations. Our goal is to develop a framework, based on these equations, to estimate k and P from the turbidity data sampled in all M wells, illustrated graphically in figure 2. 2.3. Discretization of integral equations Our turbidity data in each well are sampled at discrete times ti , so that we can think of the data from the mth well as a vector ρ̃m = (ρ̃m (t1 ), . . . , ρ̃m (tT )) with T sampling times. Given a set of basis functions {ψ1 , ψ2 , . . . , ψJ } and vector of coefficients P = (P1 , P2 , . . . , PJ ) ∈ RJ for approximating the susceptibility function (see appendices B and C) we have a model for the expected value of our data in the mth well (from equation (11)) 1 J ρ̃m (ti |k, P ) cm dxqm (x, ti |k) Pj ψj (x) (12) for all t. Given equation (4) and noting that susceptibility to lysis is heterogeneous, we can then write an equation for the fraction of cells lysed in the population dxqm (x, t|k)P (x), (6) fm (t|k, P ) = where P (x) ∈ Q is a cumulative probability function describing the fraction of lysed cells as a function of the fraction of each bond type cleaved on the cell wall. The fraction of lysed cells fm is related to the turbidity ρm through 0 j =1 which can be written as intact cell contribution lysed cell contribution ρm (t|k, P ) = ρintact nm (1 − fm (t|k, P )) + ρlysed nm fm (t|k, P ), ρ̃m (ti |k, P ) J (Am (k))ij Pj , j =1 (7) 4 (13) Phys. Biol. 7 (2010) 046002 G J Mitchell et al where 1 (Am (k))ij = cm dx qm (x, ti |k)ψj (x). we discuss later, this can also be interpreted as an algorithm for data collapse, in which the equation is ‘solved’ when curves from multiple wells collapse onto one another after the correct choice of transformation. Finally, it should be noted that parametric models for P (x) can easily be incorporated in equation (4). Consider a distribution of cell states in a population c(θ , x) and a function x) that indicates whether or not a cell is undergoing (θ, irreversible lysis as a function of the state variables θ and x. In particular, additional state variables may denote the relevance of other biological processes to lysis, e.g. the position and state of binding sites. Then, the expected susceptibility distribution can be written as dθ (θ , x)c(θ , x) . (20) P (x|c, ) = dθ c(θ , x) (14) 0 These define a set of matrix equations ρ̃m = Am (k)P (15) with our data ρ̃m ∈ RT , the susceptibility coefficients P ∈ RJ and the matrix Am (k) ∈ RT ×J . This system of M equations can be written concisely as a single matrix equation ρ̃ = A(k)P (16) with ρ̃ = (ρ̃1 , ρ̃2 , . . . , ρ̃M ) and A(k) = (A1 (k), A2 (k), . . . , AM (k)). For M > 1 this is an overdetermined system of equations linear in P . The dependence on k is nonlinear. For M = 1 the system is underdetermined. With measurements from only one well, there will always be a choice of P which solves the equation exactly for any values of K. As such, it is essential to make measurements in wells with at least two different initial relative concentrations of cells or enzymes. We wish to obtain a least-squares solution {k ∗ , P ∗ } that minimizes Gα (k, P ) = ρ̃ − A(k)P 22 + α D (2) P 22 The union of parameters of the distribution c and the function now fully parametrize the susceptibility distribution. Equation (17) can be minimized via a variable projection routine by enforcing additional equality constraints on P (x) so that the solution at each iteration satisfies equation (20). 3. Results and discussion (17) 3.1. Estimation of parameters for synthetic data subject to the monotonicity constraint D (1) P > 0. Here α is a regularization parameter [16] enforcing smoothness and D (1) and D (2) are the first- and second-order differentiation matrices (see appendix B). This specifies a nonlinear minimization problem with K + J degrees of freedom. The separability of this equation into nonlinear and linear parts reflects the natural decomposition of the problem into the chemical and physiological components. In fact, we can take advantage of this separability by forming an objective function Gα (k), called the variable projection functional [33], which reduces the number of degrees of freedom to K. Gα (k) can be thought of as the degree to which the best choice of P misfits the data for a given value of k. In biological terms, we find the most likely reaction rates by optimizing a nonlinear function conditioned on the most likely measure of susceptibility to lysis, which is obtained by solving a linear least-squares problem. Ignoring for the moment of the issue of regularization we have Gα (k, P ) → Gα (k) = (I − A(k)A+ (k))ρ̃ 22 = Φ(k)ρ̃ 22 , We summarize the results of our analysis on deterministic and stochastic Michaelis–Menten kinetics with synthetically generated data in figures 3, 4 and C2. In figure 3 we present the best fit time series along with the exact and estimated susceptibility distributions for deterministic kinetics. For deterministic reactions we solve the inverse problem in the absence of Tikhonov regularization, setting α = 0. Note that even though the fit to the time series appears exact, the estimated susceptibility distributions need not be identical. The tight fit to the turbidity time series is possible because differences in the susceptibility distribution can be compensated for by differences in kinetic constants and vice versa. However, the qualitative shapes of susceptibility distributions and quantitative values of kinetic constants are in strong agreement with the actual distributions and values used to generate these synthetic turbidity time series. As an illustration of how the lysis susceptibility distribution affects the turbidity time series we show fits on three time series with the same initial reactant concentrations and rate constants but with three susceptibility different distributions (figure 4). A tabulation of the actual and inferred rate constants (maximum likelihood values) in each case can be found in table 1. We find strong agreement between the specified kinetic constants and those estimated for the various distributions. Note that the turbidity time series in each case coincide only where the susceptibility distributions themselves intersect (corresponding to x ≈ 0.5 and t ≈ 1500 s in figure 4). If the susceptibility distributions did not intersect anywhere, then the turbidity time series would never intersect, if the kinetic rates were the same. Comparative measures of activity that rely only on turbidity at a single value will not be applicable if it is possible that there are differences in susceptibility to lysis between cultures. In figure C2 we show (18) (19) where A+ (k) is the Moore–Penrose generalized inverse and Φ(k) is the orthogonal projector onto the nullspace of A(k). Intuitively, the operator Φ(k) is a linear map from the data vector to the residual vector. We then minimize Gα (k) w.r.t. k to obtain k ∗ using fmincon from the MATLAB optimization toolbox. From k ∗ we can obtain the optimal susceptibility coefficients by P ∗ = A+ (k ∗ )ρ̃. In minimizing Gα (k) we do not explicitly compute A+ (k) or Φ(k) but rather use an iterative method (lsqr) to efficiently update the best solution for P in σ . In the order to approximate Gα (k) to within a tolerance MT ∗ + ∗ final step we again use lsqr to compute P = A (k )ρ̃. As 5 Phys. Biol. 7 (2010) 046002 0.8 turbidity ρ (b) 1 9.00 uM 7.50 uM 6.00 uM 4.50 uM 3.75 uM 1 0.6 probability of lysis (P) (a) G J Mitchell et al 0.4 0.2 exact estimated 0.8 0.6 2 0.4 1.5 0.2 0.5 1 0 0 0 1000 2000 3000 time (s) 4000 0 0 5000 0 0.5 0.2 0.4 0.6 0.8 fraction of bonds cleaved (x) 1 1 Figure 3. The inverse approach predicts chemical kinetics and lysis susceptibility. (a) The best fit turbidity (optical density) time series (line) superimposed over synthetic time series data (circles) of lysis due an action of an enzyme obeying Michaelis–Menten kinetics. The values for the actual and inferred kinetic constants are given in the ‘curved’ row of table 1. (b) The exact (solid line) and estimated (dashed) cumulative probability distributions for lysis shaping the time series in (a). Inset: the exact and estimated probability density functions. The initial substrate concentration in all cases is 10 μM. The enzyme concentrations range from 3.75 to 9.00 μM as shown above. (a) 1 probability of lysis (P) 0.8 turbidity ρ (b) 1 flat P curved P sharp P 0.6 0.4 0.2 0 0 1000 2000 3000 time (s) 4000 0.8 0.6 0.4 0.2 0 0 5000 flat P curved P sharp P 0.2 0.4 0.6 0.8 fraction of bonds cleaved (x) 1 Figure 4. Results of the inverse problem for varying susceptibilities. (a) The best fit turbidity time series, given an identical reaction time series (9.00 μM initial enzyme concentration), but different susceptibility distributions. (b) The exact (solid line) and estimated (dashed) susceptibility distributions shaping the time series in (a). 3.2. Estimation of parameters for experimental data Table 1. Estimated and exact values of kinetic constants for synthetic data. The initial condition in each well along with the corresponding turbidity time series and estimate for the susceptibility distribution are provided in figure 2. Class Actual Flat Curved Sharp k+ (μM−1 s−1 ) k− (s−1 ) kf (s−1 ) Km (μM) 1.08 × 10−2 1.03 × 10−2 9.93 × 10−3 9.99 × 10−3 5.42 × 10−1 5.43 × 10−1 5.01 × 10−1 5.06 × 10−1 6.91 × 10−3 7.07 × 10−3 6.60 × 10−3 8.06 × 10−3 5.02 × 102 5.29 × 102 5.05 × 102 5.06 × 102 Figure 5 shows the turbidity time series data and best fit obtained from the action of lysozyme on M. lysodeikticus cells for various initial enzyme concentrations in a multiwell plate, along with the estimated susceptibility distribution. M. lysodeikticus cells were prepared as described in appendix E. We determined the constants ρintact and ρlysed through independent experiments. A range of values for kinetic rates (95% confidence) was computed by fitting the likelihood function near the maximum likelihood to a normal distribution and computing the associated standard deviations. This is not a marginal likelihood for the kinetics constants, but a conditional likelihood function (i.e. conditioned on the most probable susceptibility distribution for every kinetic constant). The ranges given for forward reaction rates and binding affinities are close to values from the literature (see table 2). We estimate that egg-white lysozyme has an ≈200 μM binding constant (maximum likelihood) when acting on Micrococcus peptidoglycan, in fair agreement with prior estimates based on spectroscopic methods. One possible explanation for the observed differences between our binding rates and those reported in the literature is that our model does not take into account the position of bonds within the cell wall. At the the results of best fit susceptibility distributions for stochastic kinetics and a deterministic kinetic mechanism with dynamics that match the average values of the stochastic case. In general, stochastic reaction dynamics result in an operator which ‘blurs’ the image of the underlying susceptibility distribution to obtain a turbidity time series. As such, there are many possible solutions for susceptibility distributions that fit well to the data. Regularization of this solution results in estimates for susceptibilities which are smoother than their deterministic counterparts. Additional empirical results for convergence and stability of the method can be found in appendices C and D. 6 Phys. Biol. 7 (2010) 046002 (b) 1 1.4 1.2 probability of lysis turbidity (OD450nm) (a) G J Mitchell et al 1 0.8 0.6 0.6 0.4 0.2 0.4 0.2 0.8 0 2000 4000 time (s) 0 6000 0 0.2 0.4 0.6 0.8 fraction of bonds cleaved (x) 1 Figure 5. Parameter estimation from egg-white lysozyme experiments. (a) The best fit turbidity time series (solid line) superimposed over the data (circles) for the action of lysozyme on M. lysokeikticus cells. In each well the substrate concentration is 300 μM. Enzyme concentrations span from 65.9 nM to 4.22 μM (from top to bottom). The data for OD450 nm are plotted at every five timesteps for t < 1000 and at every 15 timesteps thereafter. The estimated kinetic rates are compared with literature values in table 2. (b) The cumulative probability distribution for lysis P (x) inferred from the time series data. Table 2. Rates of association (k+ ), dissociation (k− ), hydrolysis (kf ) and the binding constant (Km ) for the lysozyme experiment (see section 2). The corresponding turbidity time series and estimate for the susceptibility distribution are provided in figure 4. The cited literature values were obtained from (17)–(19). Estimate Literature k+ (μM−1 s−1 ) k− (s−1 ) kf (s−1 ) Km (μM) 1.41–6.19 × 10−3 – 6.05–11.6 × 10−1 – 5.42–10.1 × 10−1 1.1–17.5 × 10−1 185–354 9–84 The assay is based entirely on optical data, with enzymes acting directly on substrates attached to bacteria in microlitersized solutions and thus represents a scalable high-throughput method for quantification of the activity of lytic enzymes suitable for screening and development of anti-bacterial drugs. Other optical methods for measuring the activity of enzymes include the insertion of fluorescent markers into synthetic substrate [34–36] or surface plasmon resonance measurements to determine the binding affinities [37]. In contrast to these methods, our approach does not require the isolation of substrate from the cell. This is a critical advantage when dealing with enzymes that bind to carbohydrates or teichoic acids, as they may remain unidentified or cannot be isolated from the cell wall without loss of binding affinity. Leveraging this key innovation required the development of a computational framework for estimation of chemical parameters as well a susceptibility function. We have described a variable projection method to solve our inverse problem, but the method is also equivalent to non-parametric automated data collapse, with ‘scaling parameters’ determined by the reaction’s rate constants. The collapsed data are the susceptibility function, which is shared between wells. Parametric data collapse of turbidity time series has previously been used to quantify the dynamics of microtubule polymerization [38]. Our approach is distinct in that the turbidity scaling can be non-parametric and the time scaling has no closed form. Closer to our method is an approach developed by Battacherjee et al, based on maximizing a measure of data collapse [39]. Several approaches to data collapse by scaling arguments, including Battacherjee’s, are listed in [40]. However, none of these approaches naturally accommodate ‘stochastic scaling’ functions that arise start of the reaction, peptidoglycan bonds are not immediately available for binding and cleavage. Since our model assumes that all bonds are exposed, we may underestimate the binding rate k+ , resulting in an overestimation of the binding constant Km . The estimated susceptibility distribution implies that the cells are only loosely held together by the cell wall, as very few bonds need to be broken before lysis occurs. More specifically, we predict that only 5% of bonds are broken before 50% of cells have lysed. This is consistent with expectations of the cell preparation method, since these cells have undergone a freeze-drying process which may have compromised their integrity. In other experiments with Bacillus spp. cells [14], turbidity series data suggest that lysis seems to occur only after an initial time delay. Assuming a Michaelis–Mententype reaction mechanism, which has no significant delay for physically reasonable energy barriers, these time series suggest that substantial portions of the cell wall must be digested before lysis ensues. This observation provides further support for the idea that one must take into account both the effects of chemistry and physiology when trying to understand the dynamics of enzymatic lysis. 4. Discussion We have presented a method that allows for a statistical estimation of the microscopic details of the chemistry and physiology of bacterial lysis from macroscopic turbidity data. Our approach is based on a simultaneous estimation of reaction rate constants and a lysis susceptibility function, which is accomplished by solving an inverse scattering problem. 7 Phys. Biol. 7 (2010) 046002 G J Mitchell et al when considering non-deterministic reaction dynamics. In extending these efforts, our work generalizes the concept of optimal data collapse to include stochastic scaling and demystifies its analysis by demonstrating that it is equivalent to minimization of a nonlinear objective function constructed from a set of overdetermined equations which are separable in the component corresponding to the collapse function. Framed in this light, the effect of noise and other sources of error on optimal data collapse can be understood in terms of the stability properties of the scaling operators and the sensitivity of those operators to changes in the scaling parameters. As far as we are aware, Hunter and Asenjo [41] were the first to develop a quantitative model of microbial lysis due to lytic enzymes that accounted for the possibility of heterogeneous susceptibility to lysis. They were primarily concerned with prediction and control rather than parameter estimation and did not present a means for inferring the mechanisms of lysis from macroscopic data. Their model assumes a fixed cleavage threshold beyond which lysis occurs and incorporates heterogeneity to lysis susceptibility by allowing for a distribution of initial cell wall thicknesses. Recently Levashov et al [18] presented a model of turbidimetric assays which incorporates time delays in the degree of cell lysis after the addition of enzymes, but does not include an explicit model for the reaction dynamics. Here, the lysis susceptibility function examined is completely phenomenological and can flexibly accommodate any mechanism that results in heterogeneity. For example, currently we assume that all bonds are accessible before they are cleaved. A more realistic model, consistent with the framework developed above, would allow for spatial degrees of freedom, so that the susceptibility distribution would include information about the cleavage state at various positions within the cell wall. In this case it may be desirable to account for space explicitly in the reaction dynamics. One could also consider variation over experimental parameters such as temperature, pH or the concentration of sugars within the solution. These experimental parameters can change the reaction dynamics and the susceptibility function in a parametric or nonparametric fashion. The form of the putative lysis susceptibility function is determined by the details of the associated microscopic model of the cell wall function. Huang et al developed a model of cell wall organization and shape deformation in Gram-negative bacteria due to peptide and glycan defects [42]. They compute pore size distributions and suggest a model of outer membrane bulge formation, which leads to lysis, in terms of the basic mechanical properties of the cell wall. Developing a similar model for Gram-positive bacteria would be a first step toward predicting the form of the susceptibility function described above, which would provide insight into their mechanisms for resisting enzymatic lysis. The shape of this function could offer clues to developing better treatment schemes. For example, construction of a model that incorporates colocalization of binding sites for different enzymes or makes explicit the contribution of lateral and longitudinal stress bearing components could offer a quantitative explanation of synergistic killing observed in Streptococcus cells [43, 44]. Inasmuch as our method allows for the simultaneous estimation of both enzyme reaction kinetics and a lysis susceptibility function, it represents an opportunity for integrating chemical and physical approaches to understanding the cell wall structure. Our measure of susceptibility to lysis presents a novel quantitative probe of a basic functional property of cell walls, the explanation of which represents a new modeling challenge for biophysicists interested in the cell wall structure and function. Finally, by accounting for potential differences in the physiological states of cell in different cultures, this approach allows for the quantitative determination of enzyme activities against substrate from different cultures, which can be used to establish the reproducibility of activity measurements and for making interspecific comparisons. For these reasons we recommend that future quantitative studies adopt this method of analysis for turbidity assays. Acknowledgments The authors thank George Biros for helpful conversations and feedback on the mathematical methods used in this manuscript. They also thank Vince Fischetti for helpful early discussions that led to this work. JSW acknowledges the support of a James S McDonnell Foundation grant and the Defense Advanced Research Projects Agency under grants HR001105-1-0057 and HR0011-09-1-0055. JSW, PhD, holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund. Daniel Nelson acknowledges support from DOD grant DR080205. Appendix A. Differential equation models and numerical integration The prototypical enzyme reaction k+ kf S + E C→ H + E k− (A.1) can be cast into two different kinds of differential equation models, each of which requires different approaches for numerical integration of solutions. A.1. Deterministic mass action In the deterministic interpretation of the law of mass action the concentration of each reactant completely specifies the reaction rates so that the distribution of species at time t is given by a delta function wm (z, t|k) = δ(z − z̄m (t|k)) (A.2) with a rate equation for the mean concentration z̄˙ m = g(z̄m |k) (A.3) and some initial condition z̄m (0). For example, we can write rates for our basic enzyme reaction model in equation (12) as 8 ṡm = −k+ em sm + k− cm (A.4) ėm = −k+ em sm + (k− + kf )cm (A.5) Phys. Biol. 7 (2010) 046002 G J Mitchell et al ċm = k+ em sm − (k− + kf )cm (A.6) ḣm = kf cm (A.7) with an associated error function eJ (x). The idea is to make judicious choices of each ψj and J so the error function eJ ∈ Q\QJ is expected to be small based on prior information on the distribution (like the degree of smoothness). The size of the error can be readily calculated by |eJ | = (eJ , eJ )Q , (B.2) with z̄m = (sm , em , cm , hm ) which allows us to compute reaction trajectories given the initial concentration of each reactant. This particular rate equation does not admit closedform solutions, but we obtain estimates for the reaction coordinates at arbitrary times via numerical integration with MATLAB routines ode15s or ode45. where the inner product on Q is given by 1 dxψ(x)ψ (x). (ψ, ψ )Q = A.2. Stochastic mass action We need numerical approximations of both P (x) and its derivatives. We can write these derivatives at xi as In the stochastic interpretation of the law of mass action we consider transition rates between states which enumerate the total (integral) number of each species present. The rate of change of occupancy is written as ∂wm (z, t|k) = dz (V (z, z |k)wm ∂t × (z , t|k) − V (z , z|k)wm (z, t|k)), (A.8) ∂ k P (x) ∂x k k+ seδ(s − s + 1)δ(e − e + 1) V × δ(c − c − 1)δ(h − h ) + k− cδ(s − s − 1)δ(e − e − 1) × δ(c − c + 1)δ(h − h ) + kf cδ(s − s )δ(e − e − 1) × δ(c − c + 1)δ(h − h − 1). (B.5) (A.10) Appendix C. Choice of basis functions for deterministic kinetics The algorithm described above applies to both deterministic and stochastic reaction dynamics. In both cases the choice of basis functions is very important. For deterministic reactions we have qm (x, t|k) = δ(x − xm (t)), which gives (A.11) 1 (Am (k))ij = cm δ(x − xm (ti ))ψj (x) (C.1) 0 = cm ψj (xm (ti )). (C.2) From this equation we see that the values of the basis functions ψj evaluated at xm (ti ) and the constant factors cm determine the matrix elements and thus, implicitly, the properties of the matrices. These properties, such as stability and convergence rates, can be determined through computational experiments. Given a vector of coefficients P = (P1 , P2 , . . . , PJ ) ∈ RJ and a set of functions {ψ1 , ψ2 , . . . , ψJ } that span QJ ⊂ Q we can write the function P (x) as Pj ψj (x) + eJ (x) (B.4) x=xi With these equations, we can easily calculate an approximate derivative of P (x) through matrix-vector multiplication. These approximation schemes begin to motivate the idea that we can transform the original model with infinitely many degrees of freedom into a closely related model with finite degrees of freedom, amenable to machine calculation. Appendix B. Approximating P (x) and its derivatives P (x) = x=xi ∂ k eJ (x) + ∂x k where the elements of the first- and second-order differentiation matrices [45] are given by ∂ψj (x) ∂ 2 ψj (x) (2) D (1) = and D = . ij ij ∂x x=xi ∂x 2 x=xi (B.6) where the elements of w yield the values wm (z, t|k) at permissible coordinates given a finite total particle number and the matrix V (k) give the transitions between these coordinates. Given an initial number of substrate and enzyme particles sm(0) (0) and em we can project equation (A.11) onto a smaller subspace (0) . The resulting equation specifies a first-order of size sm(0) em linear ODE, which we integrate with a backward Euler method. J j =1 ∂ k ψj (x) Pj ∂x k ∂ k eJ (x) , =D P + ∂x k x=xi Since the distribution wm (z, t|k) is non-zero only for the integral values of z, we can rewrite equation (A.9) compactly as a single matrix-vector equation ẇm (t|k) = V (k)wm (t|k), = x=xi J (k) where V (z, z ) gives the rate of transition from the state z to the state z . Equation (A.8) can be rewritten as ∂wm (z, t|k) = dz V(z, z |k)wm (z , t|k), (A.9) ∂t where V(z, z |k) = V (z, z |k) − δ(z − z )( dz V (z , z|k)). Writing the states z = (s, c, e, h) and z = (s , c , e , h ) we have for the Michaelis–Menten reaction V (z, z |k) = (B.3) 0 C.1. Global basis (B.1) Our first basis is composed of members of Lagrange form polynomials, as described in [46]. These polynomials have j =1 9 Phys. Biol. 7 (2010) 046002 G J Mitchell et al 2 10 1 piecewise linear Lagrange polynomial 0.8 0 |e | J P(x) 10 0.6 0.4 −2 10 0.2 −4 10 0 20 40 (a) J 60 80 Estimated solution exact 0 0 100 0.2 (b) 0.4 x 0.6 0.8 1 Figure C1. (a) The error in approximating P (x) as a function of J, the number of coefficients, for both the piecewise linear basis and the global polynomial basis. (b) An illustration of convergence of the approximate solution at the various points xj for the global polynomial basis, with J = 60. Monotonicity has not been enforced here to highlight the smoothness of convergence for the unconstrained problem. the form with the normalization function J wj (x) = x − xj j =1 1.0 (C.3) (C.4) and where each point xj belongs to the set of Chebyshev points of the second kind on the interval [0, 1]. Explicitly we have π(j − 1) 1 1 xj = + cos . (C.5) 2 2 J −1 The choice of these points gives us values for the weights wj in equation (C.3) 1 (−1)j −1 , if j = 1 or j = J wj = 2 (C.6) j −1 otherwise. (−1) , 0.8 0.6 0.4 0.2 probability of lysis wj 1 (x) x − xj fraction of bonds cleaved ψj (x) = 1 0.5 0 0 0.5 1 fraction bonds cleaved 0.0 0 1000 time (s) 2000 3000 Figure C2. Time evolution of the probability density function for the number of bonds cleaved with superimposed image of equivalent deterministic kinetics (blue line). In the inset we have the inferred susceptibilities for deterministic (solid) and stochastic (dashed) kinetics. C.2. Piecewise polynomial basis C.3. Convergence and stability: linear portion Our second basis is composed of piecewise polynomial functions. The values of the coefficients of these polynomials are determined by the values of the function P (x) at the uniformly spaced points i−1 . (C.7) xi = J̃ /4 − 1 Within the interval [xi , xi+1 ] we have a polynomial of the form ⎧ aj (x − xi )3 where j mod 4 = 0 ⎪ ⎪ ⎨ where j mod 4 = 1 bj (x − xi )2 ψj (x) = (C.8) where j mod 4 = 2 cj (x − xi ) ⎪ ⎪ ⎩ where j mod 4 = 3 dj Given the correct value for k and a set of product time series xm (t), we would like to know how well our method estimates an underlying P (x) when there is a finite signal to noise ratio. For this test we model the noise in each well ξm as a Gaussian random variable with a variance σ 2 chosen to obtain a signal to noise ratio of 102 , which is a typical worst case in the experimental setup. We also set the number of wells M = 5. The rate of convergence will be determined by our choice of basis in addition to other intrinsic factors (e.g. the stability properties of the original integral equation). This convergence can be evaluated empirically for different set of basis functions, as shown in figure C1. Although we show convergence for both bases, the global polynomial basis converges faster. In general convergence will depend on the nature of the scaling functions. Stochastic reaction dynamics will result in linear operators which ‘blur’ the underlying image of the susceptibility function as shown in figure C2. and with j ∈ {1, . . . , J̃ }. The cubic terms allow us to have third-order continuity in our approximation for P (x). We can enforce this explicitly by imposing equality constraints on the zeroth, first and second derivatives of P (x) in addition to boundary conditions P (0) = 0 and P (1) = 1. With these constraints our approximate function ψ(x) is equivalent to spline interpolation of P (x) at the J̃ /4 grid points xi , such that the total number of degrees of freedom is J = J̃ /4. 10 Phys. Biol. 7 (2010) 046002 G J Mitchell et al 0 20 0.2 minimum 18 5 16 4.5 14 0.4 4 0.6 10 0.8 Gα k f 12 8 3 6 2.5 4 1 (a) 0 0.02 0.04 k+ 0.06 0.08 3.5 0.1 (b) true k = k* − 2 0 k− = k− − 0.5 r 1 1.5 Figure D1. (a) A slice of the objective Gα (k) with deterministic Michaelis–Menten kinetics (k− = constant) and M = 5. It is apparent that k is the ratio this objective function is non-convex, and includes narrow valleys. (b) A slice of the objective function Gα (rktrue ) where r = ktrue between the true solution ktrue and the coordinate k. One can see that the minimum of the objective function is not at the true solution, for any given instantiation of noise. This is due to the effects of noise in the data on the projection (k)ρ̃. Appendix D. Convergence and stability: nonlinear portion Appendix F. Definitions m = 1, 2, . . . , M z ∈ RN The existence and stability of minima of the variable projection functional Gα (k) ultimately depends on both the action of the time evolution φkt associated with the reaction model and the choices of initial conditions in the various wells wm (z, 0). Still, for a given reaction model and set of true parameters ktrue and a susceptibility function we can examine the shape of this function empirically (figure D1). It is evident that the variable projection functional Gα (k) can have narrow valleys, which may make it impossible to distinguish between different values of k given a turbidity time series. This may be due to the fact that different values of k can produce nearly equivalent reaction time series, which is always an obstacle in kinetic assays or because changes in reaction time series due to changes in k can be compensated by changes in P ∗ . x ∈ [0, 1] k ∈ RK t ∈R W φkt : W → W wm (z, 0) ∈ W wm (z, t|k) ∈ W V (z, z ) ∈ W × W V(z, z ) ∈ W × W Q Appendix E. Cell preparation XB : W → Q To prepare our cell stock we suspended .018 g lyophillized M. lysodeikticus cells (Sigma, catalog #M3770) in 1 ml PBS corresponding to 237 μM concentration of peptidoglycan monomers. This value assumes that 80% of the cell’s dry mass is due to peptidoglycan and a molecular weight of 1014Da for a peptidoglycan monomer. We mixed 0.429 g chicken egg-white lysozyme (Sigma, catalog #L6876) in 100 ml PBS, which yields a 300 μM solution assuming a molecular weight of 14313Da for lysozyme. The cell suspensions and enzyme solutions were combined in a 96 well plate. An individual well contained 100 μl of cells with seven two-fold serial dilutions of the enzyme stock, yielding initial enzyme concentrations e0 ∈ {4.21, 2.11, 1.05, 5.27 × 10−1 , 2.64 × 10−1 , 1.32 × 10−1 , 6.59 × 10−2 } (μM). The turbidity was measured through absorbance at 450 nm every 13 s on a platereading spectrophotometer (Molecular Devices). qm (x, t|k) ∈ Q P (x) ∈ Q p(x) ∈ Q fm (x) ∈ Q ρlysed , ρintact ∈ R nm ∈ R ρ̃m ∈ RT P ∈ RJ Am (k) ∈ RT ×J 11 index for wells reaction coordinate for S chemical species coordinate for the fraction of bonds cleaved vector of reaction rates time a Hilbert space of functions over the closed, connected domain ⊂ RN time evolution operator parameterized by k initial probability density function probability density function at time t density of transitions rates between states transition rates kernel in stochastic model a Hilbert space of functions over = [0, 1] normalized ‘projection’ operator describing fraction of bonds cleaved probability density function of the fraction of bonds cleaved cumulative probability distribution for probability of lysis probability density function for fraction of bonds cleaved expected fraction of cells lysed in the population specific turbidity for lysed and intact cells concentration of cells column vector of turbidity data at T time points column vector of coefficients matrix representation of integral operator parameterized by k Phys. Biol. 7 (2010) 046002 G J Mitchell et al D (1) , D (2) ∈ RJ ×J [23] Kerr D E, Plaut K, Bramley A J, Williamson C M, Lax A J, Moore K, Wells K D and Wall R J 2001 Nat. Biotechnol. 19 66–70 [24] Borysowski J, Weber-Dabrowska B and Gorski A 2006 Exp. Biol. Med. 231 366–77 [25] Fischetti V, Nelson D and Schuch R 2006 Nat.Biotechnol. 24 1508–11 [26] Cheng Q, Nelson D, Zhu S and Fischetti V A 2005 Antimicrob. Agents Chemother. 49 111–7 [27] Yoong P, Schuch R, Nelson D and Fischetti V A 2004 J. Bacteriol. 186 4808–12 [28] Daniel A, Euler C, Collin M, Chahales P, Gorelick K J and Fischetti V A 2010 Antimicrob. Agents Chemother. 54 AAC.01625–09–1612 [29] Poolman B and Glaasker E 1998 Mol. Microbiol. 29 397–407 [30] Jenzano J, Hogan S and Lundblad R 1986 J. Clin. Microbiol. 24 963–7 [31] Morsky P 1983 Anal. Biochem. 128 77–85 [32] Tarantola A 2006 Nature Phys. 2 492–4 [33] Golub G H and Pereyra V 2003 Inverse Problems 19 R1 [34] Shockman G D and Holtje J V 1994 Microbial peptidoglycan (murein) hydrolases The Bacterial Cell Wall ed J-M Ghuyson and R Hakenbeck (Amsterdam: Elsevier) chapter 7, pp 131–66 [35] Lutzner N, Patzold B, Zoll S, Stehle T and Kalbacher H 2009 Biochem. Biophys. Res. Commun. 380 554–8 [36] Lupoli T J, Taniguchi T, Wang T S, Perlstein D L, Walker S and Kahne D E 2009 J. Am. Chem. Soc. 131 18230–1 [37] Loessner M J, Kramer K, Ebel F and Scherer S 2002 Mol. Microbiol. 44 335–49 [38] Flyvbjerg H, Jobs E and Leibler S 1996 Proc. Natl Acad. Sci. USA 93 5975–9 [39] Bhattacharjee S M and Seno F 2001 J. Phys. A: Math. Gen. 34 6375–80 [40] Melchert O 2009 arXiv:0910.5403 [41] Hunter J B and Asenjo J A 1988 Biotechnol. Bioeng. 31 929–43 [42] Huang K C, Mukhopadhyay R, Wen B, Gitai Z and Wingreen N S 2008 Proc. Natl Acad. Sci. USA 105 19282–7 [43] Djurkovic S, Loeffler J M and Fischetti V A 2005 Antimicrob. Agents Chemother. 49 1225–8 [44] Loeffler J M and Fischetti V A 2003 Antimicrob. Agents Chemother. 47 375–7 [45] Trefethen L N 2000 Spectral Methods in MATLAB (Philadelphia, PA: SIAM) [46] Berrut J P and Trefethen L N 2004 SIAM Rev. 46 501–17 first- and second-order differentiation matrices ψj (x), ψj (x) ∈ Qn ⊂ Q basis function for approximate susceptibility functions References [1] Clardy J, Fischbach M and Walsh C 2006 Nat. Biotechnol. 24 1541–50 [2] Hancock R E W and Sahl H 2006 Nat. Biotechnol. 24 1551–7 [3] Salazar O and Asenjo J A 2007 Biotechnol. Lett. 29 985–94 [4] Vollmer W, Joris B, Charlier P and Foster S 2008 FEMS Microbiol. Rev. 32 259–86 [5] Rice K C and Bayles K W 2003 Mol. Microbiol. 50 729–38 [6] Nakimbugwe D, Masschalck B, Deckers D, Callewaert L, Aertsen A and Michiels C W 2006 FEMS Microbiol. Lett. 259 41–6 [7] Callewaert L and Michiels C W 2010 J. Biosci. 35 127–60 [8] Young R 1992 Microbiol. Mol. Biol. Rev. 56 430–81 [9] Nelson D, Loomis L and Fischetti V A 2001 Proc. Natl Acad. Sci. USA98 4107–12 [10] Fischetti V A 2008 Curr. Opin. Microbiol. 11 393–400 [11] Ensign J C and Wolfe R S 1965 J. Bacteriol. 90 395–402 [12] Fischetti V A, Gotschlich E C and Bernheimer A W 1971 J. Exp. Med. 133 1105–17 [13] Loeffler J M, Nelson D and Fischetti V A 2001 Science 294 2170–2 [14] Schuch R, Nelson D and Fischetti V A 2002 Nature 418 884–9 [15] Kirsch A 1991 An Introduction to the Mathematical Theory of Inverse Problems (Applied Mathematical Sciences vol 120) ed J E Marsden, L Sirovich and F John (Berlin: Springer) [16] Tarantola A 2005 Inverse Problem Theory and Methods for Model Parameter Estimation (Philadelphia, PA: SIAM) [17] Callewaert L, Vanderkelen L, Deckers D, Aertsen A, Robben J and Michiels C W 2008 Appl. Env. Microbiol. 74 4978–81 [18] Levashov P A, Sedov S A, Shipovskov S, Belogurova N G and Levashov A V 2010 Anal. Chem. USA 82 2161–3 [19] Barrett J F, Schramm V L and Shockman G D 1984 J. Bacteriol. 159 520–6 [20] Chipman D M 1971 Biochemistry 10 1714–22 [21] Banerjee S K, Kregar I, Turk V and Rupley J A 1973 J. Biol. Chem. 248 4786–92 [22] Fischetti V A, Jones K F and Scott J R 1985 J. Exp. Med. 161 1384–401 12

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