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
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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
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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.
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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
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