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Lysis dynamics and membrane oligomerizationpathways for Cytolysin A (ClyA) pore-forming toxin

M S Vaidyanathandaggera Pradeep Sathyanarayanabc Prabal K Maitid

Sandhya S Visweswariahbc and K G Ayappaab

Pore-forming toxins are known for their ability to efficiently form transmembrane pores which eventually

leads to cell lysis The dynamics of lysis and underlying self-assembly or oligomerization pathways

leading to pore formation are incompletely understood In this manuscript the pore-forming kinetics and

lysis dynamics of Cytolysin-A (ClyA) toxins on red blood cells (RBCs) are quanti1047297ed and compared with

experimental lysis data Lysis experiments are carried out on a 1047297xed mass of RBCs under isotonic

conditions in phosphate-buffered saline for different initial toxin concentrations ranging from 294ndash147

nM Kinetic models which account for monomer binding conformation and oligomerization to form the

dodecameric ClyA pore complex are developed and lysis is assumed to occur when the number of

pores per RBC (np) exceeds a critical number npc By analysing the model in a sublytic regime ( np lt npc)

the number of pores per RBC to initiate lysis is found to lie between 392 and 768 for the sequential

oligomerization mechanism and between 5300 and 6300 for the non-sequential mechanism Rupture

rates which are 1047297rst order in the number of RBCs are seen to provide the best agreement with the lysis

experiments The time constants for pore formation are estimated to lie between 1 and 20 s and

monomer conformation time scales were found to be 2ndash4 times greater than the oligomerization times

Cell rupture takes places in 100s of seconds and occurs predominantly with a steady number of pores

ranging from 515 to 11 000 on the RBC surface for the sequential mechanism Both the sequential

irreversible and non-sequential kinetics provide similar predictions of the hemoglobin release dynamics

however the hemoglobin released as a function of the toxin concentration was accurately captured only

with the sequential model Each mechanism develops a distinct distribution of mers on the surface

providing a unique experimentally observable 1047297ngerprint to identify the underlying oligomerizationpathways Our study offers a method to quantify the extent and dynamics of lysis which is an important

aspect of developing novel drug and gene delivery strategies based on pore-forming toxins

1 Introduction

Pore-forming toxins (PFTs) are a class of proteins produced by a

wide variety of organisms including bacteria1 and humans2

They have the unique property of generating pores in the

membranes of target cells3ndash5 PFTs are classied on the basis of

the secondary structure (a or b) of the pore-forming region and

usually undergo a monomer to oligomer transition that is a

pre-requisite for pore formation67 Cytolysin A (ClyA HlyE or

SheA) from E coli is a well characterized a-PFT38 and crystal

structures are available for both the water-soluble monomeric9

and membrane-associated oligomeric forms10 The monomer

of ClyA possesses 5 a helices and a hydrophobic b sheet the

ldquob-tonguerdquo that is buried within the helices More recent

crystal structure data10

indicates that the toxin oligomerizes asa dodecamer A pathway for the transition from the monomer

to the membrane-bound protomer and nally to the dodeca-

meric pore complex has been proposed based on a comparison

of the monomeric and oligomeric crystal structures10 Large

conformational changes in the N-terminus and the b sheet

regions need to occur during the monomer to oligomer tran-

sition that appear to involve initial ipping out of the b-

tongue to bind to the lipid membrane followed by larger

translocation of the N-terminus into the lipid membrane ClyA

forms cation-selective pores which have an internal diameter

of 4ndash7 nm10

a Department of Chemical Engineering Indian Institute of Science Bangalore India

E-mail ayappachemengiiscernetin Fax +91 80 23608121 Tel +91 80 22932769

b Bioengineering Programme Indian Institute of Science Bangalore India E-mail

pradeepmrdgiiscernetin Fax +91 80 23600999 Tel +91 80 22932659

c Department of Molecular Reproduction Development and Genetics Indian Institute of

Science Bangalore India E-mail sandhyamrdgiiscernetin Fax +91 80

23600999 Tel +91 80 22932659

d Department of Physics and Center for Condensed Matter Theory Indian Institute of

Science Bangalore India E-mail maitiphysicsiiscernetin Fax +91 80

22932602 Tel +91 80 22932315

dagger Present address Department of Chemical Engineering University of Texas at

Austin Te xas USA

Cite this RSC Adv 2014 4 4930

Received 16th September 2013

Accepted 22nd October 2013

DOI 101039c3ra45159c

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A variety of experimental techniques such as lytic experi-

ments gel electrophoresis site-directed mutagenesis and cryo-

electron microscopy have been used to unravel the mechanisms

of pore formation in ClyA 11ndash14 and in other widely studied PFTs

such as cholestrol-dependent toxins (CDCs)15 and S aureus a-

Hemolysin116 Despite the interest in unravelling structural and

mechanistic pathways for the action of PFTs10 the kinetics of

membrane oligomerization rates of pore formation and the

dynamics of ensuing lysis have not been the subject of muchquantitative investigation17 Pore formation kinetics and

dynamics of the release of self-quenching dye molecules from

liposomes have been quantied by Schwarz and co-workers1819

where the marker release dynamics is tted to either a single or

double exponential function with suitably motivated kinetic

models for pore formation Models which quantify the perme-

ation rates across bacterial membranes due to pore-forming

protegrin peptides20 and lysenin-induced permeation in giant

unilamellar vesicles21 have appeared in the recent literature

Recently an investigation of permeation rates due to a-Hemo-

lysin on liposomes using optical contrast microscopy and

micropipette experiments

22

reveal pore densities of about 100pores mm2

Quantifying the phenomenon of pore formation and lysis

is crucial for developing PFT-based druggene delivery thera-

pies and controlling pore formation in vesicle-based biore-

actors23 during the development of articial cells In this

study the lytic activity of ClyA is modeled based on experi-

ments carried out on red blood cells We formulate an

adsorption-kinetic model which incorporates monomer

binding conformational changes and sequential as well as

non-sequential oligomerization pathways to determine the

rate of hemoglobin released as a function of time and ClyA

concentration A rst order rupture model is used to quantify

the lysis dynamics Our model captures the experimentalhemoglobin data as a function of toxin concentration From

this observation we extract the critical number of pores per

RBC above which cell lysis occurs

2 Experimental procedure21 Expression and purication of His-tagged Cytolysin A

(ClyA)

pGS1111 plasmid containing the ClyA gene as a fusion with

glutathione S-transferase was obtained from Dr J Green

University of Sheffield UK The ClyA gene was subcloned from

pGS1111 into pPRO Ex-HTb using EcoRI and SalI to obtainpPROb ClyA containing an N-terminal hexahistidine tag E coli

BL21 endo cells transformed with pPROb ClyA were grown in

terric broth ClyA full length (ClyA FL) proteins were expressed

on induction with 500 mM isopropyl thiogalactopyranoside

Cells were lysed by sonication in buff er containing 100 mM

TrisndashHCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl

1 mM benzamidine 2 mM phenylmethylsulfonyl uoride and

10 glycerol Centrifugation was carried out at 30 000 g and the

cell-free extract was interacted with nickelndashnitrilotriacetic acid

beads Beads were washed with buff er containing 100 mM Trisndash

HCl (pH 80) 5 mM b-mercaptoethanol 500 mM NaCl 20 mM

imidazole to remove nonspecic proteins on the beads His6

ClyA was eluted in buff er containing 100 mM TrisndashHCl (pH 80)

5 mM b-mercaptoethanol 100 mM NaCl 300 mM imidazole

10 glycerol Proteins were desalted in buff er (100 mM Trisndash

HCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl and 10

glycerol)

His6 ClyA was treated with TEV protease to obtain tagless

protein 1 part of puried hexahistidine TEV protease was

taken per 30 parts by mass of ClyA and incubated overnight at 4 C TEV was separated by interacting further with Ni-

NTA beads Protein quantity was estimated by the Bradford

method24

22 Hemolysis assay

The hemolysis assay was carried out as described previously 12

Rabbit erythrocytes were washed and diluted 1 100 vv in PBS

(phosphate-buff ered saline pH 74) Aliquots of RBC suspension

were transferred to microcentrifuge tubes ClyA was added to

suitable aliquots of RBCs and incubated at 37 C in a shaking

incubator for 1 hour Lysis experiments were carried out for

ClyA concentrations ranging from 294ndash147 nM These corre-spond to 100ndash500 ng ml1 respectively since ClyA is a 34 kDa

monomer Unlysed cells and debris were sedimented by

centrifugation at 5000 rpm for 1 min Released hemoglobin in

the supernatant was quantied by spectrometric detection at

540 nm The numbers of cells remaining a er lysis were

counted in a hemocytometer

23 Turbidity assay

A suspension of rabbit erythrocytes (1 vv in phosphate-buff -

ered saline 1 ml) was treated with varying amounts of Cytolysin

A as indicated To assess turbidity 200 ml of the cell suspension

was transferred to a clear-bottomed 96-well plate and light scattering was measured at 620 nm The cells in the plate were

centrifuged at 3000 rpm for 2 min and the extent of haemolysis

was estimated by measuring the absorbance of the supernatant

at 570 nm Optical density measurements were carried out on a

Tecan Innite F50 microplate reader

3 Modeling31 Membrane binding and bulk toxin concentration

We develop a model to predict the hemoglobin release kinetics

of the RBCs as a function of initial toxin concentration The

series of steps that lead to pore formation are illustrated inFig 1 The model is developed in the mean eld framework

wherein all cells are assumed to be identical Diff usion is

assumed to be fast relative to membrane binding and oligo-

merization Assuming a protein diff usion coefficient of 1013

m2 s1 the diff usion time on the membrane is of the order of

15ndash50 ms for toxin concentrations ranging from 294ndash147 nM

Membrane binding is assumed to be irreversible and of similar

time scale to that of oligomerization The amount of hemo-

globin released due to cell lysis is signicantly larger (108

times) than that released from the pores of unlysed cells both

due to the size of the hemoglobin molecule25 (5ndash6 nm) as well as

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the small eff ective pore diameter available for transport The

inner pore diameter exposed to the cytosol is 4 nm in the crystal

structure and in a fully solvated environment the eff ective

diameter is expected to decrease further Further osmoticprotection assays of ClyA conclude that the eff ective pore sizes

range from 20ndash35 nm26 The solution is assumed to remain

isotonic as lysis proceeds since lysis did not occur in the

absence of toxin when RBCs were incubated in buff er solution

made up of fully lysed (sonicating 1 RBC (vv)) RBCs We have

assumed that the conformational change follows rst order

irreversible kinetics since the conformational step involves a

transition from a water-soluble monomer to a membrane-

inserted protomer via a series of conformational changes in the

regions around the b-tongue region of the monomer and the N-

terminus10 This is succeeded by a fast oligomerization step to

form the pore complex

The rate equation for the membrane-bound monomer

whose surface molar concentration is denoted as m is

dm

dt frac14 k aC m

ms m

Xnl frac141

pl

k dm k cm (1)

where k a is the adsorption rate constant k d is the desorption

rate constant ms is the saturated surface molar concentration

and the last term represents the rate at which the membrane-

bound monomer (m) undergoes a conformational change

to the membrane-bound protomer ( p1) with a rate constant

k c and pl is the surface molar concentration of the oligomer

containing l -mers If the bulk concentration of the toxin

monomer is constant eqn (1) is similar in form to the

LangmuirndashHinshelwood equation traditionally used to

describe the concentration of surface species undergoing

both adsorption and reaction

Since the initial toxin concentration (C in) in the aliquots is in

the range of 294ndash147 nM an additional balance is used to

describe the concentration change of toxins in solution This

yields

V sol

dC m

dt frac14

k dm k aC m

ms m

Xnl frac141

pl

ARBCN RBC (2)

where V sol denotes the volume of solution in the aliquot ARBC is

the area of a single RBC and N RBC is the number of erythrocytes

present in V sol at any instant

32 Oligomerization kinetics

Oligomerization involves the formation of dimers trimers and

higher mers from the protomer until an n-mer complex (pore)

is formed Data obtained from scanning transmission electronmicroscopy (STEM) and single-wavelength anomalous diff rac-

tion (SAD) indicate that the Cytolysin A (ClyA) pore complex

consists of n frac14 12 and 13 mers respectively1014 Oligomerization

can occur in a number of distinct kinetic pathways In Fig 2

the two main mechanisms are illustrated In the sequential

mechanism the nth mer is formed by the addition of a 1 mer to

a (n 1) mer complex In the non-sequential mechanism the

nth mer can be formed by allowed integer combinations of the

smaller mers As an example a 4 mer can be formed by a

combination of 2 + 2 mers as well as a 3 + 1 mers as illustrated

in Fig 2

If oligomerization occurs sequentially and irreversibly thereaction mechanism is

p1 thorn pl k l pl thorn1 l frac14 1 n 1 (3)

where k l is the reaction rate constant for the l th oligomerization

step If the l th oligomer is formed in an irreversible non-

sequential process the reaction mechanism is

pr thorn pl rk l pl l frac14 2 n

r frac14 1 to l =2 l even

1 to ethl 1THORN=2 l odd

for the formation of the l th oligomer In the above non-

sequential mechanism for oligomerization the number of

distinct reaction rate constants for the formation of a 12-mer

pore complex is 66 In what follows we develop the model for

the irreversible sequential mechanism We are unaware of any

Fig 1 Schematic indicating the various steps leading to pore forma-

tion The water-soluble monomer adsorbs onto the cell membrane

and undergoes a conformational change to form the membrane-

bound protomer This is followed by an oligomerization step to form

the dodecameric pore complex

Fig 2 Two possible modes of oligomerization (top) sequential olig-

omerization and (bottom) non-sequential oligomerization are shown

In sequential oligomerization a protomer is necessary for the forma-

tion of a higher oligomer whereas in the non-sequential mechanism a

higher oligomer can be formed from an allowed combination of lower

oligomers

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experiments which shed light on either of these mechanisms

Results for the reversible sequential mechanism and non-

sequential irreversible kinetics are presented later in the text

In the sequential mechanism there are 11 rate constants

Molecular simulation of hydrophobic association of small

solutes in water27 reveal that a sequential aggregation procedure

is favored during cluster formation The sequential aggregation

mechanism is also used for modelling micellar aggregation28

We assume that all the rate constants for the sequentialmechanism are identical This assumption is widely used in

sequential polymerization reactions With the assumption that

all rate constants for oligomerization (k l ) are identical a balance

on the protomer yields

d p1

dt frac14 k cm k l

Xn1

l frac141

p1 pl (4)

where the rst term on the right hand side represents the

formation of the protomer from the monomer and the other

terms represent sequential oligomerization steps wherein the

protomer binds with the other lsquomersrsquo to form the higher lsquomersrsquo

with a rate constant k l From rate considerations for dimerformation (l frac14 2) a prefactor of 12 appears in the term which

corresponds to the formation of the dimer This is a necessary

condition for satisfying the species mass balance The govern-

ing equation for the dimer (l frac14 2) is given by

d p2

dt frac14

1

2k 2 p1 p1 k 2 p1 p2 (5)

where p2 represents the concentration of dimer and k 2 repre-

sents the rate constant for the reaction The equation for the

formation of the l th oligomer (l gt 2) is

d pl

dt frac14 k l p1 pethl 1THORN eth1 dl 12THORNk l p1 pl l frac14 3 12 (6)

where d l 12 represents the Kronecker delta function The corre-

sponding number of pores per RBC is obtained using

np frac14 p12N avARBC (7)

where N av is the Avogadro number

33 Cell lysis

Every dodecamer corresponds to a stable pore in the

membrane If the rate at which lysis occurs is directly propor-

tional to the number of cells that are present at any instant of

time then cell lysis follows a rst order process On physical

grounds we further assume that cell lysis occurs only when the

number of pores exceeds a critical number of pores in each cell

Since our experiments are carried out under isotonic condi-

tions lysis is associated with rupture Lysis can be described

using the following rst order process

dx

dt frac14 klxR

np npc

(8)

where

R frac14

0 npnpc

np npc np $npc

x represents the fraction of unlysed cells at any instant of time

np is number of pores per RBC at any instant and npc is the

critical number of pores per RBC above which cell lysis occurs

The constant kl represents the decay rate constant for cell lysis

In eqn (8) the ramp function R(np npc) incorporates the

increased lysis as a function of the excess pores np npc We

also investigate other functional forms for R such as a unit step

function and a higher power dependence on np npc The

inuence of these on the model predictions are discussed laterin the text As cells lyse the number of RBCs N RBC at any

instant is

N RBC frac14 N inRBCx (9)

N inRBC is the initial number of RBCs

The rate at which hemoglobin is released from the RBCs into

solution is

dH out

dt frac14 V hrhN

inRBCkl xR

np npc

thornDhAp

ms

h H out

N RBCnp

V soll p

(10)

where the rst term represents the contribution due to lysis

(rupture) and the second term is the diff usive ux contribution

from the pores of unlysed cells In the above equation V h is the

volume of hemoglobin present in a single RBC rh is the density

of hemoglobin Dh is the diff usivity of hemoglobin l p is the

diff usion length along the pore Ap is the average area of a pore

msh H out and rh represents the saturated hemoglobin mass in

one RBC amount of hemoglobin present in the solution at any

instant of time and the density of hemoglobin respectively

4 Solution procedure While analyzing the problem it is useful to recast the equations in

suitable dimensionless forms If t frac14 t s C m frac14 C mC in m frac14 mms

and pl frac14 pl ms then eqn (1) (2) and (4)ndash(6) in dimensionless forms

are

dm

dt frac14

s

sa

C m

1 m

Xnl frac141

pl

s

sd

m s

sc

m (11a)

dC m

dt frac14

ARBCN RBCm

s

V solC inb

s

sd

m s

sa

C m

1 m

Xnl frac141

pl

(11b)

d p1

dt frac14

s

sc

m s

sl

Xn1

l frac141

p1 pl (11c)

d p2

dt frac14

s

2s2

p1 p1 s

s2

p1 p2 (11d)

d pl dt frac14

s

sl

p1 pethl 1THORN eth1 dl 12THORNs

sl

p1 pl l frac14 3 12 (11e)

From eqn (11a)ndash(11e) we can extract the following set of time

constants

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sa frac14 1

k aC inis the adsorption time constant

sd frac14 1

k dis the desorption time constant

sc frac14 1

k cis the conformational time constant

sl frac14 1

k l ms is the reaction time constant

s frac14 1

k aC in thorn k c thorn k d

(12)

41 Model parameters

Since we do not have experimental data to independently

determine the various time constants it is more convenient to

dene a ratio between time constants We dene the ratio

between conformational and adsorption times as

l frac14sc

sa

(13)

It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-

tion1014 suggesting that l gt 1 We can also rewrite the adsorption

time constant and conformational time constant in terms of l

provided we have an estimate of the time required for initiation

of pore formation Since the processes leading to pore forma-

tion occur in series the total time constant snet for pore

formation is the sum of the time constants for the individual

steps

snet frac14 sc + sa + stl + sd (14)

Using eqn (12) and (13) and with the added assumption that

desorption rate is negligible and the adsorption and reaction

time constants are similar (sa st

l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence

k a frac14 2 thorn l

snetC in(15a)

and

k c frac142 thorn l

lsnet

(15b)

Under these assumptions for a xed initial concentration of

toxins and cell mass specifying l snet and C in is sufficient to

make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be

expressed solely in terms of the constant l The values of various

system properties used in the simulation are given in Table 1

and the values of diff erent parameters are given in Table 2 For

snet the model predictions were tested for a range of values as

indicated The parameters related to the pore geometry radius

of the pore r p and length of the pore l p are obtained from the

crystal structure of the ClyA pore10 The initial number of RBCs

are counted using the hemocytometric technique Typical

liquid diff usivities are used for hemoglobin Since the diff u-

sivity only inuences the hemoglobin release in the sublytic

regime obtaining a precise value of the diff usivity is not of

special consequence

42 Simulation details

We used an explicit Euler scheme for discretizing the governing

ordinary diff erential equations and the equations were solved

with a reduced time step of 0005 (0016 s l frac14 4) Calculations

performed with a reduced time step of 0001 did not alter the

reported results We developed an in-house program using

Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers

5 Results and discussions51 Lysis experiments

The OD data from lysis experiments at 60 minutes are illus-

trated in Fig 3a as a function of the bulk monomer toxin

concentration C in The data represents an average over 5

independent experiments Based on the time evolution (Fig 3b)

data no further lysis was observed above 30 minutes for all the

toxin concentrations investigated in the study Hence data at 30

minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at

147 nM and about 10ndash15 lysis at 588 nM The data clearly

reveals that the RBC lysis occurs only above the critical toxin

concentration (npc) which we estimate at 544 nM Below this

critical concentration lies the regime of low hemolytic activity

where the absolute OD values are an order of magnitude below

values obtained with lysis indicating that leakage from pores is

not signicant Although pore formation occurs in this regime

the concentration of pores is not sufficient to initiate lysis To

further support this hypothesis we carried out turbidity assay

experiments (Fig 4) for initial toxin concentrations ranging

Table 1 Various system properties and parameters used in the

simulation In some cases only the range of parameters that were

tested are given

Area of RBC29 ( ARBC) 136 mm2

Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1

Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2

Net reaction time constant (snet ) 1ndash25 s

Table 2 Values of constants obtained from sequential and non-

sequential oligomerization

ParameterSequentialoligomerization

Non-sequentialoligomerization

l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate

constant (kl )

15ndash18 107 s1 125ndash135 107 s1

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from 294ndash294 nM The decrease in turbidity is seen to occur

simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity

decrease is also mirrored with the corresponding OD data Since

the cell mass used in the turbidity experiments is lower than

that used in the lysis experiments sublytic toxin concentrations

lie below 294 nM

52 Model predictions

In this section model predictions for the sequential irreversible

kinetics are compared with the lysis data

521 Sublytic regime

In the sublytic regime (np lt npc) the evolution of np with time is

obtained by solving eqn (11a)ndash(e) The number of pores np is

obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-

tration saturates within 10 s (Fig 5a) This saturation in np is

due to the limiting amount of toxin present in solution In

Fig 5b we plot the variation in np at saturation (20 s) with

l frac14sc

sa for diff erent initial toxin concentration assuming that

np lt npc The curve corresponding to 544 nM is tted to the

form y frac14 axb and the relation npc frac14 20033l097 is obtained

which can be used to x the value of npc for a given value of l In

order to simulate the lysis data a value of snet (eqn (14)) which

is the time constant associated with the time required for

formation of the rst pore has to be specied Initial estimates

are in the range of 1ndash25 s and we use a value of 1 s in all our

simulations unless specied We show later that our results are

Fig 3 (a) The normalized optical density values as a function of the

initial toxin concentration observed after 60 minutes during the lysis

experiments A distinct jump is observed above a toxin concentration

of 544 nM (b) Time evolution data of optical density values during

RBC lysis At 147 nM 98 lysis is observed

Fig 4 Turbidity (left axis) and lysis data (right axis) show that the

decrease in turbidity occurs simultaneously with an increase in the

OD The sublytic initial toxin concentration is less than 294 nM

(100 ng ml1)

Fig 5 Relation between l and npc in the sublytic regime (a) Number

of pores per RBC as a function of time The data plotted are for l frac14 2

and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-

dotted line) yields the relation npc frac14 20033l097

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relatively insensitive to the value of snet in this range The

parameters l npc and snet are obtained in the sublytic regime as

discussed above

522 Lysis regime

In the lysis regime in addition to the kinetic eqn (11andashe) we also

solve the lysis and hemoglobin release equations eqn (8) and

eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once

values of l npc and snet are xed in the sub-lytic regime the only

unknown parameter in the model is the value of kl (eqn (8)) In

all cases kl is xed by matching the maximum extent of lysis of

98 obtained at 147 nM Subsequent simulations are run with

diff erent values of C in to compare with the experimental data

Fig 6 illustrates the comparison between the model prediction

and the experimental OD data Since the OD varies linearly with

the amount of hemoglobin released during lysis30 we scale both

the experimental and predicted data by their respective

maximum OD to facilitate a meaningful comparison We also

carried out independent lysis experiments to verify the linear

relationship between the OD versus hemoglobin data Hence we

normalize the long time data and dene H max as the ratio of the

mass of the steady state hemoglobin released at a given C in to

the corresponding value at C in frac14 147 nM which is the highest

C in considered in the study This facilitates a comparison of

H max predicted from the model directly with the normalized OD

values For l frac14 2 the corresponding npc frac14 392 and the

comparison of H max at a value of kl frac14 18 107 s1 is

illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl

frac14 15 107 s1 is illustrated in Fig 6c Comparison of the

hemoglobin release dynamics ( H out vs time) for the corre-

sponding set of parameters are illustrated in Fig 6b and d

respectively

We observe that in this range of l (2 l 4) values and

kl 15ndash18 107 s1 the OD vs C in data is captured quite

accurately Since l is the ratio of the ClyA monomer confor-

mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that

conformation is preceded by fast adsorption followed by rapid

oligomerization The H out dynamics predicted by the model is

seen to capture the experimental data quite well (Fig 6b and d)

Upon increasing l we nd that a lower value of kl is required to

match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures

the early time release in the H out data at 147 nM quite accu-

rately and increasing snet to 24 s results in a short delay at early

times Since snet represents the time taken to form the rst pore

in situ monitoring of the hemoglobin release dynamics would be

required to determine snet more precisely In our experiments the

time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in

ice for a period of 3ndash5 minutes while the OD is determined

Experiments carried out continuously for the diff erent time

points shown in Fig 6b and d did not alter the data obtained

from the intermittent experiments Fluorescence permeation

experiments by Yamazaki and co-workers21 by lysenin (334 kDa)

induced pore formation on single giant unilamellar vesicles show

that pore formation is complete within about 10 s for toxin

concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are

similar to the time scales deduced in our model

At a toxin concentration of 147 nM a steady distribution of

mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)

Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)

during which very little lysis is observed (Fig 6a and d) Lysis is

predominantly observed a er a steady number of pores have

formed on the RBC surface This steady number of pores ranges

Fig 6 Comparison of model predictions (open circles) with experi-

mental results (open squares) The amount of hemoglobin released

(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of

hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l

between 2 and 4 is seen to accurately capture the Hmax versus C in data

[(a) and (c)] The amount of hemoglobin released is scaled with the

maximum amount to facilitate a comparison Simulations corresponding

to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines

Fig 7 The pore density is plotted as a function of time in thepost-lysis

regime The number of pores (np) per RBC ranges from 515 at 588 nM

to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392

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from 515 at 588 nM to 11 657 at 147 nM Given this situation it

is instructive to dene an eff ective lysis time constant kleff frac14

kl (np npc) where np is the steady state value of the number of

pores at a given value of C in (Fig 7) The value of kleff at 147 nM

is 2027 103 s1 which results in an eff ective lysis time

constant of 493 s

523 Oligomer and pore concentration

In both the low (lt544 nM) and high toxin ($544 nM) regimes

the amount of toxin is found to be limiting Even at the highest

toxin concentration C in frac14 147 nM the toxin in bulk solution is

depleted within 10 s Selected oligomer concentrations as a

function of time are plotted in Fig 8a and b for both high and

low toxin concentrations as predicted by the kinetic model (eqn

(4) and (6)) The pore density is illustrated in Fig 7 The gov-

erning equations for the formation of an l -mer are given in eqn

(6) Since pore formation occurs via a sequential oligomeriza-

tion mechanism a protomer (1-mer) is necessary for the

formation of all other l mers Hence a steady monomer

concentration on the RBC is achieved once the 1-mer concen-

tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the

membrane we observe that a large fraction of protomers remain

trapped as intermediate mers on the membrane At 147 nM the

number of monomers per ml is 8854 1012 The initial

number N RBC frac14 32 107 the monomers per RBC is 276

105 If all the monomers were converted to pores each RBC

would have 23 105 pores However the number of pores

formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-

cating that about 50 of the mers remain on the membrane

surface as intermediate n-mers (n frac14 1ndash11) At a sublytic

concentration of 544 nM only about 45 of the mers are

converted to pores resulting in 375 pores per RBC (Fig 5a)

The eff ect of l is more prominent at the higher toxin

concentration where both the life time and the maximum

concentration for 1-mers is found to decrease as l is increased

from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change

in the value of l results in a shi in the distribution for a

particular initial toxin concentration An increase in l implies an

increase in the conformational time relative to the adsorption

and reaction times Hence as l is increased occurrence of the

lower mers on the surface decreases due to the faster reaction

time scales relative to conformation At low bulk toxin concen-

trations (Fig 8c) the distribution of higher mers and conse-

quently the number of pores (12 mers) is very low due to the

limited supply of monomers in the system However at higher

concentration (Fig 8d) the number of monomers is no longer

the limiting factor and the distribution shi

s towards the highermers thereby increasing the number of pores on the surface

524 Parameter sensitivity

We brie y summarize the results of simulations carried out to

test the inuence of the estimated parameters on the model

predictions In the absence of monomer membrane binding

equilibria the value of saturated surface concentration ( ms) is

Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)

Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM

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unknown In order to test the inuence of ms on the model

prediction we carried out a few simulations for ms frac14 1 108

mol m2 and msfrac14 1 1010 mol m2 for various C in values For

ms frac14 1 108 mol m2 a negligible number of pores were

formed and the H max ndashC in data (Fig 3a) is underpredicted At

ms frac14 1 1010 mol m2 pore formation was extremely rapid

and little variation in pore density between C in frac14 882 nM and

147 nM was observed As a consequence H max ndashC in data (Fig 3a)

is grossly overpredicted Hence a value of ms frac14 1 10

9 molm2 was used in the simulations (Fig 6) We further note

that the amount of saturated surface concentration ms

implicitly changes the reaction rate constant k l (eqn (12))

Increasing ms eff ectively decreases the reaction rate constant

(eqn (12))

Once snet is xed npc is related to l through the relation npcfrac14

alb with the constants a and b being xed for a given initial toxin

concentration C in (Fig 5) We have found that 2 l 4 ts the

hemoglobin release data very closely (Fig 6) and although the

hemoglobin released as a function of time is slightly under-

estimated by the model the agreement is reasonable Upon

increasing l

to 7 and keeping m

s

frac14 10

9

mol m

s1

the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1

although H max versus C in data is accurately predicted the

hemoglobin versus time data is grossly underpredicted Varying

ms between 108 and 1010 mol m2 further deteriorated the

prediction Finally we point out that other functional forms of the

dependence on np npc in the cell lysis equation eqn (8) such as

the unit step function or a quadratic dependence (np npc)2 only

overestimated the H max versus C in data

53 Sequential oligomerization with reversible kinetics

In the previous discussion we present the results for the oligo-

merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is

p1 thorn pl ) k f

k b pl thorn1 l frac14 1 n 1 (16)

where k f and k b represent the forward and backward reaction

rate constants The kinetic equations are

d p1

dt frac14 k cm k f

Xn1

l frac141

p1 pl thorn k bXn1

l frac142

pl (17a)

d p2

dt frac14

1

2k f p1 p1 k f p1 p2 thorn k b p3

1

2 p2 (17b)

d pl

dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl

thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)

A reversible time constant can be dened from eqn (17) as

sb frac14 1

k b The ratio R is dened as the ratio of forward to back-

ward time constants ( R frac14 sf sb) to study the eff ect of revers-

ibility Upon examining the number of pores as a function of

time we observe that the time taken to reach a steady number of

pores is signicantly larger than the time taken to reach steady

state in the lysis experiments In order to make comparisons

with the irreversible mechanism we evaluated the number of

pores at the threshold concentration of 544 nM It is observed

that the number of pores required for lysis initially increases

and then decreases for increments in R values The distribution

of oligomers at steady state are shown for diff erent R values in

Fig 9a and b for C in frac14 147 nM The steady state concentrations

of lsquomersrsquo change from a predominantly 11-mer concentration to

a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12

mers) are signicantly higher than the intermediate lsquomerrsquo

concentrations shown in Fig 9b and range from 0142 nmol

Fig 9 Oligomer distribution as a function of the ratio of forward and

backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies

between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin

release data as a function of toxin concentration is shown forvarious R

R frac14 0 has the closest agreement with the experimental data (open

squares)

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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the

concentration of 12 mers is 024 nmol m2 and decreases with a

further increase in R The predictions using reversible sequen-

tial kinetics for the H max vs C in data (Fig 9c) indicate

greater deviation from the experimental data when compared

with R frac14 0 These results indicate that the irreversible mecha-

nism provides the best agreement with the experimental data

We point out that the critical number of pores npc in

the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2

54 Non-sequential oligomerization

In contrast to the 392 pores obtained for the critical number of

pores via the sequential mechanism a substantially larger

critical number of pores are observed via the non-sequential

oligomerization (6000 pores per cell) The mass balance for

the protomer ( p1) concentration remains identical to that of the

sequential oligomerization mechanism (eqn (4)) The governing

equations for pl (l lt l n) oligomer undergoing non-sequential

irreversible oligomerization are

d pl

dt frac14

1

2k l Xl 1

ufrac141

pu pl u eth1 dl 12THORNXn1

ufrac141

k l pl pu for 1l n (18)

where n represents the number of monomers in a pore The

relation between the critical number of pores and npc for non-

sequential oligomerization is npc frac14 5439l015 The critical

number of pores for l frac14 2 is 6035 pores which is about 20 times

greater than that obtained from a sequential oligomerization

mechanism In this scheme the hemoglobin release (Fig 10a)

data is overpredicted at intermediate toxin concentrations when

compared with the sequential oligomerization The predictions

of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)

A comparison of the tted parameters between the sequential

and non-sequential oligomerization mechanisms are given in

Table 2

Oligomer distributions obtained from the non-sequential

mechanism (Fig 11b) show an entirely diff erent trend when

compared to that obtained from the sequential mechanism

(Fig 8) In the sequential mechanism the higher lsquomersrsquo

attained a steady state once the protomer was depleted In the

non-sequential mechanism the contribution to the dodeca-

mers (12 mer) can be obtained from a large number of combi-

nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-

omer steady state concentration (Fig 11a) is about 20 s for an

initial concentration of 147 nM Similar time scales are

observed in the sequential mechanism as well

Further experiments are required to distinguish between the

various mechanisms Western Blot experiments conducted on

Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the

presence of intermediate oligomers On the other hand single-

molecule uorescence imaging of a-hemolysin on a droplet

interface bilayer showed the presence of only monomers and

heptamers (pores)32

6 Discussion and conclusions

Lysis experiments on RBCs with the ClyA pore-forming toxin

show that a threshold initial toxin concentration is required to

initiate lysis From this observation we analyzed the problem in

two regimes a low toxin concentration regime where rupture of

cells is absent and a high toxin concentration regime where

lysis occurs and hemoglobin is released Kinetic models which

accounts for monomer binding conformation (membrane-

bound monomer to protomer) and oligomerization to form the

dodecameric pore complex are developed Models which

account for sequential and non-sequential oligomerization are

tested Cell rupture is assumed to be

rst order in the number of live cells and directly proportional to the pores in excess of the

critical number of pores npc In the sublytic regime the number

of pores is found to have a power law dependence on l which is

the ratio of conformational time to the reaction time This leads

to the construction of a ldquophase diagramrdquo between the number

of pores np and l for diff erent values of the initial toxin

concentration Comparing simulations with experimental data

the range of npc was 392ndash768 for the sequential mechanism and

5300ndash6300 pores for the non-sequential mechanism for 2 l 4

The range of l values is consistent with available experimental

data on ClyA which indicates that the membrane-bound

Fig 10 (a) Model predictions from the non-sequential mechanism

Simulated Hmax C in (open circles) curves grossly overpredicts the

experimental data (open squares) (b) Simulated hemoglobin release

compares well with the experimental data Dashed line snetfrac14 1 s solid

line snet frac14 24 s

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conformational step is slower than the preceding adsorption

and subsequent oligomerization steps10

From the model we are also able to comment on the time

constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a

fast process relative to the time taken for the pore population on

a single RBC to reach steady state which is about 20ndash30 s Since

rupture kinetics is dynamic and depends on the fraction of live

cells as well as the number of pores on the cell lysis occurs in

the time scale of 10s of minutes Due to this separation of time

scales lysis is seen to occur once the number of pores has

reached a steady state This steady number of pores ranges from

515 to 11 657 as the toxin concentration ranges from 588ndash

147 nM For the non-sequential mechanism the critical

number of pores required to initiate lysis is about 20 times

higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the

irreversible sequential kinetics provides the closest match with

the hemoglobin released as a function of the initial toxin

concentration Although we observe an overprediction of the

hemoglobin release data with the non-sequential mechanism

the hemoglobin release kinetics are similar to that of the

sequential mechanism The distribution of lower oligomers is

distinctly diff erent in both cases with a negligible numbers of

lower mers observed in the non-sequential oligomerization

These diff erences in the distribution of mers off er a ngerprint

to identify the underlying mechanism for pore formation

Experiments which can determine the number of pores or the

steady state lsquomerrsquo distributions on the membrane surface will

shed light on the pathways for oligomerization and enable a

more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable

parameter Lysis experiments conducted with Vibrio cholerae El

Tor cytolysin33 and Monalysin34 show similar lysis times (in the

order of 10s of minutes) as observed in our study suggesting

similarities in the underlying kinetic pathways that lead to pore

formation and rupture The model developed in this manu-

script is generic and could be recast with some variation to

study the dynamics of other PFTs

We brie y discuss some of the limitations of the model in its

present form The model is based on the mean eld approxi-

mation where all cells are assumed to be identical and for the

purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics

of rupture with the correct functional dependence on the pore

density In general there could exist a distribution of cells with

diff erent densities of pores Preliminary experiments by varying

the number of RBCs at a xed toxin concentration led to an

increase in lysis suggesting that cell heterogeneity could be

playing a role Although a population balance model35 could

include these variations this is at an added cost of complexity A

second aspect inherent to the model is the presence of lysis

beyond the time at which steady state is observed in the

experiments (30 minutes) Once the number of pores has

Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown

for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady

state is quite large compared to the other oligomer concentrations present in the system

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reached a steady state which occurs within 30 s (Fig 7a) cell

lysis continues to occur at a xed number of pores proportional

to np npc Running the simulation to steady state would

eventually lead to lysis of all the cells in the system albeit at an

exceedingly slow rate at the lower and intermediate toxin

concentrations We have also assumed complete and irrevers-

ible binding of the monomer to the membrane and hence the

number of pores predicted represent an upper limit within the

proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to

capture the inherent time scales in the process of pore forma-

tion and rupture as well as predict the variation of hemoglobin

release as a function of the initial toxin concentration observed

in the experiments We nally point out that the eff ect of

temperature on the kinetics and lysis has not been investigated

in this work Preliminary experiments at 147 nM toxin

concentration carried out to steady state indicated a marked

drop in lytic activity for temperatures below 15 C and lysis was

not observed at 10 C Data on binding isotherms would be

required to understand temperature eff ects in these systems

Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-

vation that lysis occurs in a given window of toxin concentration

has implications while developing PFT-based drug and gene

therapy protocols In recent studies E coli used in conjunction

with radiation therapy was found to retard the growth of cancer

tumors when compared with only radiation therapy protocols36

Quantifying the dynamics and extent of lysis is an important

aspect of developing appropriate treatment protocols in these

combination therapies In other applications where pores are

used for gene delivery and in the development of articial cells

lysis must be prevented and pore formation restricted to

concentrations below the lysis threshold

Acknowledgements

We acknowledge funding from Department of Science and

Technology (DST) India under the IRHPA grant The authors

thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful

discussions on the development of the model as well as the

reviewers for their critical comments

References

1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55

733ndash

7512 J Thiery D Keefe S Boulant E Boucrot M Walch

D Martinvalet I S Goping R C Bleackley

T Kirchhausen and J Lieberman Nat Immunol 2011 12

770ndash777

3 A Ludwig C von Rhein S Bauer C Huttinger and

W Goebel J Bacteriol 2004 186 5311ndash5320

4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S

A 1976 73 467ndash471

5 M V Sousa M Richardson W Fontes and L Morhy

J Protein Chem 1994 13 659ndash667

6 G Anderluh and P Macek Toxicon 2002 40 111ndash124

7 I Iacovache F G van der Goot and L Pernot Biochim

Biophys Acta Biomembr 2008 1778 1611ndash1623

8 N R Wyborn A Clark R E Roberts S J Jamieson

S Tzokov P A Bullough T J Stillman P J Artymiuk

J E Galen L Zhao M M Levine and J Green

Microbiology 2004 150 1495ndash1505

9 A J Wallace T J Stillman A Atkins S J Jamieson

P A Bullough J Green and P J Artymiuk Cell 2000 100

265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and

N Ban Nature 2009 459 726ndash730

11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and

B E Uhlin Mol Microbiol 1999 32 1226ndash1238

12 A Atkins N R Wyborn A J Wallace T J Stillman

L K Black A B Fielding M Hisakado P J Artymiuk and

J Green J Biol Chem 2000 275 41150ndash41155

13 A Ludwig G Volkerink C von Rhein S Bauer E Maier

B Bergmann W Goebel and R Benz J Bacteriol 2010

192 4001ndash4011

14 N Eier M Vetsch M Gregorini P Ringler M Chami

A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661

15 L A Shepard O Shatursky A E Johnson and R K Tweten

Biochemistry 2000 39 10284ndash10293

16 A J Ratner K R Hippe J L Aguilar M H Bender

A L Nelson and J N Weiser J Biol Chem 2006 281

12994ndash12998

17 I Iacovache M Bischoerger and F G van der Goot Curr

Opin Struct Biol 2010 20 241ndash246

18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296

19 G Schwarz R-t Zong and T Popescu Biochim Biophys

Acta Biomembr 1992 1110 97ndash104

20 D Bolintineanu E Hazrati H T Davis R I Lehrer and

Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry

2012 51 5160ndash5172

22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz

So Matter 2013 9 3181ndash3187

23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A

2004 101 17669ndash17674

24 M M Bradford Anal Biochem 1976 72 248ndash254

25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev

Lett 1999 83 4357ndash4360

26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and

B E Uhlin Mol Microbiol 1999 32 1226ndash1238

27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969

28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty

and R Rajagopalan Langmuir 1999 15 437ndash446

29 C E McLaren G M Brittenham and V Hasselblad Am J

Physiol 1987 252 3181ndash3187

30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005

24 118ndash121

31 M Nagahama H Hara M Fernandez-Miyakawa

Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302

32 J R Thompson B Cronin H Bayley and M I Wallace

Biophys J 2011 101 2679ndash2683

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Paper RSC Advances

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7252019 c3ra45159c

httpslidepdfcomreaderfullc3ra45159c 1313

33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol

Immunol 1995 184 37ndash44

34 O Opota I Vallet-Gely R Vincentelli C Kellenberger

I Iacovache M R Gonzalez A Roussel F G van der Goot

and B Lemaitre PLoS Pathog 2011 7 e1002259

35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51

1311ndash1332

36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim

H S Bom H E Choy Y Hong and J J Min Mol Ther

2010 18 635ndash642

RSC Advances Paper

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