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

Abstract

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 quantified and compared with experimental lysis data. Lysis experiments are carried out on a fixed mass of RBCs, under isotonic conditions in phosphate-buffered saline, for different initial toxin concentrations ranging from 2.94–14.7 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 (n_p) exceeds a critical number, n_pc. By analysing the model in a sublytic regime (n_p < n_pc) 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 first 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 2–4 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 fingerprint to identify the underlying oligomerization pathways. 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.

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1. Introduction

Pore-forming toxins (PFTs) are a class of proteins produced by a wide variety of organisms including bacteria¹ and humans.² They have the unique property of generating pores in the membranes of target cells.^3–5 PFTs are classified on the basis of the secondary structure (α or β) of the pore-forming region, and usually undergo a monomer to oligomer transition that is a pre-requisite for pore formation.^6,7 Cytolysin A (ClyA, HlyE or SheA) from E. coli is a well characterized α-PFT^3,8 and crystal structures are available for both the water-soluble monomeric⁹ and membrane-associated, oligomeric forms.¹⁰ The monomer of ClyA possesses 5 α helices and a hydrophobic β sheet, the “β-tongue”, that is buried within the helices. More recent crystal structure data¹⁰ indicates that the toxin oligomerizes as a dodecamer. A pathway for the transition from the monomer to the membrane-bound protomer, and finally to the dodecameric pore complex has been proposed based on a comparison of the monomeric and oligomeric crystal structures.¹⁰ Large conformational changes in the N-terminus and the β sheet regions need to occur during the monomer to oligomer transition, that appear to involve initial flipping out of the β-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 4–7 nm.¹⁰

A variety of experimental techniques such as lytic experiments, gel electrophoresis, site-directed mutagenesis and cryo-electron microscopy have been used to unravel the mechanisms of pore formation in ClyA^11–14 and in other widely studied PFTs such as cholestrol-dependent toxins (CDCs)¹⁵ and S. aureus α-Hemolysin.^1,16 Despite the interest in unravelling structural and mechanistic pathways for the action of PFTs,¹⁰ the kinetics of membrane oligomerization, rates of pore formation, and the dynamics of ensuing lysis, have not been the subject of much quantitative investigation.¹⁷ Pore formation kinetics and dynamics of the release of self-quenching dye molecules from liposomes have been quantified by Schwarz and co-workers,^18,19 where the marker release dynamics is fitted to either a single or double exponential function with suitably motivated kinetic models for pore formation. Models which quantify the permeation 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 α-Hemolysin on liposomes using optical contrast microscopy and micropipette experiments,²² reveal pore densities of about 100 pores μm⁻².

Quantifying the phenomenon of pore formation and lysis is crucial for developing PFT-based drug/gene delivery therapies and controlling pore formation in vesicle-based bioreactors²³ during the development of artificial cells. In this study, the lytic activity of ClyA is modeled based on experiments 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 first order rupture model is used to quantify the lysis dynamics. Our model captures the experimental hemoglobin data as a function of toxin concentration. From this observation, we extract the critical number of pores per RBC above which cell lysis occurs.

Experimental procedure

1.1. Expression and purification 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 obtain pPROb ClyA containing an N-terminal hexahistidine tag. E. coli BL21 endo⁻ cells transformed with pPROb ClyA were grown in terrific broth. ClyA full length (ClyA FL) proteins were expressed on induction with 500 μM isopropyl thiogalactopyranoside. Cells were lysed by sonication in buffer containing 100 mM Tris–HCl (pH 8.0), 5 mM β-mercaptoethanol, 100 mM NaCl, 1 mM benzamidine, 2 mM phenylmethylsulfonyl fluoride and 10% glycerol. Centrifugation was carried out at 30 000 g and the cell-free extract was interacted with nickel–nitrilotriacetic acid beads. Beads were washed with buffer containing 100 mM Tris–HCl (pH 8.0), 5 mM β-mercaptoethanol, 500 mM NaCl, 20 mM imidazole to remove nonspecific proteins on the beads. His₆ ClyA was eluted in buffer containing 100 mM Tris–HCl (pH 8.0), 5 mM β-mercaptoethanol, 100 mM NaCl, 300 mM imidazole, 10% glycerol. Proteins were desalted in buffer (100 mM Tris–HCl (pH 8.0), 5 mM β-mercaptoethanol, 100 mM NaCl and 10% glycerol).

His₆ ClyA was treated with TEV protease to obtain tagless protein. 1 part of purified 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 method.²⁴

1.2. Hemolysis assay

The hemolysis assay was carried out as described previously.¹² Rabbit erythrocytes were washed and diluted 1 : 100 v/v in PBS (phosphate-buffered saline pH 7.4). 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 2.94–14.7 nM. These correspond to 100–500 ng ml⁻¹ 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 quantified by spectrometric detection at 540 nm. The numbers of cells remaining after lysis were counted in a hemocytometer.

1.3. Turbidity assay

A suspension of rabbit erythrocytes (1% v/v in phosphate-buffered saline; 1 ml) was treated with varying amounts of Cytolysin A as indicated. To assess turbidity, 200 μl 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 Infinite F50 microplate reader.

2. Modeling

2.1. 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 in Fig. 1. The model is developed in the mean field framework, wherein all cells are assumed to be identical. Diffusion is assumed to be fast relative to membrane binding and oligomerization. Assuming a protein diffusion coefficient of 10⁻¹³ m² s⁻¹, the diffusion time on the membrane is of the order of 15–50 ms for toxin concentrations ranging from 2.94–14.7 nM. Membrane binding is assumed to be irreversible and of similar time scale to that of oligomerization. The amount of hemoglobin released due to cell lysis is significantly larger (∼10⁸ times) than that released from the pores of unlysed cells, both due to the size of the hemoglobin molecule²⁵ (5–6 nm) as well as the small effective 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 effective diameter is expected to decrease further. Further, osmotic protection assays of ClyA conclude that the effective pore sizes range from 2.0–3.5 nm.²⁶ 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 buffer solution made up of fully lysed (sonicating 1% RBC (v/v)) RBCs. We have assumed that the conformational change follows first 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 β-tongue region of the monomer and the N-terminus.¹⁰ This is succeeded by a fast oligomerization step to form the pore complex.

Fig. 1 Schematic indicating the various steps leading to pore formation. 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.

The rate equation for the membrane-bound monomer, whose surface molar concentration is denoted as m, is

where k_a is the adsorption rate constant, k_d is the desorption rate constant, m^s 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 p_l 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 Langmuir–Hinshelwood 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 2.94–14.7 nM, an additional balance is used to describe the concentration change of toxins in solution. This yields

where V_sol denotes the volume of solution in the aliquot, A_RBC is the area of a single RBC and N_RBC is the number of erythrocytes present in V_sol at any instant.

2.2. 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 electron microscopy (STEM) and single-wavelength anomalous diffraction (SAD) indicate that the Cytolysin A (ClyA) pore complex consists of n = 12 and 13 mers respectively.^10,14 Oligomerization can occur in a number of distinct kinetic pathways. In Fig. 2, the two main mechanisms are illustrated. In the sequential mechanism, the n^th mer is formed by the addition of a 1 mer to a (n − 1) mer complex. In the non-sequential mechanism, the n^th 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.

Fig. 2 Two possible modes of oligomerization, (top) sequential oligomerization and (bottom) non-sequential oligomerization are shown. In sequential oligomerization a protomer is necessary for the formation of a higher oligomer whereas in the non-sequential mechanism, a higher oligomer can be formed from an allowed combination of lower oligomers.

If oligomerization occurs sequentially and irreversibly, the reaction mechanism is,

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,

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 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 aggregation.²⁸ We assume that all the rate constants for the sequential mechanism 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,

where, the first 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 ‘mers’ to form the higher ‘mers’ with a rate constant k_l. From rate considerations, for dimer formation, (l = 2), a prefactor of 1/2 appears in the term which corresponds to the formation of the dimer. This is a necessary condition for satisfying the species mass balance. The governing equation for the dimer (l = 2) is given by,where p₂ represents the concentration of dimer and k₂ represents the rate constant for the reaction. The equation for the formation of the l^th oligomer (l > 2) iswhere δ_l,12 represents the Kronecker delta function. The corresponding number of pores per RBC is obtained using

n_p = p₁₂N_avA_RBC, (7)

where N_av is the Avogadro number.

2.3. Cell lysis

Every dodecamer corresponds to a stable pore in the membrane. If the rate at which lysis occurs is directly proportional to the number of cells that are present at any instant of time then cell lysis follows a first 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 conditions, lysis is associated with rupture. Lysis can be described using the following first order process,where
x represents the fraction of unlysed cells at any instant of time, n_p is number of pores per RBC at any instant and n_pc is the critical number of pores per RBC above which cell lysis occurs. The constant κ_l represents the decay rate constant for cell lysis. In eqn (8), the ramp function [Fraktur R](n_p − n_pc) incorporates the increased lysis as a function of the excess pores, n_p − n_pc. We also investigate other functional forms for [Fraktur R] such as a unit step function and a higher power dependence on n_p − n_pc. The influence of these on the model predictions are discussed later in the text. As cells lyse, the number of RBCs, N_RBC, at any instant is

N_RBC = Nⁱⁿ_RBCx, (9)

Nⁱⁿ_RBC is the initial number of RBCs.

The rate at which hemoglobin is released from the RBCs into solution is

where the first term represents the contribution due to lysis (rupture) and the second term is the diffusive flux contribution from the pores of unlysed cells. In the above equation, V_h, is the volume of hemoglobin present in a single RBC, ρ_h is the density of hemoglobin, D_h is the diffusivity of hemoglobin, l_p is the diffusion length along the pore, A_p is the average area of a pore, m^s_h, H_out and ρ_h 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.

3. Solution procedure

While analyzing the problem, it is useful to recast the equations in suitable dimensionless forms. If t̄ = t/τ, C̄_m = C_m/C_in, m̄ = m/m^s and p̄_l = p_l/m^s then eqn (1), (2) and (4)–(6) in dimensionless forms are,

From eqn (11a)–(11e), we can extract the following set of time constants.

3.1. Model parameters

Since we do not have experimental data to independently determine the various time constants, it is more convenient to define a ratio between time constants. We define the ratio between conformational and adsorption times as,

It has been observed that the conformational times, τ_c are larger than the time for membrane binding and oligomerization^10,14 suggesting that λ > 1. We can also rewrite the adsorption time constant and conformational time constant in terms of λ, provided we have an estimate of the time required for initiation of pore formation. Since the processes leading to pore formation occur in series, the total time constant, τ_net for pore formation is the sum of the time constants for the individual steps.

τ_net = τ_c + τ_a + τ^t_l + τ_d (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 (τ_a ∼ τ^t_l = (n − 1)τ_l), the constants k_a and k_c can be expressed in terms of λ. Hence

and

Under these assumptions, for a fixed initial concentration of toxins and cell mass, specifying λ, τ_net and C_in is sufficient to make predictions for the rate at which pores are formed in the sublytic regime. With these assumptions, eqn (11) can be expressed solely in terms of the constant λ. The values of various system properties used in the simulation are given in Table 1 and the values of different parameters are given in Table 2. For τ_net, 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 pore.¹⁰ The initial number of RBCs are counted using the hemocytometric technique. Typical liquid diffusivities are used for hemoglobin. Since the diffusivity only influences the hemoglobin release in the sublytic regime, obtaining a precise value of the diffusivity is not of special consequence.

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 RBC^29 (ARBC)	136 μm^2
Volume of RBC^29 (Vh)	90 fL
Radius of pore^10 (rp)	3.5 nm
Length of pore^10 (lp)	13 nm
Diffusivity of hemoglobin (Dh)	10^−9 m^2 s^−1
Initial number of RBC (N^inRBC)	3.2 × 10^7 cells per ml
Volume of lysis assay (Vsol)	1 ml
Saturated surface concentration (m^s)	10^−9 mol m^−2
Net reaction time constant (τnet)	1–25 s

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Table 2 Values of constants obtained from sequential and non-sequential oligomerization

Parameter	Sequential oligomerization	Non-sequential oligomerization
λ	2–4	2–4
npc	392–768 pores	5300–6300 pores
Decay rate constant (κl)	1.5–1.8 × 10^−7 s^−1	1.25–1.35 × 10^−7 s^−1

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3.2. Simulation details

We used an explicit Euler scheme for discretizing the governing ordinary differential equations and the equations were solved with a reduced time step of 0.005 (0.016 s, λ = 4). Calculations performed with a reduced time step of 0.001 did not alter the reported results. We developed an in-house program using Matlab 7.0 to solve the discretized equations. Calculations were checked with a mass balance on the monomers.

4. Results and discussions

4.1. Lysis experiments

The OD data from lysis experiments at 60 minutes are illustrated 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 system. From the cell counts in the hemocytometer we find 98% lysis at 14.7 nM and about 10–15% lysis at 5.88 nM. The data clearly reveals that the RBC lysis occurs only above the critical toxin concentration (n_pc) which we estimate at 5.44 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 significant. 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 from 2.94–29.4 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 2.94 nM.

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 5.44 nM. (b) Time evolution data of optical density values during RBC lysis. At 14.7 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 2.94 nM (100 ng ml⁻¹).

4.2. Model predictions

In this section model predictions for the sequential irreversible kinetics are compared with the lysis data.

4.2.1. Sublytic regime

In the sublytic regime (n_p < n_pc) the evolution of n_p with time is obtained by solving eqn (11a)–(e). The number of pores n_p is obtained from eqn (7). From the simulations we observe that the number of pores per RBC for a given initial toxin concentration saturates within 10 s (Fig. 5a). This saturation in n_p is due to the limiting amount of toxin present in solution. In Fig. 5b, we plot the variation in n_p at saturation (20 s) with , for different initial toxin concentration assuming that n_p < n_pc. The curve corresponding to 5.44 nM is fitted to the form y = ax^b and the relation, n_pc = 200.33λ^0.97 is obtained which can be used to fix the value of n_pc for a given value of λ. In order to simulate the lysis data, a value of τ_net (eqn (14)), which is the time constant associated with the time required for formation of the first pore, has to be specified. Initial estimates are in the range of 1–25 s and we use a value of 1 s in all our simulations, unless specified. We show later that our results are relatively insensitive to the value of τ_net in this range. The parameters λ, n_pc and τ_net are obtained in the sublytic regime as discussed above.

Fig. 5 Relation between λ and n_pc in the sublytic regime. (a) Number of pores per RBC as a function of time. The data plotted are for λ = 2 and τ_net = 1 s. (b) A regression analysis of the curve at 5.44 nM (dash-dotted line) yields the relation, n_pc = 200.33λ^0.97.

4.2.2. Lysis regime

In the lysis regime, in addition to the kinetic eqn (11a–e) we also solve the lysis and hemoglobin release equations, eqn (8) and eqn (10). Cell lysis occurs, once n_p > n_pc, and the OD increases with increasing toxin concentration as shown in Fig. 3a. Once values of λ, n_pcand τ_net are fixed in the sub-lytic regime the only unknown parameter in the model is the value of κ_l (eqn (8)). In all cases κ_l is fixed by matching the maximum extent of lysis of 98% obtained at 14.7 nM. Subsequent simulations are run with different 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 lysis,³⁰ 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 define 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 = 14.7 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 λ = 2, the corresponding n_pc = 392 and the comparison of H_max at a value of κ_l = 1.8 × 10⁻⁷ s⁻¹ is illustrated in Fig. 6a. The comparison for λ = 4 (n_pc = 768) and κ_l = 1.5 × 10⁻⁷ s⁻¹ is illustrated in Fig. 6c. Comparison of the hemoglobin release dynamics (H_out vs. time) for the corresponding set of parameters are illustrated in Fig. 6b and d respectively.

Fig. 6 Comparison of model predictions (open circles) with experimental results (open squares). The amount of hemoglobin released (H_max) at the end of 30 min for (a) n_pc = 392, (c) n_pc = 768. Dynamics of hemoglobin released, (H_out) for (b) n_pc = 392, (d) n_pc = 768. A value of λ between 2 and 4 is seen to accurately capture the H_max 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 τ_net = 24 s are shown in bold lines and τ_net = 1 s in dashed lines.

We observe that in this range of λ (2 ≤ λ ≤ 4) values and κ_l ∼ 1.5–1.8 × 10⁻⁷ s⁻¹ the OD vs. C_in data is captured quite accurately. Since λ is the ratio of the ClyA monomer conformation time to the monomer adsorption time, this range of λ 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 λ we find that a lower value of κ_l is required to match the OD vs. C_in data (Fig. 6d). A value of τ_net = 1 s captures the early time release in the H_out data at 14.7 nM, quite accurately and increasing τ_net to 24 s results in a short delay at early times. Since τ_net represents the time taken to form the first pore, in situ monitoring of the hemoglobin release dynamics would be required to determine τ_net more precisely. In our experiments the time evolution of the lysis data is carried out by intermittently arresting lysis at different times and quenching the aliquots in ice for a period of 3–5 minutes, while the OD is determined. Experiments carried out continuously for the different 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 (33.4 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 ml⁻¹ and 50 s at 40 ng ml⁻¹. These are similar to the time scales deduced in our model.

At a toxin concentration of 14.7 nM, a steady distribution of mers is achieved on the time scale of 10–20 s (for a τ_net = 1 s). Within this time interval, the pore density goes through a phase of 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 after a steady number of pores have formed on the RBC surface. This steady number of pores ranges from 515 at 5.88 nM to 11 657 at 14.7 nM. Given this situation it is instructive to define an effective lysis time constant, κ_l,eff = κ_l(n_p − n_pc) where n_p is the steady state value of the number of pores at a given value of C_in (Fig. 7). The value of κ_l,eff at 14.7 nM is 2.027 × 10⁻³ s⁻¹ which results in an effective lysis time constant of 493 s.

Fig. 7 The pore density is plotted as a function of time in the post-lysis regime. The number of pores (n_p) per RBC ranges from 515 at 5.88 nM to 11 657 at 14.7 nM. λ = 2 and κ_l = 1.8 × 10⁻⁷ s⁻¹, n_pc = 392.

4.2.3. Oligomer and pore concentration

In both the low (<5.44 nM) and high toxin (≥5.44 nM) regimes, the amount of toxin is found to be limiting. Even at the highest toxin concentration, C_in = 14.7 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 governing equations for the formation of an l-mer are given in eqn (6). Since pore formation occurs via a sequential oligomerization 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 concentration reduces to zero and lag times are observed for the formation 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 14.7 nM the number of monomers per ml is 8.854 × 10¹². The initial number, N_RBC = 3.2 × 10⁷, the monomers per RBC is 2.76 × 10⁵. If all the monomers were converted to pores, each RBC would have 2.3 × 10⁵ pores. However the number of pores formed per RBC at 14.7 nM is 1.1657 × 10⁵ pores (Fig. 7) indicating that about 50% of the mers remain on the membrane surface as intermediate n-mers (n = 1–11). At a sublytic concentration of 5.44 nM only about 4.5% of the mers are converted to pores resulting in 375 pores per RBC (Fig. 5a).

Fig. 8 Model predictions for oligomer concentration profiles (p_l) for λ = 2 and κ_l = 1.8 × 10⁻⁷ s⁻¹. (a) Initial bulk concentration = 5.44 nM. (b) Initial bulk concentration = 14.7 nM. Steady state distribution of oligomers for (c) C_in = 5.44 nM, (d) C_in = 14.7 nM.

The effect of λ 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 λ is increased from 2 to 4. The steady state concentrations of various n-mers as predicted by the model are illustrated in Fig. 8c and d. A change in the value of λ results in a shift in the distribution for a particular initial toxin concentration. An increase in λ implies an increase in the conformational time relative to the adsorption and reaction times. Hence as λ 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 concentrations (Fig. 8c) the distribution of higher mers and consequently 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 shifts towards the higher mers, thereby increasing the number of pores on the surface.

4.2.4. Parameter sensitivity

We briefly summarize the results of simulations carried out to test the influence of the estimated parameters on the model predictions. In the absence of monomer membrane binding equilibria, the value of saturated surface concentration (m^s) is unknown. In order to test the influence of m^s on the model prediction, we carried out a few simulations for m^s = 1 × 10⁻⁸ mol m⁻² and m^s = 1 × 10⁻¹⁰ mol m⁻² for various C_in values. For m^s = 1 × 10⁻⁸ mol m⁻², a negligible number of pores were formed and the H_max–C_in data (Fig. 3a) is underpredicted. At m^s = 1 × 10⁻¹⁰ mol m⁻², pore formation was extremely rapid and little variation in pore density between C_in = 8.82 nM and 14.7 nM was observed. As a consequence H_max–C_in data (Fig. 3a) is grossly overpredicted. Hence a value of m^s = 1 × 10⁻⁹ mol m⁻² was used in the simulations (Fig. 6). We further note that the amount of saturated surface concentration, m^s implicitly changes the reaction rate constant k_l (eqn (12)). Increasing m^s effectively decreases the reaction rate constant (eqn (12))

Once τ_net is fixed, n_pc is related to λ through the relation n_pc = aλ^b with the constants a and b being fixed for a given initial toxin concentration C_in (Fig. 5). We have found that 2 ≤ λ ≤ 4 fits the hemoglobin release data very closely (Fig. 6) and although the hemoglobin released as a function of time is slightly underestimated by the model the agreement is reasonable. Upon increasing λ to 7, and keeping m^s = 10⁻⁹ mol m^s−1, the corresponding value of n_pc is 1349. With a value of κ_l = 1.3 × 10⁻⁷ s⁻¹, although H_max versus C_in data is accurately predicted, the hemoglobin versus time data is grossly underpredicted. Varying m^s between 10⁻⁸ and 10⁻¹⁰ mol m⁻² further deteriorated the prediction. Finally we point out that other functional forms of the dependence on n_p − n_pc in the cell lysis equation eqn (8) such as the unit step function or a quadratic dependence (n_p − n_pc)² only overestimated the H_max versus C_in data.

4.3. Sequential oligomerization with reversible kinetics

In the previous discussion we present the results for the oligomerization kinetics which was assumed to be irreversible. The sequential oligomerization model with reversible kinetics is,where k_f and k_b represent the forward and backward reaction rate constants. The kinetic equations are,

A reversible time constant can be defined from eqn (17) as. The ratio, R is defined as the ratio of forward to backward time constants (R = τ_f/τ_b) to study the effect of reversibility. 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 significantly 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 5.44 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 different R values in Fig. 9a and b for C_in = 14.7 nM. The steady state concentrations of ‘mers’ change from a predominantly 11-mer concentration to a predominant 1-mer concentration as R is varied between 0 and 1. For R < 1, we find that the concentration of pores (12 mers) are significantly higher than the intermediate ‘mer’ concentrations shown in Fig. 9b and range from 0.142 nmol m⁻² (R = 0) to 0.28 nmol m⁻² (R = 0.01). For R = 1, the concentration of 12 mers is 0.24 nmol m⁻² and decreases with a further increase in R. The predictions using reversible sequential kinetics for the H_max vs. C_in data (Fig. 9c) indicate greater deviation from the experimental data when compared with R = 0. These results indicate that the irreversible mechanism provides the best agreement with the experimental data. We point out that the critical number of pores, n_pc in the reversible framework is non-monotonic, ranging from 392 (R = 0) to 5128 (R = 1) for λ = 2.

Fig. 9 Oligomer distribution as a function of the ratio of forward and backward time constants, R = τ_f/τ_b for C_in = 14.7 nM. (a) R varies between 0 and 1. (b) R varies between 0 and 0.1. (c) Hemoglobin release data as a function of toxin concentration is shown for various R. R = 0 has the closest agreement with the experimental data (open squares).

4.4. 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 p_l (l < l ≤ n) oligomer undergoing non-sequential irreversible oligomerization are,where n represents the number of monomers in a pore. The relation between the critical number of pores and n_pc for non-sequential oligomerization is n_pc = 5439λ^0.15. The critical number of pores for λ = 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 14.7 nM (Fig. 10b) are similar when compared with the sequential oligomerization (Fig. 6b and d). A comparison of the fitted parameters between the sequential and non-sequential oligomerization mechanisms are given in Table 2.

Fig. 10 (a) Model predictions from the non-sequential mechanism. Simulated H_max − C_in (open circles) curves grossly overpredicts the experimental data (open squares). (b) Simulated hemoglobin release compares well with the experimental data. Dashed line, τ_net = 1 s, solid line, τ_net = 24 s.

Oligomer distributions obtained from the non-sequential mechanism (Fig. 11b) show an entirely different trend when compared to that obtained from the sequential mechanism (Fig. 8). In the sequential mechanism, the higher ‘mers’ attained a steady state once the protomer was depleted. In the non-sequential mechanism, the contribution to the dodecamers (12 mer) can be obtained from a large number of combinations of the lower oligomers, leading to a larger dodecamer or pore concentration. The time scale required to attain the oligomer steady state concentration (Fig. 11a) is about 20 s for an initial concentration of 14.7 nM. Similar time scales are observed in the sequential mechanism as well.

Fig. 11 Oligomer concentrations as a function of time are shown for (a) 5.44 nM and (b) 14.7 nM. Steady state oligomer concentrations are shown for (c) 5.44 nM and (d) 14.7 nM. At steady state, 1–4 mer concentrations are zero for both 5.44 and 14.7 nM. The 12-mer concentration at steady state is quite large compared to the other oligomer concentrations present in the system.

Further experiments are required to distinguish between the various mechanisms. Western Blot experiments conducted on Hemolysin E¹² and Clostridium perfringens ε toxins³¹ showed the presence of intermediate oligomers. On the other hand, single-molecule fluorescence imaging of α-hemolysin on a droplet interface bilayer showed the presence of only monomers and heptamers (pores).³²

7. 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 first order in the number of live cells and directly proportional to the pores in excess of the critical number of pores, n_pc. In the sublytic regime the number of pores is found to have a power law dependence on λ which is the ratio of conformational time to the reaction time. This leads to the construction of a “phase diagram” between the number of pores n_p and λ for different values of the initial toxin concentration. Comparing simulations with experimental data, the range of n_pc was 392–768 for the sequential mechanism and 5300–6300 pores for the non-sequential mechanism for 2 ≤ λ ≤ 4. The range of λ values is consistent with available experimental data on ClyA which indicates that the membrane-bound conformational step is slower than the preceding adsorption and subsequent oligomerization steps.¹⁰

From the model we are also able to comment on the time constants for the various processes. The time constant for pore formation 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 20–30 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 5.88–14.7 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 oligomerization. Comparison of three different 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 different in both cases with a negligible numbers of lower mers observed in the non-sequential oligomerization. These differences in the distribution of mers offer a fingerprint to identify the underlying mechanism for pore formation. Experiments which can determine the number of pores or the steady state ‘mer’ distributions on the membrane surface will shed light on the pathways for oligomerization and enable a more definite conclusion of the underlying kinetics. Further, the 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 manuscript is generic and could be recast with some variation to study the dynamics of other PFTs.

We briefly discuss some of the limitations of the model in its present form. The model is based on the mean field approximation where all cells are assumed to be identical and for the purpose of binding and oligomerization this is an adequate starting 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 different densities of pores. Preliminary experiments by varying the number of RBCs at a fixed 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 reached a steady state, which occurs within 30 s (Fig. 7a) cell lysis continues to occur at a fixed number of pores, proportional to n_p − n_pc. 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 irreversible 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 field approximation, the model proposed in this manuscript is able to capture the inherent time scales in the process of pore formation and rupture as well as predict the variation of hemoglobin release as a function of the initial toxin concentration observed in the experiments. We finally point out that the effect of temperature on the kinetics and lysis has not been investigated in this work. Preliminary experiments at 14.7 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 effects in these systems.

Our work has implications in optimizing dosage and developing novel drug delivery strategies based on PFTs. The observation 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 protocols.³⁶ 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 artificial 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.

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Footnote

† Present address: Department of Chemical Engineering, University of Texas at Austin, Texas, USA.

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