ARTICLE pubs.acs.org/Langmuir
Two-Dimensional Solvent-Mediated Phase Transformation in Lipid Membranes Induced by Sphingomyelinase Ling Chao,† Fei Chen,‡ Klavs F. Jensen,* and T. Alan Hatton* Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
bS Supporting Information ABSTRACT: The spatial pattern changes in model raft membranes during sphingomyelinase (SMase)-induced solventmediated phase transformation are characterized in terms of a model that combines three major kinetic processes suggested by experimental observations: the release of sphingomyelin (SM) by the dissolution of SM-enriched domains within the raft membrane, the diffusion of SM from the dissolution sites to the reaction site in a solvent-like fluid lipid phase, and the consumption of SM by the enzymatic reaction at this reaction site, termed an SMase feature. Such processes may be responsible for the control of morphological changes in cell membrane organization, which are suggested to influence the signal transduction through the cell membrane walls. The model predictions are shown to be consistent with our previously reported experimental results. We numerically evaluated the range of possible scenarios of spatial pattern change and provide analytical expressions for SM-domain-area change rates and total dissolution times for several limiting cases. The model results suggest that it may be possible to tune the pattern changes by adjusting the relative importance of each of the three kinetic processes, which can be discriminated through experimentally measurable time-dependent SM concentration distributions or SM-domain-area variations with time.
’ INTRODUCTION Solvent-mediated transformations between solid phases in a solvent can proceed by the growth of precipitates of a stable phase as a metastable phase dissolves,1 as has been well recognized in crystal growth.1,2 Under a number of conditions, these metastable phases may develop more quickly than the stable phase, resulting in the transient presence of metastablephase precipitates in a saturated solution. The solvent-mediated phase transformation starts later, after the stable phase has nucleated in the same solution. The growth of the stable phase consumes the dissolved substance and reduces its concentration in the solution to below saturation levels for the metastable phase, which then continues to dissolve. An understanding of the kinetic mechanism of a solvent-mediated phase transformation is highly desirable because the process is of fundamental importance and because it can have serious industrial implications because it controls the spatiotemporal phase states of the systems of interest.25 We have noted previously that the enzyme sphingomyelinase (SMase) can induce a type of membrane morphology change in 2-D lipid membranes similar to a solvent-mediated phase transformation.6 SMase, which occurs naturally in mammalian and bacterial cells, cleaves the ester bond of sphingomyelin (SM, a lipid abundant in cell membranes) to form the products ceramide (Cer) and phosphocholine. The SMase enzymatic reaction reorganizes the cell membrane heterogeneity by changing the membrane composition and can thereby influence r 2011 American Chemical Society
cellular processes.713 In our previous study, we added SMase to model raft membranes with preexisting domains rich in sphingomyelin and cholesterol (SM/Chol) and observed the formation of domains rich in sphingomyelin and ceramide (SMenriched domains) and domains rich in ceramide (Cer-enriched domains). Later, a long-lived intermediate state was observed and is attributed to the physical trapping of SM in the newly formed gel SM-enriched domains, which were found to be relatively inaccessible to SMase. The situation was resolved after the nucleation of an SMase-rich island, where SMase was hypothesized to process SM efficiently even at low SM concentration. We suggested that the SMase feature acts as a heterogeneous reaction site consuming SM and lowering its concentration in the background fluid lipid phase. The SMenriched domains dissolve once the SM in the fluidlipid phase falls below its saturation concentration. With time, as more and more SM is consumed at the SMase feature, more SM-enriched domains dissolve. After SM is converted to Cer at the SMase feature, Cer diffuses back to the fluidlipid phase, where new Cer-enriched domains nucleate and grow. We draw an analogy between the phenomena occurring after the SMase feature nucleates and a regular solvent-mediated phase transformation in which the background fluid phase is like a solvent, the SMReceived: April 28, 2011 Revised: June 6, 2011 Published: June 15, 2011 10050
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Langmuir enriched domains are the metastable phase and the Cer formed at the SMase feature due to the conversion of SM is likened to the more stable state. We have observed anisotropic dissolution pattern changes in the metastable SM-enriched domains during SMase-induced solvent-mediated phase transformation.6 Many SMase-induced phenomena have been reported in the literature,1418 but most of these studies focused only on how SMase interacts with model raft domains rich in sphingomyelin and cholesterol1518 and on how the molecular interactions between sphingomyelin and ceramide cause new domain formation.14 None of these studies addressed the dissolution or disappearance of the domains, even though dissolution ring patterns could be discerned in the reported images of membrane morphology changes induced by sphingomyelinase18,19 or by phospholipase A.20 In this work, we focus on the mechanisms of anisotropic dissolution pattern changes in the metastable SM-enriched domains during SMase-induced solvent-mediated phase transformation.6 The peculiar pattern change of membrane domains is of interest because a knowledge of lateral lipid membrane heterogeneity could provide insights into cell membrane organization.2123 It has been proposed that the segregation of various membrane domains in the cell membrane creates spatially segregated environments, resulting in suitable platforms for the signal transduction of many cellular processes.24,25 The pattern change could influence how the outside signal is transmitted to the cell through the cell membrane. This peculiar anisotropic pattern change has not been reported in regular 3-D solvent-mediated phase transformations to the best of our knowledge. Such spatiotemporal phase changes depend directly on the kinetic processes involved in the phase transformation because the relative rates of dissolution of the metastable phase and the growth of the stable phase determine the overall phase change.1,26 The diffusion process has been neglected in analyses of such systems because a uniform concentration of the transformed substance in the solvent suggests that diffusion is fast and is not a limiting factor in the overall phase transformation.27 In contrast, we have observed a clear SM concentration gradient from the dissolution site to the consumption site in the 2-D solventlipid phase during SMase-induced solvent-mediated phase transformation,6 indicating that diffusion plays an important role. The interplay between diffusion and the other two processes, dissolution and consumption, may be the reason for the peculiar pattern changes observed in our studies. We have formulated a simulation model to describe the interplay between these processes and their effects on the spatiotemporal pattern changes associated with the SM-enriched domains. The appearance and growth of the Cer-enriched domains were not included in the model because they occur after the SM is converted to Cer at the SMase-feature reaction site and do not seem to influence the SM-enriched domain pattern changes. Three experimentally observed kinetic processes have been incorporated into our model: the consumption of sphingomyelin (SM) due to the significant enzymatic reaction at an SMase feature; the dissolution of sphingomyelin-enriched (SM-enriched) domains; and the diffusion of sphingomyelin in the solventlipid phase from the dissolution site to the reaction site. The shrinkage of the SM-enriched domains with time, with rates depending on the time-varying SM concentration in the solventlipid phase, was accounted for by using a suitable boundary-tracking algorithm.28,29 The model predictions are consistent with our experimental observations.
ARTICLE
Figure 1. Schematic illustration of a membrane system at the beginning of the solvent-mediated phase transformation. The side length of the corralled membrane is 2L. The blue objects are SM-enriched domains, and the yellow object is an SMase feature. n1, n2, and n3 are the unit vectors denoting outward normal directions of the corral boundary, the SMase-feature boundary, and the SM-enriched domain boundaries, respectively. During SM-enriched domain dissolution, their boundaries move with time. For a portion of the domain boundaries with an arc length of Δs, the boundary velocity is denoted as vn3.
’ MODEL DEVELOPMENT Model Description. Figure 1 schematically illustrates a typical membrane morphology observed at the beginning of the SMaseinduced solvent-mediated phase transformation in a membrane system confined to a corral. Before the start of the phase transformation, the lipid membrane system contains SM-enriched domains (blue objects) and a background fluid lipid phase (solventlipid phase) (white background). The phase transformation is triggered by the nucleation of the SMase feature (yellow object). We assumed a 2-D geometry, with the domains taken to be circular disks and the SMase feature nucleating in the center of the membrane system. Lipids are assumed to dissolve at the boundaries of the SM-enriched domains, diffuse through the solventlipid phase, and undergo enzymatic reaction at the boundary of the SMase feature. Although this SMase feature has a 3-D structure, it connects to the solventlipid phase in the membrane through a 2-D boundary.6 The effective rate of consumption of the lipid per unit boundary length at the SMase feature is determined by the overall flux of SM to this 2-D boundary. We assume that this consumption is due primarily to the reaction at the SMase feature, which produces significant amounts of ceramide only after this feature nucleates, although small amounts of SM may also be consumed in the growth of the SMase feature itself. A more detailed characterization of the SMase feature would allow the consumption kinetics of SM at the SMase feature to be described more completely. The rate at which the boundary of an SM-enriched domain retreats with time during the phase transformation (represented as vn3 in Figure 1) is a function of the location relative to the SMase feature, the local SM concentration in the adjacent solventlipid phase, and the kinetics of the dissolution, diffusion, and reaction processes. The SMase-feature boundary does not change significantly during the phase transformation and was held constant in the simulations. Governing Equations. The dissolution rate of SM-enriched domains is influenced directly by the local SM concentration in 10051
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the adjacent solventlipid phase and the rate at which SM diffuses away from these domains. We used Fick’s law for dilute binary mixtures to describe the development of the SM spatiotemporal concentration profiles in the 2-D fluid solventlipid phase, assuming no convective flow in the solventlipid phase confined to the corralled membrane system30 ∂½SM ¼ D∇2 ½SM ∂t
are
D∇½SM 3 n1 jcorral ¼ 0
0
0
∇ C 3 n3 jSMdomain boundary ¼
νn3 ¼
kD ð½SMsat ½SMÞ ð½SMsolid ½SMsat Þ
ð6Þ
where [SM]solid is the SM concentration within the SM-enriched domains. In dimensionless form, the model equations for the normalized SM concentration in the solventlipid phase C¼
½SM ½SM0 ½SMsat ½SM0
ð7Þ
kG C ¼ DaG C ðD=LÞ
ð10Þ ð11Þ
kD ð1 CÞ¼ DaD ð1 CÞ ðD=LÞ ð12Þ
and the motion of the SM-enriched domain boundary can be expressed as 0
νn3 ¼
½SMsat kD ð1 CÞ¼ kDaD ð1 CÞ ð½SMsolid ½SMsat ÞðD=LÞ ð13Þ
The dimensionless variables introduced in these equations are τ¼
D∇½SM 3 n2 jSMasef eature ¼ RG ¼ kG ð½SM0 ½SMÞ ð4Þ
Under these boundary conditions, kG is an effective first-order enzymatic reaction rate constant per unit length of the 2-D boundary of the SMase feature, and kD is a first-order dissolution rate constant per unit length of the SM-enriched domain boundaries. The normal unit vectors, n1, n2, and n3, point outward from the corral boundary, the SMase-feature boundary, and the SM-enriched domain boundary, respectively (illustrated in Figure 1). [SM]0 is the threshold sphingomyelin concentration below which the enzymatic reaction at an SMase feature does not occur and can be equal to or larger than zero. The local rate of disappearance of SM-enriched domains is determined by the rate at which SM is released from the boundary to the solvent phase. If SM molecules do not accumulate at or diffuse tangentially along the interfaces, then the rate at which the boundary recedes (vn3) is equal to the loss of area per unit length of the boundary and can be expressed as
ð9Þ
∇ C 3 n2 jSMasef eature boundary ¼
ð3Þ
D∇½SM 3 n3 jSMdomains ¼ RD ¼ kD ð½SMsat ½SMÞ ð5Þ
C ¼ 1 at t ¼ 0 ∇ C 3 n1 jcorral ¼ 0
ð2Þ
where [SM]sat is the sphingomyelin concentration in the solvent lipid phase in equilibrium with the SM-enriched domains. For the boundary conditions, we assume no interfacial accumulation, no SM flux through the corral walls, a finite rate of dissolution of the SM at the SM-enriched domain boundaries, and a diffusion flux of SM to the SMase feature equal to the consumption rate of SM at that boundary as a result of the enzymatic reaction. We assumed the enzymatic reaction kinetics to be first-order in the local SM concentration in the solvent lipid phase relative to a threshold concentration [SM]0. The dissolution kinetics were also taken to be first order, with the dissolution driving force being the difference between the local SM concentration and the saturation concentration at the SM domain interface.
ð8Þ
0
ð1Þ
where t is time following nucleation of the SMase feature, [SM] is the sphingomyelin concentration, and D is the diffusion coefficient of SM in the solventlipid phase. The initial condition is ½SM ¼ ½SMsat at t ¼ 0
∂C 0 ¼ ∇ 2C ∂τ
t
∇ νn3 0 ; νn3 ¼ ; 1=L ðD=LÞ kG kD ; DaD ¼ ; DaG ¼ ðD=LÞ ðD=LÞ ½SMsat K ¼ ½SMsolid ½SMsat
L2 =D
;
0
∇ ¼
where τ is the normalized time, DaG is the Damk€ohler number relating the SM consumption rate to the diffusion rate (small Damk€ohler numbers correspond to slow consumption compared to diffusion), and DaD is the Damk€ohler number relating the SMenriched domain dissolution rate to the diffusion rate. K indicates the reciprocal of the SM storage capacity of the SM-enriched phase. Numerical Solution of Model Equations. The system of equations was solved using Comsol to determine the spatial concentration profiles and a Matlab code to track the movement of the domain boundaries on dissolution of the SM-enriched domains. The methodology is shown in the Supporting Information. In short, each domain boundary was discretized into a set of marker points, and the geometry of the region of interest was sent to the Comsol finite element solver to obtain the transient concentration profile, which was then used to update the location of each marker point using the equation of motion for the boundary. The marker points were redistributed over the domain boundaries at each step to increase the robustness of the simulation, to eliminate undesired topological changes, and to prevent local singularity problems.
’ RESULTS AND DISCUSSION During solvent-mediated phase transformation, the overall morphology changes depend on which of the three steps— dissolution, diffusion, or reaction—is rate-limiting. We first consider the results when only one of the three processes dominates and then consider the cases when two or all three of the processes contribute similarly to the overall dynamics of the system. 10052
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Figure 2. Modeling results with varying DaD and DaG and fixed K = 1. (a) DaD = DaG = 100 to represent a diffusion-limited system. (b) DaD = 0.01 and DaG = 1 to represent a dissolution-limited system. (c) DaD = 1 and DaG = 0.01 to represent a reaction-limited system. The cartoons on the left illustrate the SM concentration profile in the solventlipid phase away from an SMase feature (the smaller object on the y axis) and the dissolution of SMenriched domains (the other objects). The dotted lines qualitatively represent the original domain boundaries of the dissolving domains. The arrows indicate the direction of SM molecule flux. The color indicates the scaled SM concentration in the solventlipid phase, as shown in the color bar, where red indicates a higher concentration and blue indicates a lower concentration.
Table 1. Parameter Settings and Distinct Characteristics When the System Is Diffusion-Limited, Dissolution-Limited, or Reaction-Limiteda limiting process
a
DaG
DaD
dissolution features
average concentration
concentration gradient
diffusion (D/L, kG, kD)
100
100
sharp ring
medium
large
dissolution (kD , D/L, kG)
1
0.01
uniform
low
small
reaction (kG , D/L, kD)
0.01
1
relatively uniform
high
small
The averaged concentration is the spatially averaged concentration of the system when significant dissolution of SM-enriched domains occurs.
Figure 2 shows the modeling results obtained when only one of the three processes is rate-limiting; parameter values for the three cases are given in Table 1. The initial membrane morphology was set to mimic the experimental observations in membrane systems with 40 mol % SM at the beginning of the solventmediated phase transformation.6 The small, central white region is the SMase feature, and the other white regions denote the SMenriched domains, which dissolve with time. The scaled SM concentration in the solventlipid phase is indicated by the color, as shown in the color bar, where red indicates a higher concentration and blue indicates a lower concentration. The morphology evolution has several distinct characteristics when different processes are dominant, as summarized in Table 1. When diffusion is limiting (D/L , kG, kD), there is a sharp dissolution ring within which no domains exist, and the SM concentration spans the entire scaled range from [SM]sat at the dissolution site to [SM]0 at the reaction site. Beyond this ring, the domains are at their initial size and do not dissolve. Under these conditions, the time taken for the ring to move to a position
χ = R(t)/RSMase, where RSMase is the radius of the SMase feature, is derived in the Supporting Information to be ! 1 R½SMsolid + ð1 RÞ½SMsat ½ χ2 ln χ2 + ð1 χ2 Þ τ¼ 4χL 2 ½SMsat ½SM0 ð14Þ where χL = L/R SMase and R is the fractional area of the corral occupied by the SM domains. This solution is valid for ([SM]sat [SM]0 )/(R[SM]solid + (1 R)[SM]sat ) , 1. The SM domains are totally dissolved when R ≈ L (i.e., when χ = L/RSMase) so that !" 1 R½SMsolid + ð1 RÞ½SMsat τdissðdif f limÞ ¼ ln χL 2 4 ½SMsat ½SM0 1 + 1 ð15Þ χL 2 10053
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or, in dimensional terms,
!" 2 L2 R½SMsolid + ð1 RÞ½SMsat L tdissðdif f limÞ ¼ ln RSMase 4D ½SMsat ½SM0 2 RSMase + 1 ð16Þ L When the dissolution is limiting (kD , D/L, kG), the domains dissolve uniformly and the SM concentration in the solventlipid phase is relatively low, close to [SM]0. In this case, the change in the SM domain area with time is " ! #2 ½SMsat ½SM0 Adisslim ðtÞ ¼ π R0 kD ð17Þ t ½SMsolid and the SM domains are totally dissolved when ! R0 ½SMsolid tdissðdisslimÞ ¼ kD ½SMsat ½SM0
ð18Þ
where R0 is the initial domain radius. In dimensionless form, this equation can be expressed as ! ½SMsolid ð19Þ DaD τdissðdisslimÞ ¼ η ð½SMsat ½SM0 0 where η0 = R0/L. Under reaction-limiting conditions, (kG , D/R, kD), the domains dissolve relatively uniformly and the SM concentration is high at the saturation concentration [SM]sat, although there is slow growth of a depletion zone for the SM-enriched domains owing to the finite but small concentration gradient toward the SMase feature. If this slight gradient can be neglected, then the change in the area of an SM domain with time is ! k 2πR ½SM ½SM G SMase sat 0 t ð20Þ Arxnlim ðtÞ ¼ πR02 Ndomains ð½SMsolid where Ndomains is the total number of domains in the corral. The total dissolution time can therefore be estimated as ! ! ½SMsolid Ndomains R02 tdissðrxnlimÞ ¼ ð21Þ 2kG RSMase ½SMsat ½SM0 or, in dimensionless form, ψDaG τdissðrxnlimÞ ¼
! ½SMsolid η ½SMsat ½SM0 0
ð22Þ
where ψ = (R0Ndomains)/(2RSMase) is half the ratio of the total initial SM domain perimeter to that of the SMase feature. The factor of one-half, which arises naturally in the derivation, accounts for the fact that the domains decrease in size with time (i.e., the average domain size during the dissolution process appears in the definition of ψ). The timescales for the morphology changes in the three limiting cases shown in Figure 2 are very different because the normalized time used (τ) is based on the characteristic diffusion time in a corral. In a diffusion-limited system, the overall morphology change will occur on a timescale on the order of unity, but in the other two limiting cases, the values of the two
Damk€ohler numbers influence the time over which the overall morphology changes occur. Smaller Da values indicate lower rates of the limiting processes compared to diffusion and thus larger overall timescales for the non-diffusion-limited processes. The ratios of the three timescales are given by τdisslim 1 τrxnlim 1 ¼ ; ¼ ; DaD τdif flim ψDaG τdif flim
τrxnlim DaD kD ¼ ¼ τdisslim ψDaG ψkG
ð23Þ
When diffusion is nonlimiting, the overall dissolution time in the reaction-limited system is longer than that in the dissolutionlimited system, even though the Da values for the limiting steps are the same for these two cases. This difference occurs because the total boundary of the SM-enriched domains is significantly greater than that of the SMase feature, and the Da values are based on the rates per unit length of the respective interfacial boundaries. The SMase reaction-limited case should therefore be slower than the SM dissolution-limited case by a factor of approximately the ratio of their domain boundary lengths, as reflected in the parameter ψ, and is clearly evident in Figure 2. The size and shape of each SM-enriched domain change during the process and can vary depending on the location of the domains relative to the SMase feature and according to the nature of the rate-limiting process, as shown in Figure 3 for an initial morphology with ordered SM-enriched domains. The overall pattern is symmetric with respect to the origin, so only a quarter of the pattern is shown. The right-hand panels show plots of displacement versus time for the different portions of the interfaces. The rates of dissolution of the domains at each of the indicated faces can be obtained from the slopes of the curves. In a diffusion-limited system (Figure 3a), the dissolution of each domain is anisotropic, with those portions facing the SMase feature dissolving with time, whereas the far faces do not begin to dissolve significantly until the relevant domain itself is almost gone. The domains dissolve in a sequence away from the SMase feature, resulting in the sharp dissolution ring observed in the overall pattern change. For instance, the rapid dissolution of face C does not begin until domain AB is completely dissolved. Similarly, face E does not recede significantly until domain CD is lost. Note that the rate of dissolution of the B, D, and F positions is always slow because the corresponding domains dissolve fully right before their dissolution rates can increase. In a dissolution-limited system (Figure 3b), the dissolution rates are similar over the entire domain boundary, the dissolution pattern for each domain is isotropic, and all domains begin to dissolve at a similar time, resulting in a uniform dissolution pattern change as indicated by the fact that displacement curves for all regions are almost congruent. In a reaction-limited system (Figure 3c), the shape change of the distant domains is isotropic, similar to the situation in a dissolution-limited system, but is weakly anisotropic for domains close to the SMase feature; curves A and B are further apart than curves C and D, which in turn are further apart than curves E and F (which themselves are almost congruent). Combined Characteristics When More Than One of the Processes Dominates. In addition to the three single-processlimited scenarios, there are four other scenarios for the rates of the three processes: one case when all three processes are characterized by similar timescales and three cases when the rates of any two of the three processes are similar but are much slower than the rate of the third. We selected the two Da 10054
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Figure 3. Shape change of each SM-enriched domain during its dissolution when any one of three kinetic processes is limited. The round yellow object is an SMase feature, and the other objects are SM-enriched domains with different locations relative to the SMase feature. The dark lines indicate where the SM-enriched domain interface is located after each time step. The distance between two adjacent dark lines indicates the displacement of the SMenriched domain interface after the interval of each time step. The capital letters label the portions of interfaces where we measured the displacement versus time. The initial distances of A, B, C, D, E, and F from the SMase feature are set to be di, 2di, 3di, 4di, 5di, and 6di. (a) System with DaD = DaG = 100 and 0.05τ as the interval for each time step. (b) System with DaD = 0.01 and DaG = 1 and 0.5τ as the interval for each time step. (c) System with DaD = 1 and DaG = 0.01 and 5τ as the interval for each time step. All of the systems have K = 1.
Table 2. Dominant or Limiting Processes for Different Limiting Cases of DaG and DaDa DaD . 1
DaD ≈ 1
DaG .1 DaG ≈ 1
diffusion (DaD = 100, DaG = 100) reactiondiffusion (DaD = 100, DaG = 1)
dissolutiondiffusion (DaD = 1, DaG = 100) all three (DaD = 1, DaG = 10)
DaG ,1
reaction (DaD = 1, DaG = 0.01)
DaD , 1 dissolution (DaD = 0.01, DaG = 1) dissolutionreaction (DaD = 0.000167, DaG = 0.01)
a
There are seven different scenarios, and the numbers in parentheses are the parameter settings used to obtain the representative systems shown in the Results section.
parameters to fall into these four categories to obtain the representative morphology changes, as shown in Table 2. Note that the lower limiting values of DaG and DaD are not the same in the case when dissolution and reaction are both limited. To ensure that the two characteristic process times are balanced, noting that the rate constants embedded in DaG and DaD are based on the unit interfacial length, we decreased the dissolution Damk€ohler number DaD to allow for the fact that the total perimeter of the SM-enriched domains is larger than that of the SMase feature by using DaD = 1 and DaG = 10 instead of DaD = 1 and DaG = 1 to represent the system when all three processes occur at similar rates. Because diffusion is important
in this case, even though not dominant, not all SM-enriched domains start to dissolve at the same time but dissolve sequentially from the inner domains outward. Therefore, at any given time, not all of the SM-enriched domain interfaces are effective at releasing SM, and the ratio of the effective SM-enriched domain interface to the SMase-feature interface is not 60, which is the initial physical ratio but much smaller, which is the reason that we chose DaG = 10 instead of DaG = 60 for this case. The modeling results in Figure 4 show that systems with more than one limiting process exhibit a combination of the distinct characteristics of each of the single limiting cases. In the dissolution-diffusion limited system (Figure 4a), the concentration 10055
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Figure 4. Modeling results with varying DaD and DaG and fixed K = 1. (a) DaD = 1 and DaG = 100 represent a dissolution-diffusion limited system. (b) DaD = 100 and DaG = 1 represent a reaction-diffusion limited system. (c) DaD = 0.000167 and DaG = 0.01 represent a dissolution-reaction limited system. (d) DaD = 1 and DaG = 10 represent a system in which dissolution, reaction, and diffusion processes proceed at similar rates.
gradient is steeper than that observed in a dissolution-only limited process but is more gradual than that in a diffusionlimited system. In addition, the averaged concentration when most of the domains have dissolved is between the concentrations for a dissolution-limited system and a diffusion-limited system. Furthermore, the sequential dissolution of domains occurs, but the dissolution ring is not as sharp as with a diffusion-limited system. Each domain dissolves more isotropically because this process is influenced by the dissolutionlimiting factor. Similarly, the reactiondiffusion-limited system (Figure 4b) exhibits a more gradual concentration gradient and pattern change than does a diffusion-limited system (Figure 2a). The dissolution occurs primarily at high concentrations, as influenced by the reaction-limited process, and the change in the individual domains is more anisotropic than in a reactionlimited system (Figure 2c). When both dissolution and reaction are limiting (Figure 4c), the dissolution pattern is isotropic, the concentration gradient is small, and dissolution occurs primarily in the intermediate concentration range, reflecting a balance between the high concentration characteristic of a reactionlimited system and the low concentration characteristic of a dissolution-limited system. In this case, the area of an SM domain varies with time according to 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32 ! 2 u u AðtÞ 1 6t ½SMsat ½SM0 kD t 7 2 ¼ 15 4 ð2λ + 1Þ 4λ ½SMsolid πR02 2λ R0
ð24Þ
where, as before, λ = (ψDaG)/(DaD), and the overall dissolution time can be estimated from the equation ! ½SMsolid R0 tdiss ¼ ðλ + 1Þ ð25Þ ½SMsat ½SM0 kD as shown in the Supporting Information. In dimensionless form, this equation is ! ½SMsolid DaD τdissðdiss=rxnlimÞ ¼ ðλ + 1Þ η ð26Þ ð½SMsat ½SM0 Þ 0 When all three processes contribute similarly to the overall kinetics of the system (Figure 4d), a gradual dissolution pattern instead of a sharp dissolution ring develops, and the dissolution time and the concentration profile reflect the combined characteristics of the three single process-limited cases. Figure 4d is distinguished from Figure 4a in that the concentration range at which most domain dissolution occurs is higher when the reaction process becomes more important (or when DaG becomes smaller). The parameter K = ([SM]sat)/([SM]solid [SM]sat) gives the concentration of lipid in the solventlipid phase relative to the difference in lipid concentration between the SM-enriched domains and the solventlipid phase. When K f ∞, the SM content in SM-enriched domains is the same as that in the solventlipid phase (i.e., the two phases are indistinguishable), and the system should perform as if there were no SM-enriched domains in the solvent phase. However, if there is a high density 10056
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Figure 5. Modeling results with varying K in a diffusion-limited system (DaD = DaG = 100). The first row is a system in the absence of any SM-enriched domains.
of SM stored in the enriched domains, then more SM needs to be processed and delivered; therefore, more time is required for the overall process to be completed. Figure 5 shows the modeling results with varying K in a diffusion-limited system. (The results with varying K in the reaction-limited and dissolution-limited systems are in the Supporting Information, Figures S2 and S3). With K = 10 (i.e., [SM]solid = 1.1[SM]sat), the overall concentration profile change is very similar to the situation when there are no SM-enriched domains in the diffusion-limited and the reaction-limited cases (the first and second rows in Figures 5 and Supporting Information Figure S2). A comparison with the dissolution-limited system cannot be made because the dissolution rate cannot be specified in a system without domains. More importantly, we observed that the primary influence of K is on the overall processing time, which increases with decreasing K, (Figure 5, Supporting Information Figures S1 and S2), and not on the qualitative behavior of the system. Comparison of Modeling Results with the Experimentally Observed Morphology Evolution. Sharp dissolution rings have been observed in membrane systems with a 40/40/20 molar ratio of DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine)/SM (brain sphingomyelin)/Chol (cholesterol).6 To obtain the corresponding
time constant for the experimentally observed morphology evolution, we scaled the time of the morphology evolution with the diffusion characteristic time (L2/D), where L is the half-corral length of 25 μm. The diffusivity (D) of lipid molecules in the 40/ 40/20 DOPC/SM/Chol system is 1.98 μm2/s, as estimated by fluorescence recovery after photobleaching (FRAP) (details given in the Supporting Information). The morphology evolution with the corresponding times is shown in Figure 6a. The time taken to dissolve all of the SM-enriched domains in the membrane is about 50 min, which corresponds to about τ = 10. From the previous discussion, the sharp dissolution ring in the model occurs if and only if the system is diffusion-limited. In addition, when diffusion is the rate-determining step, the pattern changes with time scaled by the diffusion rate are not influenced very much by the exact values of DaD and DaG as long as both are significantly larger than unity. Therefore, we can use the system with DaD = 100 and DaG = 100 generally to represent diffusionlimited systems. In this case, the dissolution rate is determined solely by the parameter K when the model has a fixed initial geometry mimicking the experimental observation. A decrease in K results in a longer overall dissolution time because more SM is stored in the SM-enriched domains and more time is required for the overall dissolution to be completed. 10057
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Figure 6. Comparison of the experimentally observed morphology evolution with the model predictions. The white circles denote the propagation of the sharp dissolution ring in the experimental observations. (a) Experimental result from the 40/40/20 molar ratio of DOPC/brain SM/Chol membrane treated with 0.005 unit/mL SMase. The image is from one squared coral with a 50 μm side length. The bright feature is an SMase feature, where reaction occurs. The dark irregular domains are SM-enriched domains that dissolve with time. The smaller dark dots occurring later (especially obvious in the images at τ = 10 and 20) are ceramide-enriched domains, discussed elsewhere.6 (b) Modeling results for K = 0.05 in the diffusion-limited system (DaD = DaG = 100).
Figure 7. Variation of (left) the depletion zone radius with time for diffusion-limited systems and (right) the SM domain area with time as affected by the relative importance of dissolution and reaction processes in the absence of diffusional limitations. λ reflects the relative importance of the reaction and dissolution rates.
The overall dissolution patterns in a diffusion-limited system with K ≈ 0.05 are similar to the experimentally observed dissolution patterns, as shown in Figure 6, from which we can infer that the SM content in the enriched domains is about 20 times that in the solventlipid phase. Scenarios of Spatial Pattern Change. The simulation results show distinct characteristics of the spatial pattern and concentration profile changes depending on which of the three processes dominates the transitions. These pattern changes can be tuned by changing physiological parameters such as enzyme concentrations and membrane composition to adjust the relative importance of each of these three kinetic processes. The dominating processes in any particular system can, in principle, be ascertained by experimental monitoring of the temporal and spatial SM concentration distributions in the solventlipid phase and of the SM domain sizes with time; in the experimental results shown above, the process was clearly diffusion-controlled, but other patterns of change can also be obtained by adjustment of the experimental conditions. In the diffusion-limited system, a sharp dissolution ring occurs within which no domains survive but beyond which the domains do not dissolve; only domains located at this dissolution ring dissolve and the ring moves outward with time. The position of this ring as a function of time is shown in Figure 7a, where it is clear that the rate at which the depletion zone grows decreases with increasing time. In a dissolution-
limited system, the concentration is relatively uniform at, or close to, the low-concentration limit for SMase activity over the entire corral area because any SM leaving the domains is almost immediately reacted away at the SMase feature. When the reaction is limiting, the concentration is again relatively uniform, but in this case, it is at (or close to) the saturation value because any SM depleted by reaction at the SMase feature is quickly replenished by further dissolution of the SM domains. When more than one process dominates, the pattern changes and the concentration profile evolution reflects the characteristics of each of the single-process-limited cases depending on their relative importance. The analytical expressions obtained for the limiting cases indicate that the time dependence of the change in area of an SM domain depends on which is the dominating process. Figure 7b shows the variation of the area with time for different values of the parameter λ = (ψDaG)/(DaD), which reflects the relative importance of the reaction and the dissolution rates. Thus, for a reaction-limited case where λ , 1, the area of an SM domain changes linearly with time (as shown in eq 20) but parabolically for the dissolution-limited case, where λ . 1. The reason for these differences is that the perimeter of an SMenriched domain decreases when it dissolves, slowing down the overall dissolution rate, which is proportional to the perimeter of the domain. However, the perimeter of the reaction site is a constant, so the overall reaction rate does not change with time, 10058
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Langmuir and if the processes are reaction-limited, then the dissolution rate is not a factor because it adjusts to match the overall reaction rate. When the overall dissolution and reaction rates are of similar importance, the time dependence of the area change is more complicated than in the case of either the simple linear or parabolic expressions obtained when only one is dominating. The monitoring of spatial and temporal changes in the low concentrations of SM in the background solventlipid phase is technically difficult, and thus such concentration variations are not a reliable method for discriminating between the different scenarios. The functional dependence of the total SM domain area on time, which is more readily measurable experimentally, however, is a potentially more useful approach for delineating the effects of different physiological parameters on the tuning of these pattern changes. In this modeling study, we have considered only the disappearance of the SM domains and did not include the Cer domain growth. Experimentally, both processes have been observed to occur only after the nucleation of the SMase feature, with the Cer domain growth process occurring once SM is converted to Cer at the SMase-feature reaction site. Provided that the Cer generated at the SMase feature does not accumulate at the reaction site to hinder further reaction and the Cer-enriched domains do not recruit significant SM as they grow, the presence of these domains should not influence the spatial and temporal changes in SM domains and solventlipid phase concentrations with time. If desired, the formation and growth of the Cer domains with time can be incorporated readily into the model developed in this work.
’ CONCLUSIONS We have developed a model to characterize the spatial pattern changes of SM domains during sphingomyelinase (SMase)induced solvent-mediated phase transformation. The model combines three major kinetic processes—reaction, dissolution, and diffusion—and provides an enhanced understanding of the interplay among the three processes influencing the membrane morphology. The complex geometry of these membrane systems and the moving SM domain boundaries during dissolution were accounted for in the numerical solutions, and analytical expressions for SM domain total dissolution times have been derived for several limiting cases. The model predictions are shown to be consistent with our previously reported experimental results and suggest that it should be possible to tune the pattern change by adjusting the relative importance of each of the three kinetic processes; which process dominates can be determined experimentally through the measurement of the SM concentrations with time or of the functional dependence of the rate at which the domain area changes with time. In closing, it should be noted that the current results were obtained using a fixed initial membrane geometry and assuming first-order reaction and dissolution kinetics and that a more detailed analysis of geometrical factors, such as the domain number density and domain size, can be accommodated using this modeling approach. Moreover, reaction and dissolution kinetics are still unknown in the SMase-induced solventmediated phase transformation, but higher-order or more complex kinetic expressions can be incorporated directly into the model boundary conditions at the dissolution and reaction interfaces if desired.
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’ ASSOCIATED CONTENT
bS
Supporting Information. Details of the tracking algorithm used to calculate the moving boundaries of the SM domains, mathematical developments of the various limiting cases, an estimation of the SM diffusion coefficient within the solvent-like fluid lipid phase, and additional simulation results. This material is available free of charge via the Internet at http:// pubs.acs.org.
’ AUTHOR INFORMATION Present Addresses †
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan. ‡ Air Products, Allentown, Pennsylvania.
’ REFERENCES (1) Cardew, P. T.; Davey, R. J. Proc. R. Soc. London, Ser. A 1985, 398, 415–428. (2) Putnis, A.; Putnis, C. V. J. Solid State Chem. 2007, 180, 1783– 1786. (3) Scholl, J.; Bonalumi, D.; Vicum, L.; Mazzotti, M.; Muller, M. Cryst. Growth Des. 2006, 6, 881–891. (4) Wang, F.; Wachter, J. A.; Antosz, F. J.; Berglund, K. A. Org. Process Res. Dev. 2000, 4, 391–395. (5) Jaakko, A.; Paula, H.; Leena, P.; Hanna, K.; Veli Pekka, T.; Leena, C.; Jouni, H.; Jouko, Y.; Jukka, R. J. Pharm. Sci. 2006, 95, 2730–2737. (6) Chao, L.; Gast, A. P.; Hatton, T. A.; Jensen, K. F. Langmuir 2010, 26, 344–356. (7) Kolesnick, R. N.; Goni, F. M.; Alonso, A. J. Cell. Physiol. 2000, 184, 285–300. (8) Hannun, Y. A. Science (Washington, DC) 1996, 274, 1855–1859. (9) Marchesini, N.; Hannun, Y. A. Biochem. Cell Biol. 2004, 82, 27–44. (10) Chatterjee, S. Chem. Phys. Lipids 1999, 102, 79–96. (11) Kolesnick, R. N.; Kronke, M. Annu. Rev. Physiol. 1998, 60, 643– 665. (12) Levade, T.; Jaffrezou, J.-P. Biochim. Biophys. Acta, Mol. Cell Biol. Lipids 1999, 1438, 1–17. (13) Posse de Chaves, E. I. Biochim. Biophys. Acta, Biomembr. 2006, 1758, 1995–2015. (14) Castro, B. M.; de Almeida, R. F. M.; Silva, L. C.; Fedorov, A.; Prieto, M. Biophys. J. 2007, 93, 1639–1650. (15) Chiantia, S.; Kahya, N.; Ries, J.; Schwille, P. Biophys. J. 2006, 90, 4500–4508. (16) Sot, J.; Ibarguren, M.; Busto, J. V.; Montes, L. R.; Goni, F. M.; Alonso, A. FEBS Lett. 2008, 582, 3230–3236. (17) Yu, C.; Alterman, M.; Dobrowsky, R. T. J. Lipid Res. 2005, 46, 1678–1691. (18) Johnston, I.; Johnston, L. J. Biochim. Biophys. Acta, Biomembr. 2008, 1778, 185–197. (19) Johnston, I.; Johnston, L. J. Langmuir 2006, 22, 11284–11289. (20) Grainger, D. W.; Reichert, A.; Ringsdorf, H.; Salesse, C. FEBS Lett. 1989, 252, 73–82. (21) Munro, S. Cell 2003, 115, 377–388. (22) Mayor, S.; Rao, M. Traffic (Oxford, United Kingdom) 2004, 5, 231–240. (23) Shaw, A. S. Nat. Immunol. 2006, 7, 1139–1142. (24) Simons, K.; Ikonen, E. Nature (London) 1997, 387, 569–572. (25) Simons, K.; Toomre, D. Nat. Rev. Mol. Cell Biol. 2000, 1, 31–39. (26) Davey, R. J.; Cardew, P. T.; McEwan, D.; Sadler, D. E. J. Cryst. Growth 1986, 79, 648–653. (27) Jackson, K. A. Kinetic Processes: Crystal Growth, Diffusion, and Phase Transitions in Materials; Wiley-VCH: Weinheim, Germany, 2004. 10059
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(28) Chen, F.; Smith, K. A.; Hatton, T. A. AIChE 2007 Annual Meeting. (29) Sethian, J. A. Level Set Methods and Fast Marching Methods, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1999. (30) Deen, W. M. In Analysis of Transport Phenomena; Oxford University Press: New York, 1998; pp 3063.
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