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[EQUATION] [EQUATION] and [MATH] satisfies equation ( 5.53 ) in the sense that [EQUATION] for any [MATH] and [MATH] Theorem 3 Assume that |
[EQUATION] [EQUATION] and initial data [MATH] is non-negative satisfying [EQUATION] [EQUATION] then the problem ( 5.53 )–( 5.58 ) admits a unique positive solution in the sense of Definition 5.3 |
The proof of Theorem is similar to that of Theorem Acknowledgements The authors would like to thank the organizers and participants of the Industrial Problem Solving Workshop (2016, Fields Institute) where the problem studied in this article was originally proposed, especially Rob Andrews, Almut Burchard, Itamar Halevy... |
Appendix A On super- and sub-solutions The second order parabolic equation admits super- and sub-solutions. Here we show how to constract them for a class of bounded initial values [MATH] By the comparison principle this implies that [MATH] First of all, we consider the problem for the sub-solution: |
[EQUATION] where [EQUATION] with initial and boundary conditions [EQUATION] [EQUATION] for all [MATH] [EQUATION] for all [MATH] . The problem ( A.1 )–( A.3 ) has a particular solution |
[EQUATION] where [MATH] satisfies [EQUATION] The problem ( A.1 ), ( A.4 ) has a particular solution [EQUATION] where [MATH] is the solution of the following equation |
[EQUATION] If [MATH] then [EQUATION] provided [EQUATION] Next, we consider the problem for the super-solution: [EQUATION] where [EQUATION] |
with initial and boundary conditions [EQUATION] [EQUATION] for all [MATH] [EQUATION] for all [MATH] The problem ( A.5 )–( A.7 ) has a particular solution |
[EQUATION] where [MATH] satisfies [EQUATION] The problem ( A.5 ), ( A.8 ) has a particular solution [EQUATION] where [MATH] satisfies [EQUATION] Now we need to impose conditions for periodicity of sub-solutions. If [MATH] then [EQUATION] provided [EQUATION] |
# Source: arxiv 1806.02468 # Title: Spatially Resolving the Condensing Effect of Cholesterol in Lipid Bilayers # Sections: all # Downloaded: 2026-03-03T05:16:02.767704+00:00 |
Spatially Resolving the Condensing Effect of Cholesterol in Lipid Bilayers Abstract We study the effect of cholesterol on the structure of dipalmitoylphosphatidylcholine (DPPC) phospholipid bilayers. Using extensive molecular dynamics computer simulations at atomistic resolution we observe and quantify several structur... |
Introduction Lipid bilayers are fascinating materials due to their unusual physical properties and to their role as the foundation of biological membranes. While the behavior of pure bilayers is relatively well understood, that of mixed bilayers continues to be an important topic of ongoing research. Cellular membranes... |
Of fundamental interest are binary lipid bilayers that contain a single type of phospholipid and cholesterol. It has been known for nearly a century that cholesterol significantly changes the properties of lipid films Leathes ( 1925 . A prominent example is the condensing effect, which describes the decrease in lateral... |
In this contribution we spatially resolve the ordering effect of a single cholesterol molecule on lipids in a dipalmitoylphosphatidylcholine (DPPC) bilayer. We do so by analyzing extensive all-atom molecular dynamics (MD) computer simulations. By considering the infinite dilution limit we can unambiguously determine th... |
Our approach also provides novel opportunities to test phenomenological models of cholesterol–phospholipid interactions. For example, the umbrella model argues that cholesterol molecules minimize unfavorable solvent interactions by associating with phospholipids whose large polar head groups contribute to the shielding... |
Before discussing the results of our spatially-resolved analysis of single-cholesterol simulations, we first present results for the average structural properties of DPPC bilayers that contain from 0 to 50% cholesterol. We will later use these results as a basis for comparison of the single-cholesterol simulations. The... |
II Methods II.1 Computer Simulations Initial configurations for MD simulations were constructed using the Membrane Builder module of the Charmm-Gui molecular modeling server, Version 1.7 Wu et al. 2014 ); Lee et al. 2016 . Each system consisted of a symmetric lipid bilayer with [MATH] DPPC and [MATH] cholesterol molecu... |
All simulations were performed using the Gromacs 5.1 molecular dynamics simulation software Abraham et al. 2015 with a time step of 2 femtoseconds. Initial energy minimization and equilibration were performed according to Membrane Builder’s default settings. A system temperature of [MATH] was maintained by a Nose-Hoove... |
The cholesterol mole fraction [MATH] of the studied bilayers ranges from 0 to 50%. The lengths of the simulated trajectories are [MATH] for [MATH] [MATH] for [MATH] [MATH] for [MATH] , and [MATH] each for [MATH] |
II.2 Data Analysis For each simulated trajectory several globally averaged bilayer properties were computed. Bilayer thickness was calculated as the combined distance of phosphorus atoms of lipids from the two leaflets from a configuration’s bilayer mid-plane. The extent of interdigitation of the two leaflets was compu... |
[EQUATION] where [MATH] are the mass density profiles of the phospholipids in the upper and lower leaflet, respectively. The ordering of the phospholipid tails was measured by calculating the order parameter |
[EQUATION] for every carbon atom in both acyl chains. Here [MATH] is the angle between the carbon-hydrogen bond vector and the outward vector normal to the bilayer, which we take to be parallel to the coordinate system’s [MATH] -axis. This order parameter is the equivalent of the one frequently measured in deuterium NM... |
To quantify the surface area per lipid we use the two-dimensional Voronoi tessellation of each leaflet with the same procedure as Pandit and coworkers Pandit et al. 2004 . This tessellation is constructed from a chosen set of vertices. It partitions the total area into segments, each of which is assigned to a specific ... |
Uncertainties were estimated using the block average method described by Flyvbjerg and Petersen Flyvbjerg and Petersen ( 1989 and are often smaller than the point size used in the figures. |
III Results: Average bilayer properties III.1 Thickness and Interdigitation Figure shows how the thickness of the membrane, defined as the average distance between the phospholipids’ phosphorus atoms of the two leaflets, varies with the cholesterol mole fraction [MATH] . As expected, we find that the bilayer thickness ... |
The increase in membrane thickness with cholesterol content is a well-known property of phospholipid bilayers. More surprising is the slight thinning at high cholesterol levels near [MATH] . One can envision multiple potential mechanisms for this behavior. First, the two leaflets of the bilayer might interdigitate unde... |
To test the first hypothesis we we calculated the extent [MATH] of interdigitation, defined by ( ). As shown in Figure we find that [MATH] initially decreases with increasing cholesterol content, but stays essentially flat for [MATH] . This suggests that increased interdigitation is not the reason for the slight decrea... |
III.2 Head Group Orientation To test whether changes in the phospholipids’ head group tilt angle contributes to changes in bilayer thickness we calculate the probability distribution of the inclination [MATH] , defined as the angle between the vector from the phosphorus to the nitrogen atom and the outward membrane nor... |
The invariance of head group orientations with cholesterol content seems to be inconsistent with the umbrella model, which posits that the phospholipids shield cholesterol molecules from contact with water by extending their large head groups over their smaller cohabitants. Our data shows that this effect, if it exists... |
III.3 Lipid Tail Order To measure changes in the ordering of the phospholipid tails we compute the order parameter [MATH] for each carbon atom in DPPC’s two alkyl chains. As shown in Figure , we find that the entire hydrophobic tail initially becomes more ordered as the cholesterol content is increased, as expected fro... |
The partial disordering of the phospholipid tails near the bilayer center can be understood by considering the relative lengths of cholesterol and DPPC molecules. As the height of the bilayer increases, the cholesterols’ hydroxyl groups remains close to the phospholipids’ head groups to form a polar surface that intera... |
III.4 Area per Lipid Perhaps the most striking effect that cholesterol has on phospholipid bilayers, and the one that gives the condensing effect its name, is the apparent decrease in molecular area as cholesterol is added to a pure lipid bilayer. Since its discovery Leathes ( 1925 it has been measured by numerous meth... |
The most apparent measure of the condensing effect is the area per lipid [MATH] , which is given by [EQUATION] where [MATH] the ensemble average of the bilayer area and [MATH] is the total number of lipid molecules (phospholipids and cholesterol) in each leaflet. This property depends on the composition of the bilayer,... |
As pointed out by Edholm and Nagle Edholm and Nagle ( 2005 , valuable information is contained in the nonlinearity of [MATH] . Because the average bilayer area is an extensive function of the number of phospholipids and the number of cholesterol molecules, it follows from Euler’s homogenous function theorem that it can... |
[EQUATION] where [EQUATION] are the partial specific areas of phospholipids and cholesterol, respectively. These properties depend on the composition of the bilayer, and one finds that Edholm and Nagle ( 2005 |
[EQUATION] where [MATH] is the derivative of the average area per lipid with respect to cholesterol mole fraction. Figure shows our results for these quantities, where we have used a finite difference scheme to estimate the derivative [MATH] . It follows from the equations above that the partial specific area of DPPC i... |
At intermediate cholesterol concentration (10% and 14%) both [MATH] and particularly [MATH] show large fluctuations. We caution the reader that this is the region in which [MATH] has the largest curvature, and the estimator of the derivative might therefore be unreliable. A more accurate calculation would require addit... |
An alternative approach to describe the effect of cholesterol on the area occupied by the different types of lipids is to partition the total bilayer area into segments that are considered occupied by a single molecule. This can be done, for example, using a Voronoi tessellation as described in the Methods section. In ... |
[EQUATION] where [MATH] and [MATH] are the segment areas averaged over all frames and all DPPC and cholesterol molecules, respectively. This expression for the average bilayer area is superficially similar to equation ( ), even though the partial specific areas and the Voronoi areas are substantially different quantiti... |
Our results for the Voronoi areas [MATH] and [MATH] are included in Figure . While they change less dramatically with cholesterol content than their partial specific area counterparts, their variation suggests another trend: at low cholesterol levels, the average area of DPPC decreases more significantly than that of t... |
IV Results: Local phospholipid properties Having established average bilayer properties for a wide range of cholesterol concentrations we now turn to the question to what extent a single cholesterol affects the phospholipids in its environment. We address this question by calculating some of the order parameter introdu... |
We perform the analysis on the trajectory with [MATH] cholesterol and [MATH] phospholipid molecules per leaflet, and average all properties over both leaflets of the bilayer as before. While this bilayer has an average cholesterol mole fraction of 2% it is not clear whether this small system is a good approximation of ... |
Because there is only a single cholesterol molecule in the simulated bilayer there are also very few phospholipids at any given separation from the cholesterol, and there is very little data available to average over. We address this difficulty by using a much longer trajectory of [MATH] than those used in the previous... |
We begin our analysis by grouping DPPC molecules by their distance from the cholesterol molecule. The Voronoi tessellation provides a natural spatial ordering of lipids: those DPPC molecules whose Voronoi segments share an edge with that of cholesterol form the first solvation shell, those phospholipids that share an e... |
As a separate measure of local structure we also compute the cholesterol–DPPC pair correlation function [MATH] , also shown in Figure . This function shows a peak up to a separation of [MATH] that indicates enhanced local structure, whereas [MATH] adopts is plateau value of unity at larger distances. At small distances... |
Based on this information we define six distinct regions of increasing distance from the cholesterol molecule, indicated in Figure . We divide the range of the peak in the pair correlation function into three regions, and group larger distances into another set of three regions. We then calculate for each lipid the are... |
IV.1 Area per Lipid Figure shows the average Voronoi area of the phospholipids in each of the six regions of different separations from the cholesterol molecule. We find that a DPPC molecule in the first region occupies an average area of [MATH] , whereas a phospholipid in the last region with an average distance of [M... |
This data illustrates that even a single cholesterol molecule has a significant condensing effect on the phospholipids in its immediate environment. A DPPC molecule in close proximity to the cholesterol occupies a similar, and even slightly smaller, area than it would typically in a bilayer with 10% cholesterol content... |
The range of this local condensing effect is limited to one or two solvation shells of the cholesterol molecule; at distances exceeding [MATH] the phospholipids’ area is essentially indistinguishable from that in pure bilayer. |
IV.2 Lipid Tail Order A similar but weaker condensing effect can be observed when measuring the ordering of the phospholipid tails in a distance-resolved manner. As shown in Figure , DPPC molecules in the region closest to the cholesterol show a similar profile of the tail order parameter [MATH] as the average phosphol... |
Given the simple argument put forth in the previous section it is surprising that the ordering of the phospholipid tails in the immediate environment of a cholesterol molecule is less pronounced than in a bilayer with a cholesterol content of 6, let alone 10 or 16, per cent. This suggests that unlike the Voronoi area, ... |
IV.3 Head group orientation Finally we study how the presence of a single cholesterol molecule affects the orientation of the head groups of the phospholipids in its environment. As before (Figure ) we calculate the probability distribution of the inclination angle [MATH] of the phosphate–nitrogen separation vector. Ho... |
We find that the distribution of the inclination is essentially the same as in a pure phospholipid for all but the closest region to the cholesterol molecule. In the latter case the distribution is shifted to slightly larger inclination angles, suggesting that the head groups are tilted more towards the bilayer plane. |
More interesting is the distribution of the azimuth, shown in the bottom panel of Figure . We find that phospholipids in the first three regions, extending to [MATH] from the cholesterol, have a clear tendency to orient their polar head groups in the direction of the sterol. This result is a direct observation of the m... |
As is expected, phospholipids farther away from the cholesterol show no preference for any particular orientation with respect to the cholesterol, as evidenced by completely uniform probability distributions. |
Discussion Our study of lipid structure in binary DPPC/cholesterol bilayers is consistent with most previous work on the condensing effect, and it highlights several new properties that emerge in these complicated systems. Our analysis of average lipid order recapitulates the well-known effects of adding cholesterol to... |
Less well understood is the importance of leaflet interdigitation, which we find to decrease up to a composition of 30% cholesterol. Beyond that the extent of interdigitation remains approximately constant, in contrast to previous reports on unsaturated lipid bilayers. Whether this difference stems from the nature of t... |
The analysis of simulation trajectories that contain only a single cholesterol per leaflet provides a novel way to characterize the condensing effect. We find that the average area of phospholipids in the first and, to a lesser extent, in the second solvation shell of a cholesterol are lower than they would be in a pur... |
A similar but weaker ordering effect can be seen in the tail order parameter, measured as a function of distance from the cholesterol. We find that a phospholipid in close proximity displays an ordering that is typical for a bilayer containing 4% cholesterol. The fact that such a DPPC molecule, which by selection is su... |
The cholesterol also has a significant effect on the orientation of the head groups of nearby phospholipids. DPPC within about [MATH] of the cholesterol show a strong preference for orienting the choline group towards the cholesterol. This is accomplished by a rotation around the membrane normal axis rather than an inc... |
Our approach of studying bilayers with a single cholesterol molecule allows us to isolate the direct ordering effect on a nearby phospholipid from collective effects caused by interactions with other phospholipids that are themselves in contact with cholesterol, and so forth. Because there are only very few (around 6) ... |
As briefly mentioned in Section IV , a potential downside of this approach is that our simulations at very low cholesterol numbers might not be representative for larger systems with the same average cholesterol concentration. Several recent studies have highlighted the propensity of cholesterol to form transient clust... |
Acknowledgments This work was facilitated though the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington. |
# Source: arxiv 1806.02480 # Title: Pinned, locked, pushed, and pulled traveling waves in structured environments # Sections: all # Downloaded: 2026-03-02T08:58:01.467596+00:00 |
Pinned, locked, pushed, and pulled traveling waves in structured environments Abstract Traveling fronts describe the transition between two alternative states in a great number of physical and biological systems. Examples include the spread of beneficial mutations, chemical reactions, and the invasions by foreign speci... |
Introduction Propagation of waves, fronts, and pulses is a recurrent theme in natural sciences murray:mathematical_biology, . In evolution, they describe the geographic spread of a beneficial allele or the increase of fitness over time fisher:wave, kolmogorov:wave, tsimring:wave, . In ecology, they capture epidemics, i... |
Given the diverse settings in which traveling fronts occur, numerous approaches have been developed to model front propagation in specific systems. Most of these approaches can be grouped into four classes depending on whether space and time are treated as discrete or continuous (Fig. ). Discrete time typically represe... |
Patchy habitats arise naturally due to a low density of locations suitable for growth, due to turbulent flows in marine environments abraham:patchy_plankton, nelson:flow_prl_2012, , and due to internal ecological dynamics that create spatial patterns via Turing instabilities, ecological drift, and other mechanisms wils... |
I.1 Four classes of models with discrete or continuous space and time Continuous space, continuous time (CSCT) models arise naturally for chemical reactions and are typically formulated in terms of partial differential equations. In ecology and evolution, CSCT models describe organisms with overlapping generations livi... |
[EQUATION] where [MATH] is the population density that depends position [MATH] and time [MATH] [MATH] is the migration or dispersal rate; and [MATH] is the growth rate of the population. |
The only constraint on the functional form of [MATH] is that there must be a stable equilibrium at [MATH] , which is the carrying capacity. Mathematically, this constraint is expressed as [MATH] and [MATH] . An important property of [MATH] is the limit [MATH] , which is the per capita growth rate at low density. When [... |
Allee effects have profound implications for both the kinetics and evolutionary dynamics of range expansion murray:mathematical_biology, fife:allee_wave, birzu:semipushed, korolev:arrest, korolev:wave_splitting, hastings:invasion_review, roques:allee_diversity, . Most of these effects can be understood from the classif... |
fisher:wave, kolmogorov:wave, . In contrast, the dynamics of pushed waves depends on the population growth throughout the expansion front, and, therefore, the expansion velocity depends on all the details of [MATH] . As a result, there is no general expression for the velocity of pushed waves. One exception to this is ... |
[EQUATION] where [MATH] controls the strength of an Allee effect, [MATH] controls the growth rate, and [MATH] is the carrying capacity. When [MATH] , the Allee effect is strong, and [MATH] is the Allee threshold, i.e. the minimal density required for growth. The expansion velocity is known for all values of the Allee t... |
[EQUATION] Note that [MATH] marks the transition between pulled and pushed expansions. For [MATH] , the shape of the front is also known exactly aronson:allee_wave, hadeler1975travelling, fife:allee_wave, korolev:wave_splitting, . For both pulled and pushed expansions, [MATH] . We illustrate this dependence in Fig. usi... |
Continuous space, discrete time (CSDT) models describe species with strong seasonality such as annual plants that spread in a spatially homogeneous landscape. These models are typically expressed in terms of integrodifference equations that specify how population densities change from generation to generation: |
[EQUATION] where [MATH] is the population density as a function of position [MATH] at discrete generation [MATH] [MATH] is the dispersal kernel that specifies the probability of dispersal to position [MATH] from position [MATH] ; and [MATH] describes the growth dynamics. The kernels that are sufficiently short-range, e... |
To visualize the dynamics in CSDT models, we used exponential (Laplace) dispersal kernel and a simple piecewise-linear growth model, which are defined below |
[EQUATION] [EQUATION] where [MATH] quantifies dispersal distance, [MATH] is the growth rate at low densities, [MATH] is the carrying capacity, and [MATH] is the density sufficient to reach the carrying capacity. When [MATH] , the growth function is continuous, and there is no Allee effect because the per capita growth ... |
Similar to CSCT models, one can distinguish between pulled and pushed waves. Similar to CSCT models, the velocity of pulled waves depends only on the growth dynamics at low densities [MATH] . The role of dispersal is, however, more complex and [MATH] depends on all the details of [MATH] . For the Laplace kernel defined... |
[EQUATION] Because one can always rescale the spatial coordinate to eliminate [MATH] , the expansion velocity is proportional to [MATH] for both pulled and pushed waves. We illustrate this dependence in Fig. using [MATH] and [MATH] defined above. |
Discrete space, continuous time (DSCT) models are appropriate for fragmented landscapes with little or no seasonal forcing. The dynamics are described by a set of ordinary differential equations coupled by dispersal, which, in the simplest case, occurs only between the nearest patches: |
[EQUATION] The dynamics of DSCT expansions is shown in Fig. using [MATH] defined by Eq. ( ). Note that [MATH] is no longer proportional to [MATH] in either pulled or pushed waves. The velocity of pulled waves depends only on [MATH] and [MATH] and is derived in Appendix A. In contrast to CSCT and CSDT models, the veloci... |
Discrete space, discrete time (DSDT) models form the remaining class that describes seasonal growth in fragmented landscapes. In this case, the dynamics are governed by difference equations, which are also known as coupled map lattices and cellular automata: |
[EQUATION] In addition to natural populations, DSDT models capture the dynamics in some experimental studies datta:wave_splitting, gandhi:pulled_pushed, . Moreover, DSDT model underlie many simulations because discretization is frequently deployed in numerical methods. |
The dynamics of DSDT models are illustrated in Fig. using [MATH] defined by Eq. ( ). The velocity of pulled waves is derived in Appendix A and depends only on [MATH] and [MATH] . The velocity of pushed waves exhibit a striking phenomenon of velocity locking: There is a discrete set of velocities at which [MATH] remains... |
carretero:locking_physica, carretero:locking_pre, fernandez:jsp, coutinho:extended, coutinho:convolution, coutinho:monotone, turzik:stability, . A detailed study of the ecological and evolutionary implications of the locking phenomenon is the central topic of this paper. |
I.2 Locking in DSDT models Despite significant differences, the models in all four classes can describe the propagation of a traveling front. As a result, the choice of the model is often dictated not only by its relevance to a specific population, but also by its mathematical or computational tractability petrovskii:e... |
Some phenomena, however, arise most naturally only in a specific model class. One important example in invasion pinning (Fig. ), which occurs when the front gets stuck at a particular patch or landscape heterogeneity keitt:pinning, fath:pinning, . Invasion pinning requires habitat fragmentation and the presence of a st... |
Although locking of traveling fronts may seem peculiar, it has been observed for a Belousov-Zhabotinsky reaction spreading in a periodic array of periodically-driven vortices paoletti:epl_front_locking, . Moreover, the characteristic pattern of plateaus shown in Fig. was found in a variety of physical systems including... |
The theory of mode locking has been used in the context of coupled map lattices to explain the locking of invasion velocities, and some rigorous results are available on the existence and properties of locked fronts carretero:locking_physica, carretero:locking_pre, fernandez:jsp, coutinho:extended, coutinho:convolution... |
II Results II.1 Velocity locking occurs in many ecological models We studied a variety of ecological growth models to determine whether velocity locking is a generic phenomenon. Although the piecewise-linear model in Eq. ( ) is a reasonable approximation of ecological dynamics, it not clear whether velocity plateaus in... |
The Beverton-Holt model with offset beverton_holt:model, chen:allee_model, is commonly used to model population dynamics, e.g. of fisheries, and is specified by the following equation: |
[EQUATION] where [MATH] sets the carrying capacity, [MATH] sets the density at which intraspecific competition starts to significantly affect population growth, and [MATH] sets the magnitude of the Allee effect. |
Although [MATH] in the Beverton-Holt model is continuous, it has a discontinuous derivative at [MATH] . To exclude the possibility that this non-analyticity is responsible for velocity locking, we considered an infinitely differential map, known as Hill function in molecular biology hill:function, . Hill-like functions... |
[EQUATION] where [MATH] and [MATH] play the same role as in the Beverton-Holt model, and [MATH] controls the strength of an Allee effect. The Allee effect is absent for [MATH] and increases with [MATH] for [MATH] |
Both Beverton-Holt and Hill models exhibited velocity locking, which is evident from the plateaus of [MATH] shown in Fig. . Thus, velocity locking is not caused by singularities in the growth function, although velocity plateaus are larger for discontinuous or rapidly varying [MATH] . The main conclusion that we draw f... |
II.2 Locked invasions are periodic Locking of the velocity into a specific value results in a periodic and often pulsed propagation of the invasion front (Fig. ). The pulsations occur because several generations of slow growth and dispersal are necessary to reach the density at which intraspecific facilitation enables ... |
In general, locked fronts advance by [MATH] patches every [MATH] generations carretero:locking_physica, carretero:locking_pre, (Fig. ), and there are only [MATH] distinct density profiles: one for each generation from [MATH] to [MATH] . As a result, invasion velocities take only rational values, [MATH] , when measured ... |
To verify the periodic nature of locked waves, we obtained an exact solution for the invasion dynamics inside [MATH] plateau for the piecewise-linear growth model (see Appendix B). Our solution matches both the population density profiles and the locations of the velocity plateaus obtained in simulation (Fig. ). The fa... |
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