P AOX1 expression in mixed-substrate continuous cultures of Komagataella phaffii (Pichia pastoris) is completely determined by methanol consumption regardless of the secondary carbon source.

The expression of recombinant proteins by the AOX1 promoter of Komagataella phaffii is typically induced by adding methanol to the cultivation medium. Since growth on methanol imposes a high oxygen demand, the medium is often supplemented with an additional secondary carbon source which serves to reduce the consumption of methanol, and hence, oxygen. Early research recommended the use of glycerol as the secondary carbon source, but more recent studies recommend the use of sorbitol because glycerol represses P AOX1 expression. To assess the validity of this recommendation, we measured the steady state concentrations of biomass, residual methanol, and LacZ expressed from P AOX1 over a wide range of dilution rates (0.02-0.20 h-1) in continuous cultures of the Mut+ strain fed with methanol + glycerol (repressing) and methanol + sorbitol (non-repressing). We find that under these conditions, the specific P AOX1 expression rate (measured as either specific LacZ productivity or specific AOX productivity) is completely determined by the specific methanol consumption rate regardless of the type (repressing/non-repressing) of the secondary carbon source. In both cultures, the specific P AOX1 expression rate is proportional to the specific methanol consumption rate, provided that the latter is below 0.15 g/(gdw-h); beyond this threshold consumption rate, the specific P AOX1 expression rate of both cultures saturates to the same value. Analysis of the data in the literature shows that the same phenomenon also occurs in continuous cultures of Escherichia coli fed with mixtures of lactose plus repressing/non-repressing carbon sources. The specific P lac expression rate is completely determined by the specific lactose consumption rate, regardless of the type of secondary carbon source, glycerol or glucose.

Positive feedback exists and drives the glucose-mediated repression in Escherichia coli.

The expression of the lac operon of E. coli is subject to positive feedback during growth in the presence of gratuitous inducers, but its existence in the presence of lactose remains controversial. The key question in this debate is: Do the lactose enzymes, Lac permease and ß-galactosidase, promote accumulation of allolactose? If so, positive feedback exists since allolactose does stimulate synthesis of the lactose enzymes. Here, we addressed the above question by developing methods for determining the intracellular allolactose concentration as well as the kinetics of enzyme induction and dilution. We show that, during lac induction in the presence of lactose, the intracellular allolactose concentration increases with the lactose enzyme level, which implies that lactose enzymes promote allolactose accumulation, and positive feedback exists. We also show that, during lac repression in the presence of lactose + glucose, the intracellular allolactose concentration decreases with the lactose enzyme levels, which suggests that, under these conditions, the positive feedback loop turns in the reverse direction. The induction and dilution rates derived from the transient data show that the positive feedback loop is reversed due to a radical shift of the steady-state induction level. This is formally identical to the mechanism driving catabolite repression in the presence of TMG + glucose.

Inducer exclusion, by itself, cannot account for the glucose-mediated lac repression of Escherichia coli.

The lac operon of Escherichia coli is repressed several 100-fold in the presence of glucose. This repression has been attributed to cAMP receptor protein-mediated inhibition of lac transcription and EIIAGlc-mediated inhibition of lactose transport (inducer exclusion). The growing evidence against the first mechanism has led to the postulate that the repression is driven by inducer exclusion. Although inducer exclusion reduces the permease activity only 2-fold in fully induced cells, it could be more potent in partially induced cells. Here, we show that even in partially induced cells, inducer exclusion reduces the permease activity no more than 6-fold. Moreover, the repression is so small because these experiments are performed in the presence of chloramphenicol. Indeed, when glucose is added to a culture growing on glycerol and TMG, but no chloramphenicol, lac expression is repressed 900-fold. This repression is primarily due to reversal of the positive feedback loop, i.e., the decline of the intracellular TMG level leads to a lower permease level, which reduces the intracellular TMG level even further. The repression in the absence of chloramphenicol is therefore primarily due to positive feedback, which does not exist during measurements of inducer exclusion.

The Mut+ strain of Komagataella phaffii (Pichia pastoris) expresses PAOX1 5 and 10 times faster than Muts and Mut- strains: evidence that formaldehyde or/and formate are true inducers of PAOX1.

The methylotrophic yeast Komagataella phaffii is among the most popular hosts for recombinant protein synthesis. Most recombinant proteins have been expressed in the wild-type Mut+ host strain from the methanol-inducible alcohol oxidase (AOX) promoter PAOX1. Since methanol metabolism has undesirable consequences, two additional host strains, Muts (Δaox1) and Mut- (Δaox1Δaox2), were introduced which consume less methanol and reportedly also express recombinant protein better than Mut+. Both results follow from a simple model based on two widespread assumptions, namely methanol is transported by diffusion and the sole inducer of PAOX1. To test this model, we studied 14C-methanol uptake in the Mut- strain and ß-galactosidase expression in all three strains. We confirmed that methanol is transported by diffusion, but in contrast to the literature, Mut+ expressed ß-galactosidase 5- and 10-fold faster than Muts and Mut-. These results imply that methanol is not the sole inducer of PAOX1-metabolites downstream of methanol also induce PAOX1. We find that formate or/and formaldehyde are probably true inducers since both induce PAOX1 expression in Mut- which cannot synthesize intracellular methanol from formate or formaldehyde. Formate offers a promising substitute for methanol since it does not appear to suffer from the deficiencies that afflict methanol. KEY POINTS: • This is the first study to systematically compare all three Mut phenotypes as host strains. • Mut+ strain expresses 5- and 10-fold faster than Muts and Mut- strains. • Methanol is transported by diffusion in Komagataella phaffii. • Formate and formaldehyde are true and strong inducers of PAOX1 expression.

Urn models for stochastic gene expression yield intuitive insights into the probability distributions of single-cell mRNA and protein counts.

Fitting the probability mass functions from analytical solutions of stochastic models of gene expression to the single-cell count distributions of mRNA and protein molecules can yield valuable insights into mechanisms underlying gene expression. Solutions of chemical master equations are available for various kinetic schemes but, even for the basic ON-OFF genetic switch, they take complex forms with generating functions given as hypergeometric functions. Interpretation of gene expression dynamics in terms of bursts is not consistent with the complete range of parameters for these functions. Physical insights into the probability mass functions are essential to ensure proper interpretations but are lacking for models considering genetic switches. To fill this gap, we develop urn models for stochastic gene expression. We sample RNA polymerases or ribosomes from a master urn, which represents the cytosol, and assign them to recipient urns of two or more colors, which represent time intervals in which no switching occurs. Colors of the recipient urns represent sub-systems of the promoter states, and the assignments to urns of a specific color represent gene expression. We use elementary principles of discrete probability theory to solve a range of kinetic models without feedback, including the Peccoud-Ycart model, the Shahrezaei-Swain model, and models with an arbitrary number of promoter states. In the last case, we obtain a novel result for the protein distribution. For activated genes, we show that transcriptional lapses, which are events of gene inactivation for short time intervals separated by long active intervals, quantify the transcriptional dynamics better than bursts. We show that the intuition gained from our urn models may also be useful in understanding existing solutions for models with feedback. We contrast our models with urn models for related distributions, discuss a generalization of the Delaporte distribution for single-cell data analysis, and highlight the limitations of our models.

Analytical Expressions and Physics for Single-Cell mRNA Distributions of the lac Operon of E. coli.

Mechanistic models of stochastic gene expression are of considerable interest, but their complexity often precludes tractable analytical expressions for messenger RNA (mRNA) and protein distributions. The lac operon of Escherichia coli is a model system with regulatory elements such as multiple operators and DNA looping that are shared by many operons. Although this system is complex, intuition suggests that fast DNA looping may simplify it by causing the repressor-bound states of the operon to equilibrate rapidly, thus ensuring that the subsequent dynamics are governed by slow transitions between the repressor-free and the equilibrated repressor-bound states. Here, we show that this intuition is correct by applying singular perturbation theory to a mechanistic model of lac transcription with the scaled time constant of DNA looping as the perturbation parameter. We find that at steady state, the repressor-bound states satisfy detailed balance and are dominated by the looped states; moreover, the interaction between the repressor-free and the equilibrated repressor-bound states is described by an extension of the Peccoud-Ycart two-state model in which both (repressor-free and repressor-bound) states support transcription. The solution of this extended two-state model reveals that the steady-state mRNA distribution is a mixture of the Poisson and negative hypergeometric distributions, which reflects mRNAs obtained by transcription from the repressor-bound and repressor-free states. Finally, we show that the physics revealed by perturbation theory makes it easy to derive the extended two-state model equations for complex regulatory architectures.

Existence of a scaling relation in continuous cultures of Scheffersomyces stipitis: the steady states are completely determined by the ratio of carbon and oxygen uptake rates.

BACKGROUND: Recently, we showed that steady-state continuous cultures of S. stipitis follow the principles of growth on mixture of two complementary substrates. More precisely, when such cultures are fed with progressively higher concentrations of glucose s f at fixed dilution rate D = 0.1 h-1, oxygen mass-transfer coefficient k l a = 50 h-1, and oxygen solubility c o ∗ , they transition from glucose- to oxygen-limited growth through an intermediate dual-limited regime in which both glucose and oxygen are limiting, and ethanol is produced without loss of glucose. It is, therefore, of considerable interest to characterize the dual-limited regime. We found that the dual-limited regime occurs precisely when the operating parameters D, s f, k l a, and c o ∗ satisfy the relation Y os < D s f / k l a · c o ∗ < Y os ' , where Y os and Y os ' denote g of glucose consumed per g of oxygen consumed in the carbon- and oxygen-limited regimes. In this work, our goal was to determine if the above characterization of the dual-limited regime holds over a wider range of D, k l a, and to understand why the dual-limited regime is determined by the dimensionless ratio D s f / k l a · c o ∗ . RESULTS: To this end, we performed the foregoing experiments at three additional dilution rates (D = 0.07, 0.15, and 0.20 h-1) and one additional mass-transfer coefficient (k l a = 100 h-1). We find that the above characterization of the dual-limited regime is valid for these conditions as well. Furthermore, the boundaries of the dual-limited regime are determined by the dimensionless ratio D s f / k l a · c o ∗ , because the steady-state concentrations are completely determined by this ratio. More precisely, if the steady-state concentrations of biomass, glucose, oxygen, and ethanol are suitably scaled, they collapse into a single curve with D s f / k l a · c o ∗ as the independent variable. CONCLUSION: The dual-limited regime is characterized by the relation Y os < D s f / k l a · c o ∗ < Y os ' over the entire range of operating condition 0.07 h-1 ≤ D ≤ 0.20 h-1 and 50 h - 1 ≤ k l a ≤ 100 h - 1 . Since the effect of all operating parameters is embedded in the single parameter D s f / k l a · c o ∗ , the dimensionless plot provides a powerful tool to compare, with only a handful of data, various ethanol-producing strains over a wide range of operating conditions.

Quantifying the parametric sensitivity of ethanol production by Scheffersomyces (Pichia) stipitis: development and verification of a method based on the principles of growth on mixtures of complementary substrates.

Under aerobic conditions, Crabtree-negative yeasts grow but do not ferment, and under anaerobic conditions, they ferment but do not grow. It is therefore believed that fermentation by these yeasts is sensitive to small variations of the operating parameters, e.g. dilution rate D, mass transfer coefficient kla and oxygen solubility co*. However, this parametric sensitivity has never been quantified. Here, we present a method to quantify the parametric sensitivity of ethanol production in the Crabtree-negative yeast Scheffersomycesstipitis. The method is based on our experimental observation that S. stipitis cultures follow the principles of growth on mixtures of complementary substrates. Specifically, if a chemostat operating at fixed D, kla and co* is fed with progressively increasing glucose feed concentrations sf, the culture passes through three regimes. (1) At low sf, the culture is carbon-limited and no ethanol is produced. (2) At high sf, the culture is oxygen-limited and ethanol is produced, but unused glucose is lost with the effluent. (3) At intermediate sf, both glucose and oxygen are limiting, and ethanol is produced without loss of glucose. Ethanol must therefore be produced in this dual-limited regime. The dual-limited regime can be predicted by simple unstructured models. It is characterized by the relation Yos

Effector Overlap between the lac and mel Operons of Escherichia coli: Induction of the mel Operon with ß-Galactosides.

The lac (lactose) operon (which processes ß-galactosides) and the mel (melibiose) operon (which processes α-galactosides) of Escherichia coli have a close historical connection. A number of shared substrates and effectors of the permeases and regulatory proteins have been reported over the years. Until now, ß-thiogalactosides like TMG (methyl-ß-d-thiogalactopyranoside) and IPTG (isopropyl-ß-d-thiogalactopyranoside) have not generally been considered to be inducers of the mel operon. The same is true for ß-galactosides such as lactose [ß-d-galactopyranosyl-(1→4)-d-glucose], which is a substrate but is not itself an inducer of the lac operon. This report shows that all three sugars can induce the mel operon significantly when they are accumulated in the cell by Lac permease. Strong induction by ß-thiogalactosides is observed in the presence of Lac permease, and strong induction by lactose (more than 200-fold) is observed in the absence of ß-galactosidase. This finding calls for reevaluation of TMG uptake experiments as assays for Lac permease that were performed with mel+ strains.IMPORTANCE The typical textbook picture of bacterial operons is that of stand-alone units of genetic information that perform, in a regulated manner, well-defined cellular functions. Less attention is given to the extensive interactions that can be found between operons. Well-described examples of such interactions are the effector molecules shared by the lac and mel operons. Here, we show that this set has to be extended to include ß-galactosides, which have been, until now, considered not to effect the expression of the mel operon. That they can be inducers of the mel operon as well as the lac operon has not been noted in decades of research because of the Escherichia coli genetic background used in previous studies.

Protein distributions from a stochastic model of the lac operon of E. coli with DNA looping: analytical solution and comparison with experiments.

Although noisy gene expression is widely accepted, its mechanisms are subjects of debate, stimulated largely by single-molecule experiments. This work is concerned with one such study, in which Choi et al., 2008, obtained real-time data and distributions of Lac permease in E. coli. They observed small and large protein bursts in strains with and without auxiliary operators. They also estimated the size and frequency of these bursts, but these were based on a stochastic model of a constitutive promoter. Here, we formulate and solve a stochastic model accounting for the existence of auxiliary operators and DNA loops. We find that DNA loop formation is so fast that small bursts are averaged out, making it impossible to extract their size and frequency from the data. In contrast, we can extract not only the size and frequency of the large bursts, but also the fraction of proteins derived from them. Finally, the proteins follow not the negative binomial distribution, but a mixture of two distributions, which reflect the existence of proteins derived from small and large bursts.

Integration of transcriptional inputs at promoters of the arabinose catabolic pathway.

BACKGROUND: Most modelling efforts of transcriptional networks involve estimations of in vivo concentrations of components, binding affinities and reaction rates, derived from in vitro biochemical assays. These assays are difficult and in vitro measurements may not approximate actual in vivo conditions. Alternatively, changes in transcription factor activity can be estimated by using partially specified models which estimate the "hidden functions" of transcription factor concentration changes; however, non-unique solutions are a potential problem. We have applied a synthetic biology approach to develop reporters that are capable of measuring transcription factor activity in vivo in real time. These synthetic reporters are comprised of a constitutive promoter with an operator site for the specific transcription factor immediately downstream. Thus, increasing transcription factor activity is measured as repression of expression of the transcription factor reporter. Measuring repression instead of activation avoids the complications of non-linear interactions between the transcription factor and RNA polymerase which differs at each promoter. RESULTS: Using these reporters, we show that a simple model is capable of determining the rules of integration for multiple transcriptional inputs at the four promoters of the arabinose catabolic pathway. Furthermore, we show that despite the complex and non-linear changes in cAMP-CRP activity in vivo during diauxic shift, the synthetic transcription factor reporters are capable of measuring real-time changes in transcription factor activity, and the simple model is capable of predicting the dynamic behaviour of the catabolic promoters. CONCLUSIONS: Using a synthetic biology approach we show that the in vivo activity of transcription factors can be quantified without the need for measuring intracellular concentrations, binding affinities and reaction rates. Using measured transcription factor activity we show how different promoters can integrate common transcriptional inputs, resulting in distinct expression patterns. The data collected show that cAMP levels in vivo are dynamic and agree with observations showing that cAMP levels show a transient pulse during diauxic shift.

Quantitative effect and regulatory function of cyclic adenosine 5'-phosphate in Escherichia coli.

Cyclic adenosine 5'-phosphate (cAMP) is a global regulator of gene expression in Escherichia coli. Despite decades of intensive study, the quantitative effect and regulatory function of cAMP remain the subjects of considerable debate. Here, we analyse the data in the literature to show that: (a) In carbon-limited cultures (including cultures limited by glucose), cAMP is at near-saturation levels with respect to expression of several catabolic promoters (including lac, ara and gal). It follows that cAMP receptor protein (CRP) cAMP-mediated regulation cannot account for the strong repression of these operons in the presence of glucose. (b) The cAMP levels in carbon-excess cultures are substantially lower than those observed in carbon-limited cultures under these conditions, the expression of catabolic promoters is very sensitive to variation of cAMP levels. (c)=CRPcAMP invariably activates the expression of catabolic promoters, but it appears to inhibit the expression of anabolic promoters. (d) These results suggest that the physiological function of cAMP is to maintain homeostatic energy levels. In carbon-limited cultures, growth is limited by the supply of energy; the cAMP levels therefore increase to enhance energy accumulation by activating the catabolic promoters and inhibiting the anabolic promoters. Conversely, in carbonexcess cultures, characterized by the availability of excess energy, the cAMP levels decrease in order to depress energy accumulation by inhibiting the catabolic promoters and activating the anabolic promoters.

The diffusive influx and carrier efflux have a strong effect on the bistability of the lac operon in Escherichia coli.

In the presence of gratuitous inducers, the lac operon of Escherichia coli exhibits bistability. Most models in the literature assume that the inducer enters the cell via the carrier (permease), and exits by a diffusion-like process. The diffusive influx and carrier efflux are neglected. However, analysis of the data shows that in non-induced cells, the diffusive influx is comparable to the carrier influx, and in induced cells, the carrier efflux is comparable to the diffusive efflux. Since bistability entails the coexistence of steady states corresponding to both non-induced and induced cells, neither one of these fluxes can be ignored. We present a model accounting for both fluxes, and show that: (1) The thresholds (i.e., the extracellular inducer levels at which transcription turns on or off) are profoundly affected by both fluxes. The diffusive influx reduces the on threshold, and eliminates irreversible bistability, a phenomenon that is inconsistent with data. The carrier efflux increases the off threshold, and abolishes bistability at large permease activities, a conclusion that can be tested experimentally. (2) The thresholds are well approximated by simple analytical expressions obtained by considering two limiting cases (no carrier efflux and no diffusive influx). (3) The simulations are in good agreement with the data for isopropyl thiogalactoside (IPTG), but somewhat discrepant with respect to the data for thiomethyl galactoside (TMG). We discuss the potential sources of the discrepancy.

Gene regulation in continuous cultures: a unified theory for bacteria and yeasts.

During batch growth on mixtures of two growth-limiting substrates, microbes consume the substrates either sequentially (diauxie) or simultaneously. The ubiquity of these growth patterns suggests that they may be driven by a universal mechanism common to all microbial species. Recently, we showed that a minimal model accounting only for enzyme induction and dilution, the two processes that occur in all microbes, explains the phenotypes observed in batch cultures of various wild-type and mutant/recombinant cells (Narang and Pilyugin in J. Theor. Biol. 244:326-348, 2007). Here, we examine the extension of the minimal model to continuous cultures. We show that: (1) Several enzymatic trends, attributed entirely to cross-regulatory mechanisms, such as catabolite repression and inducer exclusion, can be quantitatively explained by enzyme dilution. (2) The bifurcation diagram of the minimal model for continuous cultures, which classifies the substrate consumption pattern at any given dilution rate and feed concentrations, provides a precise explanation for the empirically observed correlations between the growth patterns in batch and continuous cultures. (3) Numerical simulations of the model are in excellent agreement with the data. The model captures the variation of the steady state substrate concentrations, cell densities, and enzyme levels during the single- and mixed-substrate growth of bacteria and yeasts at various dilution rates and feed concentrations. This variation is well approximated by simple analytical expressions that furnish deep physical insights. (4) Since the minimal model describes the behavior of the cells in the absence of cross-regulatory mechanisms, it provides a rigorous framework for quantifying the effect of these mechanisms. We illustrate this by analyzing several data sets from the literature.

Immunostaining evidence for PI(4,5)P2 localization at the leading edge of chemoattractant-stimulated HL-60 cells.

It is well known that in fMLP-stimulated neutrophils, phosphatidyl inositol 3,4,5-trisphosphate [PI(3,4,5)P3] localizes at the leading edge of the cells. However, no effort has been made to study the PI 4,5-bisphosphate [PI(4,5)P2] distribution in these cells. In fact, it has been suggested that PI(4,5)P2 is unlikely to localize, as its basal level is orders of magnitude higher than that of PI(3,4,5)P3. We developed an optimized immunostaining protocol for studying the endogenous distribution of PI(4,5)P2 in neutrophil-like HL-60 cells. We show that PI(4,5)P2 localizes sharply at the leading edge with an intensity gradient similar to that for PI(3,4,5)P3. The enzymes for the production of PI(4,5)P2, namely, PI5KIalpha and PI5KIgamma, were also found to localize at the leading edge, further supporting our finding that PI(4,5)P2 localizes at the leading edge. These results imply that complementary regulation of PI3K and phosphate and tensin homolog (PTEN) is not the sole or dominant mechanism of PI(3,4,5)P3 polarization in HL-60 cells.

Bistability of the lac operon during growth of Escherichia coli on lactose and lactose+glucose.

The lac operon of Escherichia coli can exhibit bistability. Early studies showed that bistability occurs during growth on TMG/succinate and lactose+glucose, but not during growth on lactose. More recently, studies with lacGFP-transfected cells show bistability during growth on TMG/succinate, but not during growth on lactose and lactose+glucose. In the literature, these results are invariably attributed to variations in the destabilizing effect of the positive feedback generated by induction. Specifically, during growth on TMG/succinate, lac induction generates strong positive feedback because the permease stimulates the accumulation of intracellular TMG, which in turn, promotes the synthesis of even more permease. This positive feedback is attenuated during growth on lactose because hydrolysis of intracellular lactose by beta-galactosidase suppresses the stimulatory effect of the permease. It is attenuated even more during growth on lactose + glucose because glucose inhibits the uptake of lactose. But it is clear that the stabilizing effect of dilution also changes dramatically as a function of the medium composition. For instance, during growth on TMG/succinate, the dilution rate of lac permease is proportional to its activity, e, because the specific growth rate is independent of e (it is completely determined by the concentration of succinate). However, during growth on lactose, the dilution rate of the permease is proportional to e2 because the specific growth rate is proportional to the specific lactose uptake rate, which in turn, proportional to e. We show that: (a) This dependence on e2 creates such a strong stabilizing effect that bistability is virtually impossible during growth on lactose, even in the face of the intense positive feedback generated by induction. (b) This stabilizing effect is weakened during growth on lactose+glucose because the specific growth rate on glucose is independent of e, so that the dilution rate once again contains a term that is proportional to e. These results imply that the lac operon is much more prone to bistability if the medium contains carbon sources that cannot be metabolized by the lac enzymes, e.g., succinate during growth on TMG/succinate and glucose during growth on lactose+glucose. We discuss the experimental data in the light of these results.

Coarse-grained molecular dynamics simulations of phase transitions in mixed lipid systems containing LPA, DOPA, and DOPE lipids.

The mechanisms that mediate biomembrane shape transformations are of considerable interest in cell biology. Recent in vitro experiments show that the chemical transformation of minor membrane lipids can induce dramatic shape changes in biomembranes. Specifically, it was observed that the addition of DOPA to DOPE has no effect on the stability of the bilayer structure of the membrane. In contrast, the addition of LPA to DOPE stabilizes the bilayer phase of DOPE, increasing the temperature of a phase transition from the bilayer to the inverted hexagonal phase. This result suggests that the chemical conversion of DOPA to LPA is sufficient for triggering a dramatic change in the shape of biomembranes. The LPA/DOPA/DOPE mixture of lipids provides a simple model system for understanding the molecular events driving the shape change. In this work, we used coarse-grained molecular dynamics simulations to study the phase transitions of this lipid mixture. We show that despite the simplicity of the coarse-grained model, it reproduces the experimentally observed phase changes of: 1), pure LPA and DOPA with respect to changes in the concentration of cations; and 2), LPA/DOPE and DOPA/DOPE mixtures with respect to temperature. The good agreement between the model and experiments suggests that the computationally inexpensive coarse-grained approach can be used to infer macroscopic membrane properties. Furthermore, analysis of the shape of the lipid molecules demonstrates that the phase behavior of single-lipid systems is consistent with molecular packing theory. However, the phase stability of mixed lipid systems exhibits significant deviations from this theory, which suggests that the elastic energy of the lipids, neglected in the packing theory, plays an important role.

Role of molecular tilt in thermal fluctuations of lipid membranes.

Long-wavelength thermal fluctuations of lipid membranes are adequately described by the Helfrich elastic model. On the other hand, fluctuations of wavelengths comparable with bilayer thickness exhibit significant deviations from the prediction of the elastic model and are typically assumed to be dominated by microscopic surface tension due to protrusion of lipid molecules into the solvent. We present evidence that the short-wavelength modes of a lipid membrane are dominated by fluctuations of the tilt of lipid molecules with respect to the membrane normal rather than the microscopic surface tension. We obtain an expression for the spectral intensity of the thermal membrane fluctuations by appealing to the Hamm-Kozlov model, which accounts for both membrane bending and lipid tilt contributions to the total membrane energy but neglects the contributions of the microscopic surface tension. The tilt and the bending fluctuations obtained from our coarse-grained molecular dynamics simulations of a dipalmitoylphosphatidylcholine lipid bilayer show good agreement with the theory. Furthermore, the obtained tilt and bending moduli are in close agreement with experimentally determined values. The magnitude of the microscopic protrusion tension estimated from our simulations is significantly smaller than that of the tilt modulus. These results indicate that the membrane fluctuations can be adequately described by a macroscopic elastic model down to scales of interlipid distance provided one accounts for the tilt fluctuations.