Fluctuations In Number Of Molecules Research Articles

Free energy dissipation enhances spatial accuracy and robustness of self-positioned Turing pattern in small biochemical systems.

Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Turing proposed a general mechanism for pattern formation exemplified by a reaction-diffusion model with two chemical species in a large system. However, in small biological systems such as a cell, the existence of multiple Turing patterns and strong noise can lower the spatial order. Recently, a modified reaction-diffusion model with an additional chemical species is shown to stabilize the Turing pattern. Here, we study non-equilibrium thermodynamics of this three-species reaction-diffusion model to understand the relationship between energy cost and the performance of self-positioning. By using computational and analytical approaches, we show that beyond the onset of pattern formation the positioning error decreases as energy dissipation increases. In a finite system, we find that a specific Turing pattern exists only within a finite range of total molecule number. Energy dissipation broadens this range, which enhances the robustness of Turing pattern against molecule number fluctuations in living cells. The generality of these results is verified in a realistic model of the Muk system underlying DNA segregation in Escherichia coli, and testable predictions are made for the dependence of the accuracy and robustness of the spatial pattern on the ATP/ADP ratio.

Quantification of fluctuations from fluorescence correlation spectroscopy experiments in reaction-diffusion systems.

Fluorescence correlation spectroscopy (FCS) is commonly used to estimate diffusion and reaction rates. In FCS the fluorescence coming from a small volume is recorded and the autocorrelation function (ACF) of the fluorescence fluctuations is computed. Scaling out the fluctuations due to the emission process, this ACF can be related to the ACF of the fluctuations in the number of observed fluorescent molecules. In this paper the ACF of the molecule number fluctuations is studied theoretically, with no approximations, for a reaction-diffusion system in which the fluorescence changes with binding and unbinding. Theoretical ACFs are usually derived assuming that fluctuations in the number of molecules of one species are instantaneously uncorrelated to those of the others and obey Poisson statistics. Under these assumptions, the ACF derived in this paper is characterized only by the diffusive timescale of the fluorescent species and its total weight is the inverse of the mean number of observed fluorescent molecules. The theory is then scrutinized in view of previous experimental results which, for a similar system, gave a different total weight and correct estimates of other diffusive timescales. The total weight mismatch is corrected by assuming that the variance of the number of fluorescent molecules depends on the variance of the particle numbers of the other species, as in the variance decomposition formula. Including the finite acquisition time in its computation, it is shown that the ACF depends on various timescales of the system and that its total weight coincides with the one obtained with the variance decomposition formula. This calculation implies that diffusion coefficients of nonobservable species can be estimated with FCS experiments performed in reaction-diffusion systems. Ways to proceed in future experiments are also discussed.

Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise.

It is well known that the kinetics of an intracellular biochemical network is stochastic. This is due to intrinsic noise arising from the random timing of biochemical reactions in the network as well as due to extrinsic noise stemming from the interaction of unknown molecular components with the network and from the cell's changing environment. While there are many methods to study the effect of intrinsic noise on the system dynamics, few exist to study the influence of both types of noise. Here we show how one can extend the conventional linear-noise approximation to allow for the rapid evaluation of the molecule numbers statistics of a biochemical network influenced by intrinsic noise and by slow lognormally distributed extrinsic noise. The theory is applied to simple models of gene regulatory networks and its validity confirmed by comparison with exact stochastic simulations. In particular, we consider three important biological examples. First, we investigate how extrinsic noise modifies the dependence of the variance of the molecule number fluctuations on the rate constants. Second, we show how the mutual information between input and output of a network motif is affected by extrinsic noise. And third, we study the robustness of the ubiquitously found feed-forward loop motifs when subjected to extrinsic noise.

Fluctuations, Correlations and the Estimation of Concentrations inside Cells.

Information transmission in cells occurs quite accurately even when concentration changes are “read” by individual binding sites. In this paper we study ligand number and site occupancy fluctuations when ligands diffuse and react going beyond the analyses that focus on their asymptotic decay. In this way we show that, for immobile binding sites, fluctuations in the number of bound molecules decay on a relatively fast scale before the asymptotic behavior kicks in. This result can explain the observed co-existence of highly fluctuating instantaneous transcriptional activities with accumulated mRNA concentrations that have relatively small noise levels. We also show that the initial stages of the decay in the bound molecule number fluctuations have one or two characteristic timescales depending on the concentration of free molecules. This transition can explain the changes in enzyme activity observed at the single molecule level.

Spatially Multiplexed Imaging: Fluorescence Correlation Spectroscopy for Efficient Measurement of Molecular Diffusion at Solid-Liquid Interfaces.

Fluorescence correlation spectroscopy (FCS) has become an important technique for the characterization of molecular dynamics, especially at interfaces. Fluorescence correlation spectroscopy provides both temporal and spatial resolution for measuring fast processes at equilibrium through analysis of noise in fluorescence intensities from the statistical fluctuations in a small number of molecules. The small molecular populations produce very low-level fluorescence signals, where time-averaging the fluorescence autocorrelation function is needed to generate reasonable signal-to-noise (S/N) ratios. Recently imaging cameras have been adapted to FCS measurements of molecular dynamics at interfaces (membranes and surfaces) through the use of electron-multiplying charge-coupled device (EM-CCD) detectors for acquisition of fluorescence from addressable areas on the detector. This approach provides a major advantage over traditional focused-spot FCS by allowing electronic control over the location and area of the acquired region on the sample surface. Imaging-FCS can also provide a spatial multiplexing advantage through its ability to measure intensity data from larger areas in parallel with no loss of time resolution. In this work, this multiplexing advantage is exploited to determine molecular diffusion rates from the simultaneous measurement of multiple areas on a surface, the autocorrelation traces from which are averaged to improve the S/N ratio. As proof of concept, the diffusion of 1,1'-dioctadecyl-3,3,3'3'-tetramethylindocarbocyanine perchlorate (DiI) on a C18-modified interface was measured using this multiplexed method and compared to autocorrelation data acquired from a single spot. Due to the slow thermal recovery of the EM-CCD that inhibits fast time-averaging, spatial multiplexing in imaging-FCS provides an eightyfold time savings to reach the same S/N ratio as multiple (time-averaged) measurements from a single spot.

Concentration Estimates from Counting Individual Molecules

The transmission of information in cells usually involves changes in concentration that need to be read by different cell components. Very often small molecule numbers are involved so that fluctuations play a relevant role. In this work we study molecule number fluctuations when molecules diffuse and react. Building upon our previous works on the analysis of Fluorescence Correlation Spectroscopy experiments we derive a formula for the observation time that is needed to estimate concentrations with a given precision out of counting individual molecules. In this way we determine that fluctuations in the number of bound molecules can be averaged out on a relatively fast timescale due to correlations and that they are described by either one or two characteristic timescales depending on the concentration of free molecules. We show that how our results can be used to understand the transformation from a highly fluctuating instantaneous transcriptional activity into the expected output concentration relatively fast and to analyze the dynamics of enzymes at the single molecule level.

Quantum non-demolition detection of polar molecule complexes: dimers, trimers, tetramers

An optical nondestructive method for in situ detection of the bound states of ultracold polar molecules is developed. It promises a minimally destructive measurement scheme up to a physically exciting quantum non-demolition (QND) level. The detection of molecular complexes beyond simple pairs of quantum particles (dimers, known, e.g., from the BEC–BCS theory) is suggested, including three-body (trimer) and four-body (tertramer) complexes trapped by one-dimensional tubes. The intensity of the scattered light is sensitive to the molecule number fluctuations beyond the mean-density approximation. Such fluctuations are very different for different complexes, which leads to radically different light scattering. This type of research extends ‘quantum optics of quantum gases’ to the field of ultracold molecules. Merging the quantum optical and ultracold gas problems will advance experimental efforts towards the study of the light–matter interaction at its ultimate quantum level, where the quantizations of both light and matter are equally important.

Relaxation Kinetics by Fluorescence Correlation Spectroscopy: Determination of Kinetic Parameters in the Presence of Fluorescent Impurities.

The use of Fluorescence Correlation Spectroscopy (FCS) in combination with Förster Resonance Energy Transfer (FRET) is gaining popularity as a tool to investigate kinetics in equilibrium conditions. The technique is based on the study of fluorescence fluctuations in small numbers of molecules, and is particularly well-suited to investigate conformational dynamics in biopolymers. In practice, its applicability is often hindered by the presence of certain impurities such as partially labeled biomolecules, excess of free fluorophore, or partially dissociated multi-subunit complexes. Here, we show that the simultaneous measurement of the fluctuations in the donor and acceptor intensities allows the determination of the kinetic relaxation time of the reaction in the presence of donor-only particles when cross-talk is negligible, or in cases where all species have the same diffusion coefficient. Theoretical predictions are supported with the results of Monte Carlo simulations, and demonstrate that the applicability of the technique is more general than previously thought.

Interlinked dual-time feedback loops can enhance robustness to stochasticity and persistence of memory

Multiple interlinked positive feedback loops shape the stimulus responses of various biochemical systems, such as the cell cycle or intracellular Ca2+ release. Recent studies with simplified models have identified two advantages of coupling fast and slow feedback loops. This dual-time structure enables a fast response while enhancing resistances of responses and bistability to stimulus noise. We now find that (1) the dual-time structure similarly confers resistance to internal noise due to molecule number fluctuations, and (2) model variants with altered coupling, which better represent some specific biochemical systems, share all the above advantages. We also develop a similar bistable model with coupling of a fast autoactivation loop to a slow loop. This model's topology was suggested by positive feedback proposed to play a role in long-term synaptic potentiation (LTP). The advantages of fast response and noise resistance are also present in this autoactivation model. Empirically, LTP develops resistance to reversal over approximately 1h . The model suggests this resistance may result from increased amounts of synaptic kinases involved in positive feedback.

Bistable MAP kinase activity: a plausible mechanism contributing to maintenance of late long-term potentiation

Bistability of MAP kinase (MAPK) activity has been suggested to contribute to several cellular processes, including differentiation and long-term synaptic potentiation. A recent model (Markevich NI, Hoek JB, Kholodenko BN. J Cell Biol 164: 353-359, 2004) predicts bistability due to interactions of the kinases and phosphatases in the MAPK pathway, without feedback from MAPK to earlier reactions. Using this model and enzyme concentrations appropriate for neurons, we simulated bistable MAPK activity, but bistability was present only within a relatively narrow range of activity of Raf, the first pathway kinase. Stochastic fluctuations in molecule numbers eliminated bistability for small molecule numbers, such as are expected in the volume of a dendritic spine. However, positive-feedback loops have been posited from MAPK up to Raf activation. One proposed loop in which MAPK directly activates Raf was incorporated into the model. We found that such feedback greatly enhanced the robustness of both stable states of MAPK activity to stochastic fluctuations and to parameter variations. Bistability was robust for molecule numbers plausible for a dendritic spine volume. The upper state of MAPK activity was resistant to inhibition of MEK activation for >1 h, which suggests that inhibitor experiments have not sufficed to rule out a role for persistent MAPK activity in the maintenance of long-term potentiation (LTP). These simulations suggest that persistent MAPK activity and consequent upregulation of translation may contribute to LTP maintenance and to long-term memory. Experiments using a fluorescent MAPK substrate may further test this hypothesis.

Fluorescence Fluctuation Spectroscopic Approaches to the Study of a Single Molecule Diffusing in Solution and a Live Cell without Systemic Drift or Convection: A Theoretical Study

Reentries of a single molecule in the confocal, femtoliter-sized probe region (about 10(-16) L and less) are significant because during measurement times they give rise to fluctuation phenomena such as molecule number fluctuations at the single-molecule level in solution without immobilization or hydrodynamic focusing. These fluctuations are the fundamental physical process on which, for example, fluorescence correlation spectroscopy and two-color fluorescence cross-correlation spectroscopy are based. The reentries of just one molecule in the confocal probe region are theoretically examined in this original article using a hidden, continuous-time Markov model. The system is not set up to have systemic drift or convection. It is found that the reentries obey certain conditions and analytical expressions for the reentry probabilities are obtained first. In particular, the time constant of the mean value and the variance of the reentry probabilities are obtained. The fractions of non-meaningful reentries and meaningful reentries are found for these experimental situations. Therewith, the concentration dependence of the meaningful time that one can study bimolecular reactions of the selfsame molecule in the confocal probe region is derived for the first time. The meaningful time in the probe volume is proportional to the diffusion time of the selfsame molecule and related inversely to the size of the given confocal probe volume. For small molecules, i.e. small diffusion times at a given size of the confocal probe region, one needs lower concentrations of molecules of the same kind in the bulk phase, whereas large molecules can be studied at higher concentrations. The selfsame molecule scenario is compared with the molecular scenario that a second molecule enters the probe volume at random as a function of the meaningful time. The analytical solutions of the physical reentry model (mechanism) hold for the one-, two- (membrane), or three- (solution, live cell) dimensional Brownian motion.

Random timing in signaling cascades

Mol Syst Biol. 3: 81 Cells often rely on precise temporal coordination and can go through great troubles to time events in response to internal signals and environmental cues, as demonstrated by circadian clocks, cell‐cycle control, and morphogenesis. But for any given set of signals and cues there can also be substantial random variability from cell to cell. The origin of such variation is understood in general terms—chemical reactions involve random collisions between diffusing molecules—but its extent cannot be inferred from first principles. Most single‐cell studies have also focused on fluctuations in molecule numbers (Elowitz et al , 2002; Ozbudak et al , 2002), and there are rather few quantitative measurements (Bean et al , 2006) of intracellular timing even in the best‐studied model systems. In this issue of Molecular Systems Biology , Stavans and co‐workers (Amir et al , 2007) address this problem in an insightful study of how temporal fluctuations propagate along the lytic cascade of bacteriophage λ. When λ phage infects a bacterium, it chooses between two paths: it either hijacks cellular resources, overproduces phage particles, and busts the cell open (lysis), or it integrates into the host chromosome and protects the cell from further phage infection (lysogeny). Lysogenic phage then quietly replicates with the chromosome until DNA damage activates the RecA protein, which degrades the λ repressor CI and derepresses phage promoters pL and pR. Promoter pL drives the expression of an anti‐terminator for pR expression, allowing read‐through to lytic downstream genes. A drop in CI thus only triggers lysis if it lasts long enough for pR to remain derepressed by the time the anti‐terminator reaches a high enough concentration to allow read‐through. This …

‘True’ single-molecule molecule observations by fluorescence correlation spectroscopy and two-color fluorescence cross-correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) and two-color fluorescence cross-correlation spectroscopy (FCCS) are a measure of fluctuations of detected light as a fluorescence molecule diffuses through a femtoliter detection volume caused by a tightly focused laser and confocal optics. Fluorescence from a single molecule can easily be distinguished from the slight background associated with a femtoliter of solvent. At a solution concentration of about 1 nM, the probability that there is an analyte molecule in the probe volume is less than one. Although fluorescence from individual molecules is collected, the data are analyzed by autocorrelation or two-color cross-correlation functions that are the average of thousands of molecules. Properties of single molecules are not obtained. I have been working on problems and opportunities associated with very dilute solutions. The molecule in the confocal probe volume is most probably the molecule that just diffused out, turned around, and diffused back in, i.e., reentered. For the first time, some theoretical results of the novel theory of the meaningful time are presented that enable study of just one single molecule over extended periods of times without immobilization or hydrodynamic focusing. Reentries that may also be called reoccurrences or encounters of a single molecule are significant because during measurement times they give rise to fluctuation phenomena such as molecule number fluctuations. Likewise, four criteria have been developed that can be used to verify that there is only one “selfsame” molecule in the laser probe volume during the experiment: (Földes–Papp, Z., 2006. What it means to measure a single molecule in a solution by fluorescence fluctuation spectroscopy. Exp. Mol. Pathol. 80 (3) 209–218).

A Reduced Model Clarifies the Role of Feedback Loops and Time Delays in the Drosophila Circadian Oscillator

Although several detailed models of molecular processes essential for circadian oscillations have been developed, their complexity makes intuitive understanding of the oscillation mechanism difficult. The goal of the present study was to reduce a previously developed, detailed model to a minimal representation of the transcriptional regulation essential for circadian rhythmicity in Drosophila. The reduced model contains only two differential equations, each with time delays. A negative feedback loop is included, in which PER protein represses per transcription by binding the dCLOCK transcription factor. A positive feedback loop is also included, in which dCLOCK indirectly enhances its own formation. The model simulated circadian oscillations, light entrainment, and a phase-response curve with qualitative similarities to experiment. Time delays were found to be essential for simulation of circadian oscillations with this model. To examine the robustness of the simplified model to fluctuations in molecule numbers, a stochastic variant was constructed. Robust circadian oscillations and entrainment to light pulses were simulated with fewer than 80 molecules of each gene product present on average. Circadian oscillations persisted when the positive feedback loop was removed. Moreover, elimination of positive feedback did not decrease the robustness of oscillations to stochastic fluctuations or to variations in parameter values. Such reduced models can aid understanding of the oscillation mechanisms in Drosophila and in other organisms in which feedback regulation of transcription may play an important role.

Intrinsic noise in gene regulatory networks.

Cells are intrinsically noisy biochemical reactors: low reactant numbers can lead to significant statistical fluctuations in molecule numbers and reaction rates. Here we use an analytic model to investigate the emergent noise properties of genetic systems. We find for a single gene that noise is essentially determined at the translational level, and that the mean and variance of protein concentration can be independently controlled. The noise strength immediately following single gene induction is almost twice the final steady-state value. We find that fluctuations in the concentrations of a regulatory protein can propagate through a genetic cascade; translational noise control could explain the inefficient translation rates observed for genes encoding such regulatory proteins. For an autoregulatory protein, we demonstrate that negative feedback efficiently decreases system noise. The model can be used to predict the noise characteristics of networks of arbitrary connectivity. The general procedure is further illustrated for an autocatalytic protein and a bistable genetic switch. The analysis of intrinsic noise reveals biological roles of gene network structures and can lead to a deeper understanding of their evolutionary origin.

Fluctuations in Repressor Control: Thermodynamic Constraints on Stochastic Focusing

The influence of fluctuations in molecule numbers on genetic control circuits has received considerable attention. The consensus has been that such fluctuations will make regulation less precise. In contrast, it has more recently been shown that signal fluctuations can sharpen the response in a regulated process by the principle of stochastic focusing (SF) (Paulsson et al., 2000, Proc. Natl. Acad. Sci. USA. 97:7148–7153). In many cases, the larger the fluctuations are, the sharper is the response. Here we investigate how fluctuations in repressor or corepressor numbers can improve the control of gene expression. Because SF is found to be constrained by detailed balance, this requires that the control loops contain driven processes out of equilibrium. Some simple and realistic out-of-equilibrium steps that will break detailed balance and make room for SF in such systems are discussed. We conclude that when the active repressors are controlled by corepressor molecules that display large (“coherent”) number fluctuations or when corepressors can be irreversibly removed directly from promoter-bound repressors, the response in gene activity can become significantly sharper than without intrinsic noise. A simple experimental design to establish the possibility of SF for repressor control is suggested.

Predicting Temporal Fluctuations in an Intracellular Signalling Pathway

We used a newly developed stochastic-based program to predict the fluctuations in numbers of molecules in a chemotactic signalling pathway of coliform bacteria. Specifically, we examined temporal changes in molecules of CheYp, a cytoplasmic protein known to influence the direction of rotation of the flagellar motor. Signalling molecules in the vicinity of a flagellar motor were represented as individual software objects interacting according to probabilities derived from experimentally-observed concentrations and rate constants. The simulated CheYp molecules were found to undergo random fluctuations in number about an average corresponding to the deterministically calculated concentration. Both the relative amplitude of the fluctuations, as a proportion of the total number of molecules, and their average duration, increased as the simulated volume was reduced. In a simulation corresponding to 10% of the volume of a bacterium, the average duration of fluctuations was found to be 80.7 ms, which is much shorter than the observed alternations between clockwise and counter-clockwise rotations of tethered bacteria (typically 2.6 s). Our results are therefore not in agreement with a simple threshold-crossing model for motor switching. However, it is possible to filter the CheYp fluctuations to produce temporal distributions closer to the observed swimming behaviour and we discuss the possible implications for the control of motor rotation.

The statistical mechanical theory of concentration fluctuations in mixtures

A general theory of concentration fluctuations in mixtures is derived. The resulting expressions, which include the Kirkwood–Buff theory as a particular case, relate different partial molar properties to a single combination of molecule number fluctuations within differently defined subsystems. The formalism, therefore, gives rise to a common definition for partial molar energies, volumes, enthalpies, and entropies in terms of molecule number fluctuations exclusively. Likewise, a second function of molecule number fluctuations describes both the mixture’s isobaric specific heat as well as its isothermal compressibility. The partial molar energy and enthalpy definitions, moreover, suggest new methods for the study of mixture properties via computer simulations.

Fluctuations in number of molecules in several liquids including water. The role of attractive intermolecular interactions in anomalies of water

The temperature dependence of fluctuation in number expressed by 〈( ΔN) 2〉 / 〈 N〉 has been examined for water, hexane, toluene, carbon disulfide under isochoric conditions. The fluctuation in water is four to six times larger than that in the other molecular liquids because of the attractive forces. At a low liquid density, the fluctuation has a minimum against temperature in water and hexane, but it increases monotonically in toluene and carbon disulfide. Upon compression the temperature dependence of the fluctuation becomes positive for all liquids except water. It is verified that the negative temperature dependence is caused by the attractive intermolecular interactions. The minimum appears in systems where attractive interactions are important as a result of a relatively low molecular packing. Water is anomalous in the sense that the attractive nature of the molecular field cannot be lost easily by an increase in density which may be absorbed by the open structure.