Fluorescence correlation spectroscopy with high-order and dual-color correlation to probe nonequilibrium steady states

Understanding underlying physical mechanism reveals early warning indicators and key elements for adaptive infections disease networks

The study of infectious diseases holds significant scientific and societal importance, yet current research on the mechanisms of disease emergence and prediction methods still face challenging issues. This research uses the landscape and flux theoretical framework to reveal the non-equilibrium dynamics of adaptive infectious diseases and uncover its underlying physical mechanism. This allows the quantification of dynamics, characterizing the system with two basins of attraction determined by gradient and rotational flux forces. Quantification of entropy production rates provides insights into the system deviating from equilibrium and associated dissipative costs. The study identifies early warning indicators for the critical transition, emphasizing the advantage of observing time irreversibility from time series over theoretical entropy production and flux. The presence of rotational flux leads to an irreversible pathway between disease states. Through global sensitivity analysis, we identified the key factors influencing infectious diseases. In summary, this research offers valuable insights into infectious disease dynamics and presents a practical approach for predicting the onset of critical transition, addressing existing research gaps.

Thermodynamic cost for precision of general counting observables

We analytically derive universal bounds that describe the tradeoff between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the waiting times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting-time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

Early warning indicators of war and peace through the landscapes and flux quantifications

War and peace, spanning history, deeply affect society, economy, and individuals. Grasping their dynamics is vital to lessen harm and foster global peace. Yet, quantifying them remains hard. Our goal is to create a simple qualitative model using landscape and flux theory, exploring war and peace mechanisms. In this symmetric network, they appear as separate attraction basins, dynamically shifting. Analyzing landscape shape gives insights into global stability. Near critical points, indicators like cross correlations, autocorrelation times, and flickering frequency surge, as warnings. We also calculate the irreversible path between war and peace due to rotational flux. Global sensitivity analysis identifies history's role in system stability. In summary, our research unveils a way to understand war and peace complexities, enhancing knowledge of key elements that lead to conflict, aiding resolution.

Thermodynamic bounds on time-reversal asymmetry

Quantifying irreversibility of a system using finite information constitutes a major challenge in stochastic thermodynamics. We introduce an observable that measures the time-reversal asymmetry between two states after a given time lag. Our central result is a bound on the time-reversal asymmetry in terms of the total cycle affinity driving the system out of equilibrium. This result leads to further thermodynamic bounds on the asymmetry of directed fluxes, on the asymmetry of finite-time cross-correlations, and on the cycle affinity of coarse-grained dynamics.

Global dynamics, thermodynamics and non-equilibrium origin of bifurcations for single neuron dynamics

The understanding of neural excitability and oscillations in single neuron dynamics remains incomplete in terms of global stabilities and the underlying mechanisms for phase formation and associated phase transitions. In this study, we investigate the mechanism of single neuron excitability and spontaneous oscillations by analyzing the potential landscape and curl flux. The topological features of the landscape play a crucial role in assessing the stability of resting states and the robustness/coherence of oscillations. We analyze the excitation characteristics in Class I and Class II neurons and establish their relation to biological function. Our findings reveal that the average curl flux and associated entropy production exhibit significant changes near bifurcation or phase transition points. Moreover, the curl flux and entropy production offer insights into the dynamical and thermodynamical origins of nonequilibrium phase transitions and exhibit distinct behaviors in Class I and Class II neurons. Additionally, we quantify time irreversibility through the difference in cross-correlation functions in both forward and backward time, providing potential indicators for the emergence of nonequilibrium phase transitions in single neurons.

Nonequilibrium indicator for the onset of epileptic seizure

The occurrence of spontaneous bursts of uncontrolled electrical activity between neurons can disrupt normal brain function and lead to epileptic seizures. Despite extensive research, the mechanisms underlying seizure onset remain unclear. This study investigates the onset of seizures from the perspective of nonequilibrium statistical physics. By analyzing the probability flux within the framework of the nonequilibrium potential-flux landscape, we establish a connection between seizure dynamics and nonequilibrium. Our findings demonstrate that the degree of nonequilibrium is sensitive to the onset of epileptic seizures. This result offers an alternative perspective on assessing seizure onset in epilepsy.

Thermodynamic Bound on the Asymmetry of Cross-Correlations

The principle of microscopic reversibility says that, in equilibrium, two-time cross-correlations are symmetric under the exchange of observables. Thus, the asymmetry of cross-correlations is a fundamental, measurable, and often-used statistical signature of deviation from equilibrium. Here we find a simple and universal inequality that bounds the magnitude of asymmetry by the cycle affinity, i.e., the strength of thermodynamic driving. Our result applies to a large class of systems and all state observables, and it suggests a fundamental thermodynamic cost for various nonequilibrium functions quantified by the asymmetry. It also provides a powerful tool to infer affinity from measured cross-correlations, in a different and complementary way to the thermodynamic uncertainty relations. As an application, we prove a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert [Phys. Rev. E 95, 062409 (2017)PRESCM2470-004510.1103/PhysRevE.95.062409]. We also derive a thermodynamic bound on directed information flow in a biochemical signal transduction model.

Non-equilibrium early-warning signals for critical transitions in ecological systems

Complex systems can exhibit sudden transitions or regime shifts from one stable state to another, typically referred to as critical transitions. It becomes a great challenge to identify a robust warning sufficiently early that action can be taken to avert a regime shift. We employ landscape-flux theory from nonequilibrium statistical mechanics as a general framework to quantify the global stability of ecological systems and provide warning signals for critical transitions. We quantify the average flux as the nonequilibrium driving force and the dynamical origin of the nonequilibrium transition while the entropy production rate as the nonequilibrium thermodynamic cost and thermodynamic origin of the nonequilibrium transition. Average flux, entropy production, nonequilibrium free energy, and time irreversibility quantified by the difference in cross-correlation functions forward and backward in time can serve as early warning signals for critical transitions much earlier than other conventional predictors. We utilize a classical shallow lake model as an exemplar for our early warning prediction. Our proposed method is general and can be readily applied to assess the resilience of many other ecological systems. The early warning signals proposed here can potentially predict critical transitions earlier than established methods and perhaps even sufficiently early to avert catastrophic shifts.

Reaction cycle of operating pump protein studied with single-molecule spectroscopy

Biological machinery relies on nonequilibrium dynamics to maintain stable directional fluxes through complex reaction cycles. For such reaction cycles, the presence of microscopically irreversible conformational transitions of the protein, and the accompanying entropy production, is of central interest. In this work, we use multidimensional single-molecule fluorescence lifetime correlation spectroscopy to measure the forward and reverse conformational transitions of bacteriorhodopsin during trans-membrane H + pumping. We quantify the flux, affinity, enthalpy and entropy production through portions of the reaction cycle as a function of temperature. We find that affinity of irreversible conformational transitions decreases with increasing temperature, resulting in diminishing flux and entropy production. We show that the temperature dependence of the transition affinity is well fit by the Gibbs-Helmholtz relation, allowing the DH trans to be experimentally extracted.

Unifying deterministic and stochastic ecological dynamics via a landscape-flux approach

The frequency distributions can characterize the population-potential landscape related to the stability of ecological states. We illustrate the practical utility of this approach by analyzing a forest-savanna model. Savanna and forest states coexist under certain conditions, consistent with past theoretical work and empirical observations. However, a grassland state, unseen in the corresponding deterministic model, emerges as an alternative quasi-stable state under fluctuations, providing a theoretical basis for the appearance of widespread grasslands in some empirical analyses. The ecological dynamics are determined by both the population-potential landscape gradient and the steady-state probability flux. The flux quantifies the net input/output to the ecological system and therefore the degree of nonequilibriumness. Landscape and flux together determine the transitions between stable states characterized by dominant paths and switching rates. The intrinsic potential landscape admits a Lyapunov function, which provides a quantitative measure of global stability. We find that the average flux, entropy production rate, and free energy have significant changes near bifurcations under both finite and zero fluctuation. These may provide both dynamical and thermodynamic origins of the bifurcations. We identified the variances in observed frequency time traces, fluctuations, and time irreversibility as kinematic measures for bifurcations. This framework opens the way to characterize ecological systems globally, to uncover how they change among states, and to quantify the emergence of quasi-stable states under stochastic fluctuations.

The dynamic and thermodynamic origin of dissipative chaos: chemical Lorenz system

Chaos appears widely in various chemical and physical systems and is often accompanied by nonequilibrium due to its dissipative nature. However, it is still not clear how dissipative chaos is influenced by nonequilibrium conditions. Here, we study chaos from the perspective of nonequilibrium dynamics by considering a chemical Lorenz system. We found that its nonequilibrium nature can be quantified from the steady-state probability flux in the state space. The dynamic origin for the onset and offset of dissipative chaos was from the sudden appearance and disappearance of such nonequilibrium fluxes. Meanwhile, the dissipation associated with the flux as quantified by the entropy production rate provides the thermodynamic origin of dissipative chaos. Sharp changes in the degree of nonequilibrium also provide alternative quantitative indicators for the onset and offset of dissipative chaos.

Elimination of fast variables in stochastic nonlinear kinetics

A reduced chemical scheme involving a small number of variables is often sufficient to account for the deterministic evolution of the concentration of the main species contributing to a reaction. However, its predictions are questionable in small systems used, for example in fluorescence correlation spectroscopy (FCS) or in explosive systems involving strong nonlinearities such as autocatalytic steps. We make precise dynamical criteria defining the validity domain of the quasi-steady-state approximation and the elimination of a fast concentration in deterministic dynamics. Designing two different three-variable models converging toward the same two-variable model, we show that the variances and covariance of the fluctuations of the slow variables are not correctly predicted using the two-variable model, even in the limit of a large system size. The more striking weaknesses of the reduced scheme are figured out in mesoscaled systems containing a small number of molecules. The results of two stochastic approaches are compared and the shortcomings of the Langevin equations with respect to the master equation are pointed out. We conclude that the description of the fluctuations and their coupling with nonlinearities of deterministic dynamics escape reduced chemical schemes.

Dynamics of the HIV Gag Lattice Detected by Localization Correlation Analysis and Time-Lapse iPALM

Immature human immunodeficiency virus (HIV) virions have a lattice of Gag and Gag-Pol proteins anchored to the lumen of their envelope. Using electron microscopy, we demonstrate that HIV virus-like particles (VLPs) assembled by the viral protein Gag and tagged at its C-terminus with the fluorescent protein Dendra2 have the same morphology and size as the VLPs assembled using only HIV Gag. We characterize the photophysical properties of Dendra2 and demonstrate that 60% of Dendra2 molecules can be photoswitched and reliably counted in our interferometric photoactivated localization microscopy (iPALM) setup. We further perform iPALM imaging on immobilized HIV Gag-Dendra2 VLPs and demonstrate that we can localize and count 900–1600 Dendra2 molecules within each immobilized VLP with a single-molecule localization precision better than (10 nm)³. Our molecular counts correspond to 1400–2400 Gag-Dendra2 proteins incorporated within each VLP. We further calculate temporal correlation functions of localization data, which we present as localization correlation analysis, and show dynamics within the lattice of immobilized VLPs in the timescale of 10–100 s. We further use our localization data to reconstruct time-lapse iPALM images of the Gag-Dendra2 lattice within the lumen of immobilized VLPs. The iPALM time-lapse images show significant lattice dynamics within the lumen of VLPs. Addition of disuccinimidyl suberate to the VLPs completely abrogated these dynamics as observed in both localization correlation analysis and time-lapse iPALM. In a complementary approach, we utilized HaXS8 cross-linking reactions between Halo and SNAP proteins and verified lattice dynamics in purified VLPs incorporating 10% Gag-SNAP, 10% Gag-Halo, and 80% Gag proteins. The HIV Gag lattice, along with the structural lattice of other enveloped viruses, has been mostly considered static. Our study provides an important tool to investigate the dynamics within these enveloped viruses.

Curl Flux as a Dynamical Origin of the Bifurcations/Phase Transitions of Nonequilibrium Systems: Cell Fate Decision Making

The underlying interactions in physical and biological systems often lead to a variety of behaviors and emergent states or phases. Under certain conditions, these phases can be transformed from one to another. The phase transition behaviors can be described by the bifurcation or catastrophe where different stable/unstable states can branch out or meet together with the birth of the new and death of the old states. Despite significant efforts, how the bifurcation and catastrophe actually occur dynamically and the associated mechanisms for nonequilibrium systems are still not very clear. As an example, we study the underlying mechanism of cell differentiation through bifurcations. Cell differentiation is one of the key fate decision-making processes that a cell faces. It is crucial for the development of multicellular organisms. Under induction, gene regulation changes, or stochastic fluctuations, the cell fate decision-making processes can exhibit different types of bifurcations or phase transitions. In order to understand the underlying mechanism, it is crucial to find out where and how the bifurcation occurs. However, this is still largely unknown. In this study, we found that the average of the curl flux as a major component of the driving force for the dynamics in addition to the landscape gradient and the associated entropy production rate both reach maximum near the bifurcation. This indicates that the curl flux and entropy production rate may provide the dynamical and thermodynamic origins of the bifurcation/catastrophe or phase transitions for cell differentiation and this possibly applies to many other nonequilibrium active systems.

Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis-Menten enzyme kinetics

The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.

Interferometric fluorescence cross correlation spectroscopy

Measuring transport properties like diffusion and directional flow is essential for understanding dynamics within heterogeneous systems including living cells and novel materials. Fluorescent molecules traveling within these inhomogeneous environments under the forces of Brownian motion and flow exhibit fluctuations in their concentration, which are directly linked to the transport properties. We present a method utilizing single photon interference and fluorescence correlation spectroscopy (FCS) to simultaneously measure transport of fluorescent molecules within aqueous samples. Our method, within seconds, measures transport in thousands of homogenous voxels (100 nm)³ and under certain conditions, eliminates photo-physical artifacts associated with blinking of fluorescent molecules. A comprehensive theoretical framework is presented and validated by measuring transport of quantum dots, associated with VSV-G receptor along cellular membranes as well as within viscous gels.

Probing the intracellular refractive index and molecular interaction of gold nanoparticles in HeLa cells using single particle spectroscopy

BACKGROUND

We have introduced a novel method to quantify the intracellular refractive index (RI) of living cells and determine the molecular interaction of two interacting molecules using single particle spectroscopy. The advantages of this proposed technique over fluorescence-based imaging techniques is that it does not require any contrasting agent and it does not blink and bleach. Instead, our technique provides a non-destructive, non-invasive, high-resolution imaging of live cells.

METHODS

To verify our technique, we initially tested our approach for a dielectric medium where gold nanoparticles (AuNPs) were embedded in a polyvinyl alcohol (PVA) matrix, which was then extended to the cellular environment. In the dielectric medium, we identified the single particle and dimer and determined the interparticle distance of AuNPs using confocal laser scattering microscopy. We also determined the single particle RI from dark-field scattering microscopy images, which was confirmed with Mie theory and finite-difference time-domain (FDTD) simulated results. The single particle spectroscopy and microscopy technique was then extended to determine the intracellular RI and biomolecular interaction inside living cells using hyperspectral imaging and dark-field scattering microscopy.

RESULTS

The novelty of the paper lies in the demonstration of a direct and accurate method to probe the intracellular RI and molecular interaction focused on single particle analysis whereas previous demonstrations were based on AuNP ensembles. Optically acquired single particle and dimer images was verified by correlated SEM images also optical spectrum with analytical models and FDTD simulations for both the dielectric and cellular environment. We reported the interparticle distance of AuNPs inside HeLa cells and intracellular refractive index, which was also confirmed with Mie Theory and extensive FDTD simulations.

CONCLUSION

Moreover, we believe that our in-depth plasmonic NP-based alternate imaging technique will provide a new insight in monitoring cellular dynamics and tracking the targeted NPs within live cells, enabling us to use plasmonic NPs as an intracellular biosensor.

Exploring the Underlying Mechanisms of the Xenopus laevis Embryonic Cell Cycle

The cell cycle is an indispensable process in proliferation and development. Despite significant efforts, global quantification and physical understanding are still challenging. In this study, we explored the mechanisms of the Xenopus laevis embryonic cell cycle by quantifying the underlying landscape and flux. We uncovered the Mexican hat landscape of the Xenopus laevis embryonic cell cycle with several local basins and barriers on the oscillation path. The local basins characterize the different phases of the Xenopus laevis embryonic cell cycle, and the local barriers represent the checkpoints. The checkpoint mechanism of the cell cycle is revealed by the landscape basins and barriers. While landscape shape determines the stabilities of the states on the oscillation path, the curl flux force determines the stability of the cell cycle flow. Replication is fundamental for biology of living cells. We quantify the input energy (through the entropy production) as the thermodynamic requirement for initiation and sustainability of single cell life (cell cycle). Furthermore, we also quantify curl flux originated from the input energy as the dynamical requirement for the emergence of a new stable phase (cell cycle). This can provide a new quantitative insight for the origin of single cell life. In fact, the curl flux originated from the energy input or nutrition supply determines the speed and guarantees the progression of the cell cycle. The speed of the cell cycle is a hallmark of cancer. We characterized the quality of the cell cycle by the coherence time and found it is supported by the flux and energy cost. We are also able to quantify the degree of time irreversibility by the cross correlation function forward and backward in time from the stochastic traces in the simulation or experiments, providing a way for the quantification of the time irreversibility and the flux. Through global sensitivity analysis upon landscape and flux, we can identify the key elements for controlling the cell cycle speed. This can help to design an effective strategy for drug discovery against cancer.

Kinetics of biochemical sensing by single cells and populations of cells

We investigate the collective stationary sensing using N communicative cells, which involves surface receptors, diffusive signaling molecules, and cell-cell communication messengers. We restrict the scenarios to the signal-to-noise ratios (SNRs) for both strong communication and extrinsic noise only. We modified a previous model [Bialek and Setayeshgar, Proc. Natl. Acad. Sci. USA 102, 10040 (2005)PNASA60027-842410.1073/pnas.0504321102] to eliminate the singularities in the fluctuation correlations by considering a uniform receptor distribution over the surface of each cell with a finite radius a. The modified model enables a simple and rigorous mathematical treatment of the collective sensing phenomenon. We then derive the scaling of the SNR for both juxtacrine and autocrine cases in all dimensions. For the optimal locations of the cells in the autocrine case, we find identical scaling for both two and three dimensions.

The dynamic mechanism of noisy signal decoding in gene regulation

Experimental evidence supports that signaling pathways can induce different dynamics of transcription factor (TF) activation, but how an input signal is encoded by such a dynamic, noisy TF and further decoded by downstream genes remains largely unclear. Here, using a system of stochastic transcription with signal regulation, we show that (1) keeping the intensity of the signal noise invariant but prolonging the signal duration can both enhance the mutual information (MI) and reduce the energetic cost (EC); (2) if the signal duration is fixed, the larger MI needs the larger EC, but if the signal period is fixed, there is an optimal time that the signal spends at one lower branch, such that MI reaches the maximum; (3) if both the period and the duration are simultaneously fixed, increasing the input noise can always enhance MI in the case of transcription regulation rather than in the case of degradation regulation. In addition, we find that the input noise can induce stochastic focusing in a regulation-dependent manner. These results reveal not only the dynamic mechanism of noisy signal decoding in gene regulation but also the essential role of external noise in controlling gene expression levels.

Using Multiorder Time-Correlation Functions (TCFs) To Elucidate Biomolecular Reaction Pathways from Microsecond Single-Molecule Fluorescence Experiments

Recent advances in single-molecule fluorescence imaging have made it possible to perform measurements on microsecond time scales. Such experiments have the potential to reveal detailed information about the conformational changes in biological macromolecules, including the reaction pathways and dynamics of the rearrangements involved in processes, such as sequence-specific DNA "breathing" and the assembly of protein-nucleic acid complexes. Because microsecond-resolved single-molecule trajectories often involve "sparse" data, that is, they contain relatively few data points per unit time, they cannot be easily analyzed using the standard protocols that were developed for single-molecule experiments carried out with tens-of-millisecond time resolution and high "data density." Here, we describe a generalized approach, based on time-correlation functions, to obtain kinetic information from microsecond-resolved single-molecule fluorescence measurements. This approach can be used to identify short-lived intermediates that lie on reaction pathways connecting relatively long-lived reactant and product states. As a concrete illustration of the potential of this methodology for analyzing specific macromolecular systems, we accompany the theoretical presentation with the description of a specific biologically relevant example drawn from studies of reaction mechanisms of the assembly of the single-stranded DNA binding protein of the T4 bacteriophage replication complex onto a model DNA replication fork.

Fluctuations in reactive networks subject to extrinsic noise studied in the framework of the chemical Langevin equation

Theoretical and experimental studies have shown that the fluctuations of in vivo systems break the fluctuation-dissipation theorem. One can thus ask what information is contained in the correlation functions of protein concentrations and how they relate to the response of the reactive network to a perturbation. Answers to these questions are of prime importance to extract meaningful parameters from the in vivo fluorescence correlation spectroscopy data. In this paper we study the fluctuations of the concentration of a reactive species involved in a cyclic network that is in a nonequilibrium steady state perturbed by a noisy force, taking into account both the breaking of detailed balance and extrinsic noises. Using a generic model for the network and the extrinsic noise, we derive a chemical Langevin equation that describes the dynamics of the system, we determine the expressions of the correlation functions of the concentrations, and we estimate the deviation of the fluctuation-dissipation theorem and the range of parameters in which an effective temperature can be defined.

A novel hybrid single molecule approach reveals spontaneous DNA motion in the nucleosome

Structural dynamics of nucleic acid and protein is an important physical basis of their functions. These motions are often very difficult to synchronize and too fast to be clearly resolved with the currently available single molecule methods. Here we demonstrate a novel hybrid single molecule approach combining stochastic data analysis with fluorescence correlation that enables investigations of sub-ms unsynchronized structural dynamics of macromolecules. Based on the method, we report the first direct evidence of spontaneous DNA motions at the nucleosome termini. The nucleosome, comprising DNA and a histone core, is the fundamental packing unit of eukaryotic genes that must be accessed during various genome transactions. Spontaneous DNA opening at the nucleosome termini has long been hypothesized to enable gene access in the nucleosome, but has yet to be directly observed. Our approach reveals that DNA termini in the nucleosome open and close repeatedly at 0.1-1 ms(-1). The kinetics depends on salt concentration and DNA-histone interactions but not much on DNA sequence, suggesting that this dynamics is universal and imposes the kinetic limit to gene access. These results clearly demonstrate that our method provides an efficient and robust means to investigate unsynchronized structural changes of DNA at a sub-ms time resolution.

Determination of reaction flux from concentration fluctuations near a Hopf bifurcation

Small open chemical systems, typically associated with far-from-equilibrium, nonlinear stochastic dynamics, offer the appropriate framework to elucidate biological phenomena at the cellular scale. Stochastic differential equations of Langevin-type are employed to establish the relation between the departure from equilibrium and the time cross-correlation functions of concentration fluctuations for chemical species susceptible to oscillate. Except in the immediate vicinity of the Hopf bifurcation, the results are in agreement with simulations of the chemical master equation but always differ from the prediction obtained for linear deterministic dynamics. In general, the magnitude of the asymmetry of time correlation functions definitely depends on the reaction flux circulating in an open system but also on the details of the nonlinearities of deterministic dynamics.

Statistics and Related Topics in Single-Molecule Biophysics

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra- and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steady-state probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.

Vectorized data acquisition and fast triple-correlation integrals for Fluorescence Triple Correlation Spectroscopy

Fluorescence Correlation Spectroscopy (FCS) is widely used to quantitate reaction rates and concentrations of molecules in vitro and in vivo. We recently reported Fluorescence Triple Correlation Spectroscopy (F3CS), which correlates three signals together instead of two. F3CS can analyze the stoichiometries of complex mixtures and detect irreversible processes by identifying time-reversal asymmetries. Here we report the computational developments that were required for the realization of F3CS and present the results as the Triple Correlation Toolbox suite of programs. Triple Correlation Toolbox is a complete data analysis pipeline capable of acquiring, correlating and fitting large data sets. Each segment of the pipeline handles error estimates for accurate error-weighted global fitting. Data acquisition was accelerated with a combination of off-the-shelf counter-timer chips and vectorized operations on 128-bit registers. This allows desktop computers with inexpensive data acquisition cards to acquire hours of multiple-channel data with sub-microsecond time resolution. Off-line correlation integrals were implemented as a two delay time multiple-tau scheme that scales efficiently with multiple processors and provides an unprecedented view of linked dynamics. Global fitting routines are provided to fit FCS and F3CS data to models containing up to ten species. Triple Correlation Toolbox is a complete package that enables F3CS to be performed on existing microscopes.

Global parameter identification of stochastic reaction networks from single trajectories

We consider the problem of inferring the unknown parameters of a stochastic biochemical network model from a single measured time-course of the concentration of some of the involved species. Such measurements are available, e.g., from live-cell fluorescence microscopy in image-based systems biology. In addition, fluctuation time-courses from, e.g., fluorescence correlation spectroscopy (FCS) provide additional information about the system dynamics that can be used to more robustly infer parameters than when considering only mean concentrations. Estimating model parameters from a single experimental trajectory enables single-cell measurements and quantification of cell-cell variability. We propose a novel combination of an adaptive Monte Carlo sampler, called Gaussian Adaptation (GaA), and efficient exact stochastic simulation algorithms (SSA) that allows parameter identification from single stochastic trajectories. We benchmark the proposed method on a linear and a non-linear reaction network at steady state and during transient phases. In addition, we demonstrate that the present method also provides an ellipsoidal volume estimate of the viable part of parameter space and is able to estimate the physical volume of the compartment in which the observed reactions take place.

Quantitation of ten 30S ribosomal assembly intermediates using fluorescence triple correlation spectroscopy

The self-assembly of bacterial 30S ribosomes involves a large number of RNA folding and RNA-protein binding steps. The sequence of steps determines the overall assembly mechanism and the structure of the mechanism has ramifications for the robustness of biogenesis and resilience against kinetic traps. Thermodynamic interdependencies of protein binding inferred from omission-reconstitution experiments are thought to preclude certain assembly pathways and thus enforce ordered assembly, but this concept is at odds with kinetic data suggesting a more parallel assembly landscape. A major challenge is deconvolution of the statistical distribution of intermediates that are populated during assembly at high concentrations approaching in vivo assembly conditions. To specifically resolve the intermediates formed by binding of three ribosomal proteins to the full length 16S rRNA, we introduce Fluorescence Triple-Correlation Spectroscopy (F3CS). F3CS identifies specific ternary complexes by detecting coincident fluctuations in three-color fluorescence data. Triple correlation integrals quantify concentrations and diffusion kinetics of triply labeled species, and F3CS data can be fit alongside auto-correlation and cross-correlation data to quantify the populations of 10 specific ribosome assembly intermediates. The distribution of intermediates generated by binding three ribosomal proteins to the entire native 16S rRNA included significant populations of species that were not previously thought to be thermodynamically accessible, questioning the current interpretation of the classic omission-reconstitution experiments. F3CS is a general approach for analyzing assembly and function of macromolecular complexes, especially those too large for traditional biophysical methods.

The spectroscopic basis of fluorescence triple correlation spectroscopy

We have developed fluorescence triple correlation spectroscopy (F3CS) as an extension of the widely used fluorescence microscopy technique fluorescence correlation spectroscopy. F3CS correlates three signals at once and provides additional capabilities for the study of systems with complex stoichiometry, kinetic processes, and irreversible reactions. A general theory of F3CS was developed to describe the interplay of molecular dynamics and microscope optics, leading to an analytical function to predict experimental triple correlations of molecules that freely diffuse through the tight focus of the microscope. Experimental correlations were calculated from raw fluorescence data using triple correlation integrals that extend multiple-tau correlation theory to delay times in two dimensions. The quality of experimental data was improved by tuning specific spectroscopic parameters and employing multiple independent detectors to minimize optoelectronic artifacts. Experiments with the reversible system of freely diffusing 16S rRNA revealed that triple correlation functions contain symmetries predicted from time-reversal arguments. Irreversible systems are shown to break these symmetries, and correlation strategies were developed to detect time-reversal asymmetries in a comprehensive way with respect to two delay times, each spanning many orders of magnitude in time. The correlation strategies, experimental approaches, and theory developed here enable studies of the composition and dynamics of complex systems using F3CS.

Fluorescence correlation spectroscopy: past, present, future

In recent years fluorescence correlation spectroscopy (FCS) has become a routine method for determining diffusion coefficients, chemical rate constants, molecular concentrations, fluorescence brightness, triplet state lifetimes, and other molecular parameters. FCS measures the spatial and temporal correlation of individual molecules with themselves and so provides a bridge between classical ensemble and contemporary single-molecule measurements. It also provides information on concentration and molecular number fluctuations for nonlinear reaction systems that complement single-molecule measurements. Typically implemented on a fluorescence microscope, FCS samples femtoliter volumes and so is especially useful for characterizing small dynamic systems such as biological cells. In addition to its practical utility, however, FCS provides a window on mesoscopic systems in which fluctuations from steady states not only provide the basis for the measurement but also can have important consequences for the behavior and evolution of the system. For example, a new and potentially interesting field for FCS studies could be the study of nonequilibrium steady states, especially in living cells.

Intrinsic noise alters the frequency spectrum of mesoscopic oscillatory chemical reaction systems

Mesoscopic oscillatory reaction systems, for example in cell biology, can exhibit stochastic oscillations in the form of cyclic random walks even if the corresponding macroscopic system does not oscillate. We study how the intrinsic noise from molecular discreteness influences the frequency spectrum of mesoscopic oscillators using as a model system a cascade of coupled Brusselators away from the Hopf bifurcation. The results show that the spectrum of an oscillator depends on the level of noise. In particular, the peak frequency of the oscillator is reduced by increasing noise, and the bandwidth increased. Along a cascade of coupled oscillators, the peak frequency is further reduced with every stage and also the bandwidth is reduced. These effects can help understand the role of noise in chemical oscillators and provide fingerprints for more reliable parameter identification and volume measurement from experimental spectra.

Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation

Circadian rhythms with a period of ~24 h, are natural timing machines. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. However, the global and physical natures of the oscillation under fluctuations are still not very clear. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. The potential landscape of the network is uncovered and has a global Mexican-hat shape. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.

Temporal correlation of photons following frequency up-conversion

We demonstrate an approach to measure temporal correlations of photons in the near infrared range using frequency up-conversion. In this approach, the near infrared signal photons are converted into the visible range, in which highly efficient silicon avalanche photodiodes are used to perform the temporal correlation measurements. A coherent light source and a pseudo-thermal light source were used in the experiment. The results are in agreement with theoretical values and those obtained from measurements directly made using superconducting nanowire single photon detectors. We conclude that the temporal correlation (up to 4th order) of photons was preserved in the frequency up-conversion process. We further theoretically and experimentally studied the influence of the dark counts on the measurement. The setup uses commercially available components and achieves high total detection efficiency (~26%).

Event ordering in live-cell imaging determined from temporal cross-correlation asymmetry

We use the temporal asymmetry of the cross-correlation function to determine the temporal ordering of spatially localized cellular events in live-cell multichannel fluorescence imaging. The analysis is well suited to noisy, stochastic systems where the temporal order may not be apparent in the raw data. The approach is applicable to any biochemical reaction not in chemical equilibrium, including protein complex assembly, sequential enzymatic processes, gene regulation, and other cellular signaling events. As an automated quantitative measure, this approach allows the data to be readily interpreted statistically with minimal subjective biases. We first test the technique using simulations of simple biophysical models with a definite temporal ordering. We then demonstrate the approach by extracting the temporal ordering of three proteins-actin, sorting nexin 9, and clathrin-in the endocytic pathway.

Communications: Can one identify nonequilibrium in a three-state system by analyzing two-state trajectories?

For a three-state Markov system in a stationary state, we discuss whether, on the basis of data obtained from effective two-state (or on-off) trajectories, it is possible to discriminate between an equilibrium state and a nonequilibrium steady state. By calculating the full phase diagram we identify a large region where such data will be consistent only with nonequilibrium conditions. This regime is considerably larger than the region with oscillatory relaxation, which has previously been identified as a sufficient criterion for nonequilibrium.

FRET fluctuation spectroscopy of diffusing biopolymers: contributions of conformational dynamics and translational diffusion

The use of fluorescence correlation spectroscopy (FCS) to study conformational dynamics in diffusing biopolymers requires that the contributions to the signal due to translational diffusion are separated from those due to conformational dynamics. A simple approach that has been proposed to achieve this goal involves the analysis of fluctuations in fluorescence resonance energy transfer (FRET) efficiency. In this work, we investigate the applicability of this methodology by combining Monte Carlo simulations and experiments. Results show that diffusion does not contribute to the measured fluctuations in FRET efficiency in conditions where the relaxation time of the kinetic process is much shorter than the mean transit time of the molecules in the optical observation volume. However, in contrast to what has been suggested in previous work, the contributions of diffusion are otherwise significant. Neglecting the contributions of diffusion can potentially lead to an erroneous interpretation of the kinetic mechanisms. As an example, we demonstrate that the analysis of FRET fluctuations in terms of a purely kinetic model would generally lead to the conclusion that the system presents complex kinetic behavior even for an idealized two-state system.

Diffusion, transport, and cell membrane organization investigated by imaging fluorescence cross-correlation spectroscopy

Cell membrane organization is dynamic and is assumed to have different characteristic length scales. These length scales, which are influenced by lipid and protein composition as well as by the cytoskeleton, can range from below the optical resolution limit (as with rafts or microdomains) to far above the resolution limit (as with capping phenomena or the formation of lipid "platforms"). The measurement of these membrane features poses a significant problem because membrane dynamics are on the millisecond timescale and are thus beyond the time resolution of conventional imaging approaches. Fluorescence correlation spectroscopy (FCS), a widely used spectroscopic technique to measure membrane dynamics, has the required time resolution but lacks imaging capabilities. A promising solution is the recently introduced method known as imaging total internal reflection (ITIR)-FCS, which can probe diffusion phenomena in lipid membranes with good temporal and spatial resolution. In this work, we extend ITIR-FCS to perform ITIR fluorescence cross-correlation spectroscopy (ITIR-FCCS) between pixel areas of arbitrary shape and derive a generalized expression that is applicable to active transport and diffusion. ITIR-FCCS is applied to model systems exhibiting diffusion, active transport, or a combination of the two. To demonstrate its applicability to live cells, we observe the diffusion of a marker, the sphingolipid-binding domain (SBD) derived from the amyloid peptide Abeta, on live neuroblastoma cells. We investigate the organization and dynamics of SBD-bound lipid microdomains under the conditions of cholesterol removal and cytoskeleton disruption.

Revival of high-order fluorescence correlation analysis: generalized theory and biochemical applications

Analysis of high-order correlations in fluorescence fluctuation spectroscopy was developed in the late 1980s but since then has been replaced by alternative brightness analysis methods. However, high-order correlation has important advantages in many experiments. We present a new cumulant-based formalism of high-order correlation that greatly simplifies data analysis. The new formalism is used to derive general expressions for variance of high-order correlations that show good agreement with experiment in a model system of fluorescently labeled DNA oligomers. A simulation of binary systems in which both diffusion time and brightness are varied illustrates clearly that high-order analysis has better sensitivity to brightness than fluorescence correlation spectroscopy (FCS). These results have implications for analysis of isomerization reactions and dual-beam FCS with flow. We also demonstrate that high-order correlations can detect photobleaching in the observation volume. The application of this formalism to many FCS-based experiments allows more accurate analysis in addition to describing more molecular parameters.

High-order temporal coherences of chaotic and laser light

We demonstrate a new approach to measuring high-order temporal coherences that uses a four-element superconducting nanowire single-photon detector. The four independent, interleaved single-photon-sensitive elements parse a single spatial mode of an optical beam over dimensions smaller than the minimum diffraction-limited spot size. Integrating this device with four-channel time-tagging electronics to generate multi-start, multi-stop histograms enables measurement of temporal coherences up to fourth order for a continuous range of all associated time delays. We observe high-order photon bunching from a chaotic, pseudo-thermal light source, measuring maximum third- and fourth-order coherence values of 5.87 +/- 0.17 and 23.1 +/- 1.8, respectively, in agreement with the theoretically predicted values of 3! = 6 and 4! = 24. Laser light, by contrast, is confirmed to have coherence values of approximately 1 for second, third and fourth orders at all time delays.

Determination of the six rate constants of a three-state enzymatic network and a noninvasive test of detailed balance

The Michaelis-Menten mechanism is unanimously recognized by experimentalists and theoreticians as the reference model for the description of enzymatic catalysis. The recent explosion in the diversity of fluorescent probes solves the problem of in situ observation of proteins and the experimental investigation of enzymatic dynamics, which determines the Michaelis constant or a small number of relaxation times, is becoming more and more common. We propose a protocol for the full characterization of enzyme kinetics in the framework of the Michaelis-Menten mechanism. The method relies on the measurement of the oscillation amplitude of the enzymatic concentrations, when the biological medium is submitted to a temperature modulation of a few degrees. Analytical expressions of all the rate constants as functions of the concentration amplitudes are derived. The noninvasive character of the perturbation and the assessable uncertainty on the rate constant values make an in situ test of detailed balance possible.

Resonant response to temperature modulation for enzymatic dynamics characterization

We consider enzymes involved in a three-state Michaelis-Menten kinetics and submitted to well-chosen temperature modulations of small amplitude. From the first-order amplitudes of concentration oscillations, we design a response function that is maximum for targeted values of the chemical relaxation times. This resonant function can be used to screen a large set of enzymes and identify the one governed by the desired kinetics. The method gives access to all the dynamical parameters of the targeted enzyme without resorting to a fit. We show how to estimate the precision of this parameter determination and give some hints for experimental validation.