European Biophysics Journal

Although rhizosphere microbiomes are known to enhance plants resistance to water stress, it is believed that only fungi actively contribute to the transport and uptake of water. We investigated the biomechanical impact of bacterial motility on water transport in soil by combining surface tension measurements and water infiltration experiments in soil microcosms. We observed that flagellar-based motility in Bacillus subtilis cells reduces the apparent surface tension of fluids by up to 15%. The effect reported depends on cell density and swimming speed, confirming its biomechanical origin, and was able to accelerate water infiltration and rewetting of soil. We conclude that Bacillus subtilis facilitates soil water transport through the deformation of air water interfaces in pores.

Aims/hypothesisQuantitative nanoscale analysis of insulin secretory granules (ISGs) in human pancreatic tissue has been limited by the lack of imaging methods that combine high resolution with large-scale sampling. We aimed to establish expansion microscopy (ExM) as a platform for in situ, quantitative analysis of ISG organisation in human {beta}-cells and to assess whether type 2 diabetes (T2D) is associated with alterations in granule size, abundance or spatial organisation. MethodsWe applied Magnify ExM to PFA-fixed, paraffin-embedded pancreatic tissue sections from 6 human donors, 3 non-diabetic (ND) and 3 T2D, enabling super-resolution optical imaging of insulin-labelled granules. Insulin-positive structures were segmented and analysed using a morphometric pipeline to quantitatively assess size, shape and spatial features. Granule clustering was quantified based on combined area and roundness criteria. ResultsThe diameter distribution of highly circular granules was similar between ND and T2D samples and estimates of granule number per cell indicated only a modest reduction in T2D ([~]25%). In contrast, mapping insulin-positive structures in a roundness-area space revealed a marked enrichment of large, irregular objects consistent with granule clustering in T2D. The fraction of clustered granules was significantly increased in T2D and strongly inversely correlated with insulin stimulation index (r = -0.85). Conclusions/interpretationThese results establish expansion microscopy as a powerful platform for quantitative nanoscale analysis of human pancreatic tissue and identify altered spatial organisation of insulin granules, rather than marked granule depletion, as a prominent feature associated with {beta}-cell dysfunction in T2D. Research in contextO_ST_ABSWhat is already known about this subject?C_ST_ABSO_LI{beta}-cell dysfunction in type 2 diabetes is often attributed to reduced insulin content or {beta}-cell loss. C_LIO_LIInsulin secretory granules (ISGs) have been characterised ultrastructurally, but quantitative analysis in human tissue remains limited. C_LIO_LISuper-resolution approaches, including expansion microscopy, are emerging tools for nanoscale imaging in biological tissues. C_LI What is the key question?O_LIIs {beta}-cell dysfunction in type 2 diabetes associated with depletion of insulin granules or with altered spatial organisation? C_LI What are the new findings?O_LIInsulin granule size distribution is largely preserved in type 2 diabetes, with only a modest reduction in granule number per cell. C_LIO_LIA significant increase in insulin granule clustering is observed in diabetic {beta}-cells. C_LIO_LIGranule clustering is strongly inversely correlated with insulin secretion in the same donor tissues. C_LI How might this impact on clinical practice in the foreseeable future?O_LIIdentifying altered granule organisation as a feature of {beta}-cell dysfunction may help refine the understanding of disease mechanisms and guide future strategies targeting {beta}-cell function. C_LI

With the development of microfluidics, it has now become possible to assess the susceptibility of bacteria to antibiotics at the single-cell level instead of relying on population measurements. Such studies are particularly relevant when the growth of bacterial population in the presence of antibiotics is heterogeneous. Here, we build a model to describe such a case, and apply it to experimental measurements on a small population of E. Coli exposed to ciprofloxacin, a drug which is well known for triggering a bistable response.

As the duration of space flights increases, so does the need to optimize off-planet microbial growth. Microbes can both be unintentionally brought into space and cause human disease or be intentionally harnessed for on-site bioengineering functions. However, optimizing microbial growth is challenging due to an insufficient understanding of how microbial communities are affected by the extraterrestrial environment. To address this gap, we have modified a previously developed model for cell growth in microgravity. By improving the functional form used for cell growth as well as the code usability, we enable further research into how microbial communities are influenced by gravity. Applying this model to isolate individual effects of gravity on cell growth indicates that a lack of gravity-driven flow decreases cell growth in microgravity, while the absence of sedimentation increases cell growth in microgravity. These opposite effects likely contribute to the system-dependent effects of microgravity observed experimentally.

Quantum kernel methods offer a potential advantage for classification tasks in high-dimensional feature spaces, yet their practical benefit critically depends on how input features are prepared. We compare five dimensionality reduction strategies--principal component analysis (PCA), Gaussian random projection (RP Gaussian), sparse random projection (RP Sparse), partial least squares (PLS), and uniform manifold approximation and projection (UMAP) -- as pre-processing steps for quantum kernel support vector machines (SVMs) applied to trabecular bone classification from synthetic micro-computed tomography (micro-CT) data. Using a custom procedural generator based on Gaussian random field zero-crossings, we produced 500 synthetic trabecular bone volumes with controlled morphometric properties such as bone volume fraction (BV/TV), trabecular thickness (Tb.Th), number (Tb.N) and spacing (Tb.Sp). Texture features extracted from grayscale slices are reduced to 8-dimensional quantum circuit inputs via each method, then classified using both classical radial basis function (RBF)-SVMs and quantum kernel SVMs with ZZ feature maps on a statevector simulator, both evaluated with 5 x 5 repeated stratified cross-validation (25 folds). Our results show that UMAP is the only reduction method where the quantum kernel remains competitive with the classical baseline. Under repeated cross-validation, UMAP showed a +0.032 accuracy gap favouring the quantum kernel (Dietterich 5 x 2 CV p = 0.177); however, validation on 10 fully independent datasets--each with independently generated samples, separate reduction fits, and separate kernel matrices -- reversed the sign to -0.030 (paired t-test p = 0.123; Wilcoxon p = 0.193; quantum wins 3/10 datasets), indicating that the apparent advantage was likely inflated by fold dependence. Nevertheless, UMAPs gap remains small and non-significant in both analyses, whereas all linear methods (PCA, RP Gaussian, PLS) show substantial quantum deficits of -0.090 to -0.116 across BV/TV classification, with PCA and PLS remaining significant under corrected tests (5 x 2 CV p = 0.004 and p = 0.007 respectively). We additionally evaluate quantum kernel ridge regression for continuous morphometric prediction, finding that ZZ quantum kernels fail uniformly at regression (negative R2 for all methods except PLS at 4 qubits), suggesting that the ZZ kernel captures decision boundaries but not smooth metric structure. These findings provide practical guidance for feature engineering in near-term quantum machine learning pipelines and demonstrate that the choice of dimensionality reduction can determine whether quantum kernels remain competitive with classical baselines.

Significant cellular processes, including protein sorting, signal transduction, and pathogen entry, amongst others, are associated with membrane microdomains, also known as lipid rafts. Lipid rafts, due to their unique biophysical properties compared to their surrounding environment, which stem from their distinct lipid and protein profiles, have garnered interest in methods and techniques that tune their coexisting liquid-ordered/liquid-disordered state, aiming to disrupt or destabilize them. Since cholesterol stabilizes the membrane domain, cholesterol-depleting compounds like cyclodextrin can be used to destabilize and disrupt the membrane rafts. Overall, given the membrane rafts importance in biological processes, it is crucial to understand the biophysical factors that influence its stability. In this study, we present a new method for disrupting and dissolving lipid rafts in a model system of phase-separated supported lipid bilayer (SLB) patches composed of DOPC, DPPC, and cholesterol. Using fluorescence microscopy to monitor the liquid ordered (Lo) and liquid disordered (Ld) phases of the SLB patches, we observed that adding DOPC liposomes causes a transformation of the co-existing Ld and Lo phases into a single-phase bilayer. On the other hand, adding liposomes that match the lipid content of the phase-separated SLB patch increase the areas of the existing Ld and Lo phases. This work also offers a new method for redistributing raft-localized molecules, confirmed by tracking the redistribution of cholera toxin bound to GM1 after domain dissolution with DOPC liposomes. The work describes an alternative method for dynamically altering membrane composition and dissolving domains via liposome addition, rather than lipid depletion or exchange.

Vertex models are widely used within the field of developmental biology to study tissue morphogenesis. These models are well-suited for modeling deformation at the cellular level where movement is driven by local forces. However, understanding how these microscopic movements coordinate to yield macroscopic phenomena such as the shapes of entire tissues remains a challenge. Here we study a top-down approach using differentiable programming on a simplified vertex model of a laminar tissue, and investigate whether the attributes of individual cells can be tuned to make the mesh as a whole acquire a predefined shape. We let the mesh evolve according to simple rules defined by the input to each polygon, and evaluate the resulting shape against a target boundary. Additionally, we show how the high degeneracy of the output can be reduced by constraining the polygon distributions: first, by adding simple penalties on tissue-wide attributes; and second, by dividing the tissue into regions, within which we bias the attributes toward characteristic values. Our study shows how a simple vertex model can be combined with differentiable programming to model developing tissues, and provides insight into the way individual cells must coordinate to yield macroscopic phenomena such as pre-programmed shapes.

The molten globule (MG) state is an intermediate in the unfolding pathway of proteins, typically triggered by denaturing agents such as urea, extreme pH, high pressure, or heat. The microscopic details of such states are far from understood. Here we study the MG states in protein Hen Egg-White Lysozyme (PDB ID: 1AKI) using microscopic constant pH molecular dynamics (CpHMD) simulations and experiments across a wide pH range. We observe that the titratable residues act as key drivers of conformational fluctuations, promoting the emergence of MG states at extreme pH. These states display partial unfolding, and small global structural changes (< 7% deviation). Hydration around the fluctuating acidic residues shows reduced water density and weakened hydrogen bonding at low pH. At high pH, hydration around acidic residues increases relative to pH = 7, whereas hydration around basic residues decreases. The translational and rotational dynamics of hydration water also exhibit pronounced pH dependence: the translational diffusion coefficient (Dtrans) increases linearly with decrease in pH in acidic medium and increases linearly with increasing pH in the basic regime. The rotational diffusion (Drot) shows similar dependencies on pH except a break at pH {approx} 4 corresponding to acidic residue pKa values. Our results may be useful to identify ligand binding of lysozyme in extreme pH conditions.

The Poisson-Nernst-Planck (PNP) system is an accurate model of electrodiffusion of ionic species. It is commonly used in situations where nanoscale resolution is required, for instance close to ion channels in the membranes of biological cells. The inherent stiffness of the equations has made them challenging to solve and has limited the applicability of the system. In particular, the time step required for stable solutions has typically needed to be very short (nanoseconds), which makes simulations on the time scale of an action potential (milliseconds) difficult. Recently, it has been observed that avoiding operator splitting and instead solving the concentration equations and the electrostatic equation in a coupled manner relaxes the time-step limitation considerably. However, no theoretical explanation of this observation has been provided. Here, we aim to explain why the coupled scheme allows much larger time steps. We illustrate the mechanism by considering special cases that define necessary, but not sufficient, conditions for stability. We also show that these conditions remain relevant for the fully coupled PNP model in 3D.

Invertebrate vision relies on bistable visual pigments flipping upon photon absorption between rhodopsin and metarhodopsin states. In living butterflies, the UV-VIS absorption spectra of rhodopsin and metarhodopsin, respectively with 11-cis and all-trans isomers of 3-hydroxy-retinal (A3) chromophore, can be conveniently recorded from the eyeshine, the light reflected from the compound eye after passing twice through the light-guiding rhabdoms. * Here, a microscope coupled with a broadband LED source and a microspectrometer was used to record photorelaxations reported in eyeshine reflection spectra. Fitting temporal exponential relaxations to log-reflectance arrays yielded transient and baseline spectra that are analogous to absorbance difference and sum, respectively. Both types of spectra were subjected to singular value decomposition and to fitting of templated visual pigment absorption spectra. * The compound eye of the high brown fritillary Fabriciana adippe was exposed to a series of second-long broadband light pulses, causing photorelaxations with time constants between 40 and 120 ms that led to 80% metarhodopsin in equilibrium. The transient and baseline spectra were fitted with pigment templates, estimating the alpha peak wavelength 547-552 nm for rhodopsin and 496-501 nm for metarhodopsin. The metarhodopsin to rhodopsin alpha peak absorbance ratio 1.25-1.35 is consistent with the isosbestic wavelength at 530 nm. The second isosbestic wavelength indicates that rhodopsin beta (UV) peak absorbs more strongly than metarhodopsin below 405 nm. * Baseline spectra, which were not explicitly analysed in previous studies, enable concatenation of exposures, monitor long-term changes of pigment, and enhance the estimation of beta peak parameters. * The method can be directly used in many butterflies and could be adapted to other insects, particularly fruitflies, facilitating studies of the relation between the visual pigment spectra and the opsin sequences. Spectroscopic results can be complemented with physiologically measured photoreceptor spectral sensitivity datasets and analysed with the same global fitting procedure.

The patch-clamp experimental technique is widely used to study the electrical properties of ion channels in biological and artificial lipid membranes. The key to the high quality of the experiments is the manufacturing of glass pipettes that provide highly electrically resistant contact between the edge of the pipette tip and the lipid bilayer. Preparation of the pipettes is particularly challenging for studies of the mitochondrial membranes due to the need for very small pipette tip sizes. Here, we present a robust procedure for producing pipettes suitable for experiments with native mitochondrial membranes. This procedure involves a two-step approach: initial fabrication of relatively large glass micropipettes using a standard micropipette puller, followed by tip refinement using a microforger to achieve smooth glass surface and reduced opening size. Pipette tip diameters and surface structure were examined using field emission - scanning electron microscopy (FE-SEM) imaging to assess the effects of variable parameters on pipette geometry and size. The resulting pipettes were validated in patch-clamp recording of the mitochondrial inner membranes. This approach enables the reproducible production of optimized pipettes for mitochondrial patch-clamp experiments, improving the quality and throughput of electrophysiological recordings of the mitochondrial ion channels.

Serine-aspartate repeat-containing protein D (SdrD) is a Staphylococcus aureus cell wall-anchored, calcium-binding adhesin member of the MSCRAMM Sdr subfamily that may contribute to bacterial adhesion and virulence. S. aureus is the most common cause of periprosthetic joint infection (PJI). Population-level distribution and sequence diversity of SdrD among clinical PJI isolates have not been systematically characterized, and the SdrD binding mechanism is still not well understood. To address these gaps, sdrD alleles were queried across 156 newly sequenced PJI isolates and compared to publicly available S. aureus genomes, and nucleotide- and protein-level phylogenies of the sdrCDE locus constructed. The SdrD crystal structure from S. aureus JH1 was determined, with solution small-angle X-ray scattering (SAXS) and molecular dynamics (MD) simulations, and assessment of conformational changes with calcium depletion. Three dominant sdrD subtypes were defined, associating with USA300, JH1, and TCH60; the JH1 sdrD subtype was predominant among PJI isolates. Structural studies showed that the conformation of individual domains and interdomain organization of the multidomain SdrD have limited flexibility in solution, and that the calcium-binding B domain retains its core fold under conditions of calcium depletion. Together, the findings presented support functional diversification among Sdr family members in mediating host attachment and inform a re-evaluation of the ligand-binding mechanism previously proposed for SdrD. AUTHOR SUMMARYStaphylococcus aureus is the leading cause of infections that develop around joint implants (periprosthetic joint infection, PJI). This bacterium has a large arsenal of surface proteins that allow it to stick to human tissues and implanted devices. This work focused on one such protein, SdrD, which has been linked to implant-associated infections but the structure and diversity of which among patients with PJI had not been well characterized. The genetic sequences of SdrD were analyzed across thousands of bacterial genomes, including those from patients with PJI. Distinct genetic variants of the protein were found, one of which was particularly common with PJI. The three-dimensional structure of SdrD was determined at atomic resolution and solution small-angle X-ray scattering (SAXS) and molecular dynamics used to study how it moves and responds to changes in its environment. Contrary to what was previously described, SdrD was shown to be relatively rigid. These findings change how SdrDs mechanism of action should be considered, potentially informing design strategies to block bacterial attachment before infection takes hold.

Due to recent technological advances, in situ structural cell biology is becoming a high throughput microscopy technique as all the steps of the workflow, from sample preparation to data analysis, are executed faster, more reliable and more reproducible. Sample thinning by cryoFIB-SEM is an essential tool in preparing electron transparent lamellae of biological specimens suitable for further characterization by cryoET. Modern cryoFIB-SEM instruments can be operated remotely and are capable of automated and unsupervised lamellae preparation. To take full advantage of these developments they need a constant supply of LN2 to maintain cryogenic conditions inside the microscope chamber. Here, we introduce a custom automated LN2 refill system that is compatible with gas cooled cryostages, supports long-term cryoFIB-SEM operations and liberates the user from highly repetitive and manual work. We believe this solution can be utilized with other cryoSEM or cryoFIB-SEM devices requiring N2 gas-flow cooling.

Understanding how biological shape and movement interact with surrounding fluids represents a fundamental challenge at the intersection of biology, physics, and engineering. Fish locomotion exemplifies this challenge: body morphology and swimming kinematics together determine the hydrodynamic forces and flow structures that enable efficient propulsion and maneuverability. Whereas biologists have long sought to connect morphological variation to swimming performance, traditional morphometric approaches provide limited insight into the fluid mechanical consequences of shape differences. Similarly, although computational fluid dynamics can reveal detailed flow physics, simulating hydrodynamics across diverse and dynamic morphologies remains prohibitively expensive for systematic investigation. To bridge this gap, we introduce a data-driven framework that connects fish body shape dynamics to hydro-dynamic performance through compact morphospace parameterization and reduced-order modeling. Using CFD simulations of 15 fish species from the Digital Life Project database (www.digitallife3d.org/3d-model), we generate hydrodynamic datasets capturing the shape-flow relationship. Principal Component Analysis (PCA) extracts four dominant shape parameters from dorsal body profiles, which are then integrated into an Inverse-Design with Dynamic Mode Decomposition (ID-DMD) framework to model the resulting fluid dynamics. The resulting modal analysis suggests that locomotion strategies emerge from specific shape-flow interactions. We further demonstrate the frameworks utility through single- and multi-objective shape optimization, showing how it enables efficient exploration of the morphology-hydrodynamics relationship. This approach offers a novel analysis and design tool for understanding how biological form and motion interact with fluid mechanics, with applications ranging from bio-inspired vehicle development to evolutionary biomechanics.

Hygroscopic surfaces act as local vapor sinks that reshape the condensation field around them, but whether distributed biological structures do the same has not been investigated. We have established that hyphae of fungal colonies functionally behave as vapor sinks, creating a dry region of width{delta} around themselves when placed on a cooled substrate. In addition, the radial distribution of droplet sizes steepens during condensation, and the rate at which droplets evaporate locally after chamber drying increases. In order to quantify this behavior, we employed a combination of time-resolved imaging and survival analysis to determine how long individual droplets persist on the surface surrounding the colony. These data were used to derive three quantitative measures of the vapor-sink effect. Each measure was found to be directly proportional to the vapor-sink strength of the substrate, as calibrated against NaCl-agar hydrogels of known water activity (LOOCV RMSE = 0.031 for recovered aw). These findings were consistent across three fungal genera (35 experiments), and all species fell along calibration lines defined by the hydrogel standards. This result is consistent with a diffusion-limited vapor-depletion framework. The measured genus-level{delta} ratios agreed to within 6% of predictions from structural absorbing capacity, and field measurements on Gymnosporangium-infected apple leaves were consistent with the same signatures under natural conditions. These results establish a non-contact method for inferring the material properties of thin hygroscopic biological surfaces from their condensation patterns.

Bacterial communities grow as dynamic populations that respond to their environments. A clinically relevant example is the inactivation of beta-lactam antibiotics by intracellular beta-lactamase in E. faecalis resistant strains. In these populations, resistant bacteria act as antibiotic sinks, detoxifying the environment and allowing sensitive bacteria to survive treatment through a cooperative interaction. In this work, we study strongly coupled planktonic and biofilm populations of mixed sensitive-resistant E. faecalis bacteria under antibiotic stress using fluorescent microscopy. The presence of resistant bacteria in the system benefits both resistant and sensitive cells, leading to mixed planktonic and biofilm populations at super-inhibitory drug concentrations. We show that a beta-lactam antibiotic with or without the addition of a beta-lactam inhibitor can lead to a population inversion effect, characterized by a non-monotonic relation between initial and final fractions of resistant bacteria. The effect is observed in both the planktonic and biofilm populations and is modulated by the total initial cell density. A well-mixed model with competition mediated by resource sharing and cooperation from global degradation of toxins predicts the experimentally observed behavior. These observations suggest underlying population-level mechanisms that are largely independent of biofilm spatial structure.

Coarse-grained structure-based models (CG-SBMs; or G[o] models) are simplified potential energy functions of biomolecules or biomolecular complexes that encode their structure. Molecular dynamics simulations of such SBMs have been successfully used to study long time-scale dynamics such as protein and RNA folding, and large conformational transitions of biomolecular complexes. SBMs have several advantages: (1) Their MD simulations are computationally inexpensive, making extensive sampling easily accessible to many researchers. (2) They are easy to modify and can be adapted for the specific biomolecular problem that needs to be investigated. However, the force-fields of SBMs are not usually included in commonly used biomolecular simulation packages resulting in a barrier to their use. Here, we present SuBMIT (Structure Based Models Input Toolkit; https://github.com/sglabncbs/submit), a toolkit for generating coarse-grained SBM input files for performing MD simulations with GROMACS and OpenMM/OpenSMOG. Simulations whose input files can be generated using the different flavors of CG-SBMs present in SuBMIT include the folding and conformational ensembles of proteins with intrinsically disordered regions, 3D-domain-swapping in proteins and the dynamics of RNA-protein assemblies (e.g., simple RNA viruses).

Sarcomeres, the basic repeating unit of striated muscle, are joined together by crosslinked actin filaments found at the boundaries of muscle sarcomeres, termed Z-discs. Z-discs play a key role in cardiac signalling and disease, however, the arrangement and function of many of the proteins present in the Z-disc remain to be understood. Here, we determined the organisation of 3 key proteins, ZASP, [a]-Actinin-2 and the Z1Z2 epitope of titin, located within the Z-disc. We fluorescently labelled these proteins in cardiac myofibrils using Adhirons specific to each protein and used interferometric photoactivated localization microscopy (iPALM) to obtain the 3D position of these proteins to a high precision (<10nm in x,y,z). We then used PERPL (Pattern Extraction from Relative Positions of Localisations) to analyse patterns in the relative positions of the proteins and reveal their underlying organisation. This analysis revealed that ZASP and [a]-Actinin-2 have a similar repeating organisation, but that the organisation of Z1Z2 is different.

GFAP is a type III intermediate filament primarily found within astrocytes and is known to maintain proper cell structure and mechanical strength. Mutations in GFAP are implicated in the pathology of Alexander disease, a neurodegenerative disease characterized by cytoplasmic inclusions of protein, known as Rosenthal fibers. GFAP has a typical type III intermediate filament domain structure, consisting of a highly conserved alpha-helical rod domain bracketed by an intrinsically disordered N-terminal head and C-terminal tail domains. While the general domain organization of monomeric GFAP and the assembly process for higher order quaternary structures are known, we lack an atomic resolution mechanistic understanding of GFAP assembly into mature filaments. Understanding the structure of GFAP filaments and how mutations disrupt this structure will provide vital information into how mutations produce Alexander disease pathology. As a first step towards a mechanistic description, we characterized GFAP wild type tetrameric and filamentous assemblies using solid state NMR and compared the results to those obtained from an assembly-deficient GFAP mutant. For wild-type GFAP, we observe surprisingly uniform rigid alpha helical structure and can spectroscopically resolve highly mobile intrinsically disordered regions in the filament assemblies. Wild type tetramers show increased mobility, likely arising from the head and tail domains. Mutation of the highly conserved cysteine at position 294 to serine results in an inability to form full-length filament assemblies. We show that the rigid regions of the C294S mutant assemblies largely remain structurally consistent with wild type tetrameric assemblies but differ from wild-type filament assemblies. There is an increase in highly mobile regions for the C294S mutant relative to the wild-type. Our results provide a foundation for developing solid state NMR approaches to characterize intermediate filament assembly mechanisms and the interfering effect of disease mutations.

1Correlations between cellular variables, such as gene-expression levels, provide insights into regulatory mechanisms. We focus here on correlations between mRNA and protein levels and re-examine previously derived analytical predictions. We test this prediction on single-cell E. coli data and see substantial disagreement. We hypothesize that this discrepancy arises from the assumption of constant cell volume and develop a theoretical framework for mRNA-protein correlations in growing and dividing cells. Within this framework, we derive an analytical expression for mRNA- protein correlations and show that explicit incorporation of growth and division substantially alters these correlations. The resulting relation is invariant to upstream transcriptional dynamics, and we validate it using stochastic simulations across multiple gene-regulatory architectures. Finally, we show that the derived predictions are consistent with the E. coli data.