We study the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets. More precisely, we establish a formula for the Assouad spectrum of all Gatzouras-Lalley carpets as the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Our formula implies a number of novel properties for the Assouad spectrum not previously observed for dynamically invariant sets; in particular, the Assouad spectrum can be a non-trivial differentiable function on the entire domain (0,1) and can be strictly concave on open intervals. Our proof introduces a general framework for covering arguments using techniques developed in the context of multifractal analysis, including the method of types from large deviations theory and Lagrange duality from optimisation theory.

We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in ℝ^d. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set X itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if X additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.

Tumor Necrosis Factor (TNF) is a trimeric cytokine that exists in soluble (sTNF) and membrane-bound (mTNF) forms, both of which regulate immune responses through interactions with their cognate receptors. sTNF has been the subject of extensive biophysical investigations, leading to a mechanistic model in which a symmetric arrangement of the trimer promotes receptor signaling, whereas asymmetric conformations inhibit this function. In contrast, the structural and energetic landscape of mTNF remains largely under-explored. Here, we combined multi-microsecond unbiased molecular dynamics and Metadynamics-Adaptive Biasing Force simulations to characterize mTNF embedded in a physiologically relevant lipid bilayer. We show that in the absence of membrane engagement, the extracellular domain (ECD) dynamically samples multiple asymmetric conformations similar to those observed in sTNF. The association of the ECD with the membrane, mediated primarily by the basic residues R78–R82, R107, R108, R120, and R207, restricts conformational heterogeneity and stabilizes the symmetric state. By quantifying the energetic effects of ECD–membrane association, we demonstrate that the symmetric state in the membrane-bound ECD is stabilized over asymmetric conformations to a greater extent than in sTNF. This energetic effect, together with the spatial confinement imposed by the lipid bilayer, may explain the previously reported reduction in affinity of certain biologics for mTNF. In conclusion, our work elucidates how the membrane influences the structure, dynamics, and energetics of TNF. These mechanistic insights could guide future efforts to design mTNF-selective inhibitors that account for both membrane constraints and the energetic modulation of the ECD conformational landscape. ### Competing Interest Statement The authors have declared no competing interest.

In a recent article, Rapaport showed that the there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the 𝒞^2 topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the dual IFS which we believe has significant interest in its own right. As an application we also characterise when an analytic IFS can be conjugated to a self-similar IFS.

Intermediate dimensions were introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford–McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions dimθ⁡Λ of any Bedford–McMullen carpet Λ for the whole spectrum of θ∈[0,1], in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in θ, and the function θ↦dimθ⁡Λ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent.

We propose a new way of utilizing normal modes to study protein conformational transitions. Instead of considering individual modes independently, we show that a weighted mixture of low-frequency vibrational modes can reveal dynamic information about the conformational mechanism in more detail than any single mode can. The weights in the mixed mode, termed the allosteric covibrational mode, are determined using a simple model where the conformational transition is viewed as a perturbation of the coupled harmonic oscillator associated with either of the two conformations. We demonstrate our theory in a biologically relevant example of high pharmaceutical interest involving the V617F mutation of Janus 2 tyrosine kinase (JAK2).

In this paper, a sponge in Rd is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the Lq spectrum of any self-affine measure defined on a sponge for all q is an element of R. Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions, the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier-Young formula for Hausdorff dimension.

Conformational samplingof complex biomolecules is an emergingfrontier in drug discovery. Advances in lab-based structural biologyand related computational approaches like AlphaFold have made greatstrides in obtaining static protein structures for biologically relevanttargets. However, biology is in constant motion, and many importantbiological processes rely on conformationally driven events. Conventionalmolecular dynamics (MD) simulations run on standard hardware are impracticalfor many drug design projects, where conformationally driven biologicalevents can take microseconds to milliseconds or longer. An alternativeapproach is to focus the search on a limited region of conformationalspace defined by a putative reaction coordinate (i.e., path collectivevariable). The search space is typically limited by applying restraints,which can be guided by insights about the underlying biological processof interest. The challenge is striking a balance between the degreeto which the system is constrained and still allowing for naturalmotions along the path. A plethora of restraints exist to limit thesize of conformational search space, although each has drawbacks whensimulating complex biological motions. In this work, we present athree-stage procedure to construct realistic path collective variables(PCVs) and introduce a new kind of barrier restraint that is particularlywell suited for complex conformationally driven biological events,such as allosteric modulations and conformational signaling. The PCVpresented here is all-atom (as opposed to C-alpha or backbone only)and is derived from all-atom MD trajectory frames. The new restraintrelies on a barrier function (specifically, the scaled reciprocalfunction), which we show is particularly beneficial in the contextof molecular dynamics, where near-hard-wall restraints are neededwith zero tolerance to restraint violation. We have implemented ourPCV and barrier restraint within a hybrid sampling framework thatcombines well-tempered metadynamics and extended-Lagrangian adaptivebiasing force (meta-eABF). We use three particular examples of highpharmaceutical interest to demonstrate the value of this approach:(1) sampling the distance from ubiquitin to a protein of interestwithin the supramolecular cullin-RING ligase complex, (2) stabilizingthe wild-type conformation of the oncogenic mutant JAK2-V617F pseudokinasedomain, and (3) inducing an activated state of the stimulator of interferongenes (STING) protein observed upon ligand binding. For examples 2and 3, we present statistical analysis of meta-eABF free energy estimatesand, for each case, code for reproducing this work.

Abstract We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb {R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for $d \geqslant 4$ . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen carpets we completely characterise the family of probability vectors which minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford-McMullen carpets before.

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a precise formula for $\dim_{\theta}\Lambda$ is particularly challenging when $\Lambda$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_{\theta}\Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta<1$, moreover, the derivative of the upper bound is strictly positive at $\theta=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempton.

Conformational sampling of complex biomolecules is an emerging frontier in drug discovery. Indeed, advances in lab-based structural biology and related computational approaches like AlphaFold have made great strides in obtaining static protein structures. However, biology is in constant motion and many important biological processes rely on conformationally-driven events. Unrestrained molecular dynamics (MD) simulations require that the simulated time be comparable to the real time of the biological processes of interest, rendering pure MD impractical for many drug design projects, where conformationally-driven biological events can take microseconds to milliseconds or longer. An alternative approach is to accelerate the sampling of specific motions by applying restraints, guided by insights about the underlying biological process of interest. A plethora of restraints exist to limit the size of conformational search space, although each has drawbacks when simulating complex biological motions. In this work, we introduce a new kind of restraint for molecular dynamics simulations (MD) that is particularly well suited for complex conformationallydriven biological events, such as protein-ligand binding, allosteric modulations, conformational signalling, and membrane permeability. The new restraint, which relies on a barrier function (the scaled reciprocal function) is particularly beneficial to MD, where hard-wall restraints are needed with zero tolerance to restraint violation. We have implemented this restraint within a hybrid sampling framework that combines metadynamics and extended-Lagrangian adaptive biasing force (meta-eABF). We use two particular examples to demonstrate the value of this approach: (1) quantification of the approach of E3-loaded ubiquitin to a protein of interest as part of the Cullin ring ligase and (2) membrane permeability of heterobi-functional degrader molecules with a large degree of conformational flexibility. Future work will involve extension to additional systems and benchmarking of this approach compared with other methods.

The protein STING (stimulator of interferon genes) is a central regulator of the innate immune system and plays an important role in antitumor immunity by inducing the production of cytokines such as type I interferon (IFN). Activation of STING stems from the selective recognition of endogenous cyclic dinucleotides (CDNs) by the large, polar, and flexible binding site, thus posing challenges to the design of small molecule agonists with drug-like physicochemical properties. In this work we present the design of SNX281, a small molecule STING agonist that functions through a unique self-dimerizing mechanism in the STING binding site, where the ligand dimer approximates the size and shape of a cyclic dinucleotide while maintaining drug-like small molecule properties. SNX281 exhibits systemic exposure, STING-mediated cytokine release, strong induction of type I IFN, potent in vivo antitumor activity, durable immune memory, and single-dose tumor elimination in mouse models via a C max -driven pharmacologic response. Bespoke computational methods – a combination of quantum mechanics, molecular dynamics, binding free energy simulations, and artificial intelligence – were developed during the course of the project to design SNX281 by explicitly accounting for the unique self-dimerization mechanism and the large-scale conformational change of the STING protein upon activation. Over the course of the project, we explored millions of virtual molecules while synthesizing and testing only 208 molecules in the lab. This work highlights the value of a multifaceted computationally-driven approach anchored by methods tailored to address target-specific problems encountered along the project progression from initial hit to the clinic.

We define a new metric between natural numbers induced by the 8oo norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function Loo(N), which tabulates the cumulative sum of distances between consecutive natural numbers up to N in this new metric. Our main result is to identify the positive and finite limit of the sequence Loo(N)/N as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for K = 1, 2 or 3 and omega 0, . . . , omega K >= 2 the following asymptotic density holds

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for the box dimension of self-affine carpets of Gatzouras-Lalley and of Bara\'nski type to their higher dimensional sponge analogues. In addition to a closed form, we also obtain a variational formula which resembles the Ledrappier-Young formula for Hausdorff dimension.

Targeted protein degradation (TPD) has emerged as a powerful approach in drug discovery for removing (rather than inhibiting) proteins implicated in diseases. A key step in this approach is the formation of an induced proximity complex, where a degrader molecule recruits an E3 ligase to the protein of interest (POI), facilitating the transfer of ubiquitin to the POI and initiating the proteasomal degradation process. Here, we address three critical aspects of the TPD process: 1) formation of the ternary complex induced by a degrader molecule, 2) conformational heterogeneity of the ternary complex, and 3) assessment of ubiquitination propensity via the full Cullin Ring Ligase (CRL) macromolecular assembly. The novel approach presented here combines experimental biophysical data—in this case hydrogen-deuterium exchange mass spectrometry (HDX-MS, which measures the solvent exposure of protein residues)—with all-atom explicit solvent molecular dynamics (MD) simulations aided by enhanced sampling techniques to predict structural ensembles of ternary complexes at atomic resolution. We present results demonstrating the efficiency, accuracy, and reliability of our approach to predict ternary structure ensembles using the bromodomain of SMARCA2 (SMARCA2 BD ) with the E3 ligase VHL as the system of interest. The simulations reproduce X-ray crystal structures – including prospective simulations validated on a new structure that we determined in this work (PDB ID: 7S4E) – with root mean square deviations (RMSD) of 1.1 to 1.6 Å. The simulations also reveal a structural ensemble of low-energy conformations of the ternary complex within a broad energy basin. To further characterize the structural ensemble, we used snapshots from the aforementioned simulations as seeds for Hamiltonian replica exchange molecular dynamics (HREMD) simulations, and then perform 7.1 milliseconds of aggregate simulation time using Folding@home. The resulting free energy surface identifies the crystal structure conformation within a broad low-energy basin and the dynamic ensemble is consistent with solution-phase biophysical experimental data (HDX-MS and small-angle x-ray scattering, SAXS). Finally, we graft structures from the ternary complexes onto the full CRL and perform enhanced sampling simulations, where we find that differences in degradation efficiency can be explained by the proximity distribution of lysine residues on the POI relative to the E2-loaded ubiquitin. Several of the top predicted ubiquitinated lysine residues are validated prospectively through a ubiquitin mapping proteomics experiment.

We construct a family of planar self-affine carpets with overlaps using lower triangular matrices in a way that generalizes the original Gatzouras–Lalley carpets (Gatzouras and Lalley 1992 Indiana Univ. Math. J. 41 533–68) defined by diagonal matrices. Assuming the rectangular open set condition, Barański (2008 Discrete Continuous Dyn. Syst. A 21 1015–23) proved for this construction that for typical parameters, which can be explicitly checked, the inequalities between the Hausdorff, box and affinity dimension of the attractor are strict. We generalize this result to overlapping constructions, where we allow complete columns to be shifted along the horizontal axis (as in Fraser and Shmerkin (2016 Ergod. Theor. Dyn. Syst. 36 2463–81) and Pardo-Simón (2019 Ergod. Theor. Dyn. Syst. pp 733–63) or allow parallelograms to overlap within a column in a transversal way. Our main result is to show sufficient conditions under which these overlaps do not cause a drop of the dimension of the attractor. Several examples are provided to illustrate the results, including a self-affine smiley, a family of self-affine continuous curves, examples with overlaps and an application of our results to some three-dimensional systems.

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise H\"older exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise H\"older exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, our results apply for de Rham's curve.