A thermomechanical framework for reconciling the effects of ultraviolet radiation exposure time and wavelength on connective tissue elasticity

Augmentation of the mechanical properties of connective tissue using ultraviolet (UV) radiation—by targeting collagen cross-linking in the tissue at predetermined UV exposure time \((t)\) and wavelength \((\lambda )\) —has been proposed as a therapeutic method for supporting the treatment for structural-related injuries and pathologies. However, the effects of \(\lambda \) and \(t\) on the tissue elasticity, namely elastic modulus \((E)\) and modulus of resilience \((u_\mathrm{Y})\) , are not entirely clear. We present a thermomechanical framework to reconcile the \(t\) - and \(\lambda \) -related effects on \(E\) and \(u_\mathrm{Y}\) . The framework addresses (1) an energy transfer model to describe the dependence of the absorbed UV photon energy, \(\xi \) , per unit mass of the tissue on \(t\) and \(\lambda \) , (2) an intervening thermodynamic shear-related parameter, \(G\) , to quantify the extent of UV-induced cross-linking in the tissue, (3) a threshold model for the \(G\) versus \(\xi \) relationship, characterized by   \(t_\mathrm{C}\) —the critical \(t\) underpinning the association of \(\xi \) with \(G\) —and (4) the role of \(G\) in the tissue elasticity. We hypothesized that \(G\) regulates \(E\) (UV-stiffening hypothesis) and \(u_\mathrm{Y}\) (UV-resilience hypothesis). The framework was evaluated with the support from data derived from tensile testing on isolated ligament fascicles, treated with two levels of \(\lambda \) (365 and 254 nm) and three levels of \(t\) (15, 30 and 60 min). Predictions from the energy transfer model corroborated the findings from a two-factor analysis of variance of the effects of \(t\) and \(\lambda \) treatments. Student’s t test revealed positive change in \(E\) and \(u_\mathrm{Y}\) with increases in \(G\) —the findings lend support to the hypotheses, implicating the implicit dependence of UV-induced cross-links on \(t\) and \(\lambda \) for directing tissue stiffness and resilience. From a practical perspective, the study is a step in the direction to establish a UV irradiation treatment protocol for effective control of exogenous cross-linking in connective tissues.     

Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

We introduce a mathematical model of the in vivo progression of Alzheimer’s disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of \(\upbeta \) -amyloid plaques based on the concentrations of A \(\upbeta \)  oligomers, PrP^C proteins, and the A \(\upbeta \) - \(\times \) -PrP^Ccomplex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of \(\upbeta \) -amyloid is constant and also when it is described by a power law.     

Transmission dynamics for vector-borne diseases in a patchy environment

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross–MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number \(\mathcal{R }_0\) is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of \(\mathcal{R }_0\) . It is shown that the disease free equilibrium is asymptotically stable if \(\mathcal{R }_0<1\) , and it is unstable if \(\mathcal{R }_0>1\) ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for \(\mathcal{R }_0\) are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for \(\mathcal{R }_0\) in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, \(\mathcal{R }_0\) is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.     

Modulation of the cAMP Response by G\alpha _i and G\beta \gamma : A Computational Study of G Protein Signaling in Immune Cells

Cyclic AMP is important for the resolution of inflammation, as it promotes anti-inflammatory signaling in several immune cell lines. In this paper, we present an immune cell specific model of the cAMP signaling cascade, paying close attention to the specific isoforms of adenylyl cyclase (AC) and phosphodiesterase that control cAMP production and degradation, respectively, in these cells. The model describes the role that G protein subunits, including G \(\alpha _s\) , G \(\alpha _i\) , and G \(\beta \gamma \) , have in regulating cAMP production. Previously, G \(\alpha _i\) activation has been shown to increase the level of cAMP in certain immune cell types. This increase in cAMP is thought to be mediated by \(\beta \gamma \) subunits which are released upon G \(\alpha \) activation and can directly stimulate specific isoforms of AC. We conduct numerical experiments in order to explore the mechanisms through which G \(\alpha _i\) activation can increase cAMP production. An important conclusion of our analysis is that the relative abundance of different G protein subunits is an essential determinant of the cAMP profile in immune cells. In particular, our model predicts that limited availability of \(\beta \gamma \) subunits may both \((i)\) enable immune cells to link inflammatory G \(\alpha _i\) signaling to anti-inflammatory cAMP production thereby creating a balanced immune response to stimulation with low concentrations of PGE2, and \((ii)\) prohibit robust anti-inflammatory cAMP signaling in response to stimulation with high concentrations of PGE2.     

Quantifying interspecific spatial overlap in aquatic macrophyte communities

Levins’s asymmetrical α index quantifies between species overlap over resources more realistically than similar-purpose single-value indices. The associated community-wide \(\bar \alpha\) index expresses the degree of “species packing”. Both indices were formulated upon competing animal (i.e., mobile) organisms and are independent of population densities. However, overlap over resources for nonmobile organisms such as plants may have an impact even below carrying capacity. The proposed \(\hat \alpha\) index, based on Levins’s α index, quantifies spatial overlap for plants integrating information on species spatial distribution and crowding conditions. The \(\hat \alpha\) index is specifically designed for plant distribution data collected in discrete plots with density expressed as percent coverage (%cover) of substratum. We also propose a community-wide \({\hat \alpha_{\text{c}}}\) index, conceptually analogous to \(\bar \alpha\) , but furnished with a measure of dispersion (se \({\hat \alpha_{\text{c}}}\) ). Species importance within the community is inferred from comparisons of pairwise \(\hat \alpha\) ’s with \({\hat \alpha_{\text{c}}}\) . The \(\hat \alpha\) and \({\hat \alpha_{\text{c}}}\) indices correlate closely and exponentially with plant density, and correct apparent over- and underestimations of interaction intensity at low and very high crowding by Levins’s α and \(\bar \alpha\) , respectively. Index application to aquatic plant communities gave results consistent with within-community and general ecological patterns, suggesting a high potential of the proposed \(\hat \alpha\) and \({\hat \alpha_{\text{c}}}\) indices in basic and applied macrophyte ecological studies and management.     

Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: equilibrium shape,motion transition under shear flow,and elongation during tank-treading motion

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ( \(\alpha =0\) ), biconcave disk shape ( \(\alpha =1\) ) and their intermediate shapes ( \(0<\alpha <1\) ) with the nonuniformity parameter \(\alpha \) , while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of \(2.5\times 10^{-6}\,\hbox {N}/\hbox {m}\) and an out-of-plane bending rigidity of \(2.0\times 10^{-19}\,\hbox {N}\cdot \hbox {m}\) of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for \(\alpha >0.22\) and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for \(0.05<\alpha <0.34\) ; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the \(\alpha \) value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for \(0.22<\alpha <0.34\) . The condition \(0.05<\alpha <0.22\) precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of \(\alpha \) and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane’s natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.     

Individual-Based Model for Quorum Sensing with Background Flow

Quorum sensing is a wide-spread mode of cell–cell communication among bacteria in which cells release a signalling substance at a low rate. The concentration of this substance allows the bacteria to gain information about population size or spatial confinement. We consider a model for \(N\) cells which communicate with each other via a signalling substance in a diffusive medium with a background flow. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior concentration \(u\) of the signalling substance, coupled with \(N\) ODEs for the masses \(a_i\) of the substance within each cell. The cells are balls of radius \(R\) in \(\mathbb {R} ^3\) , and under some scaling assumptions we formally derive an effective system of \(N\) ODEs describing the behaviour of the cells. The reduced system is then used to study the effect of flow on communication in general, and in particular for a number of geometric configurations.     

On linear birth-and-death processes in a random environment

We study the probability of extinction for single-type and multi-type continuous-time linear birth-and-death processes in a finite Markovian environment. The probability of extinction is equal to 1 almost surely if and only if the basic reproduction number \(R_0\) is \(\le 1\) , the key point being to identify a suitable definition of \(R_0\) for such a result to hold.     

A locally convergent rotationally invariant particle swarm optimization algorithm

Several well-studied issues in the particle swarm optimization algorithm are outlined and some earlier methods that address these issues are investigated from the theoretical and experimental points of view. These issues are the: stagnation of particles in some points in the search space, inability to change the value of one or more decision variables, poor performance when the swarm size is small, lack of guarantee to converge even to a local optimum (local optimizer), poor performance when the number of dimensions grows, and sensitivity of the algorithm to the rotation of the search space. The significance of each of these issues is discussed and it is argued that none of the particle swarm optimizers we are aware of can address all of these issues at the same time. To address all of these issues at the same time, a new general form of velocity update rule for the particle swarm optimization algorithm that contains a user-definable function \(f\) is proposed. It is proven that the proposed velocity update rule guarantees to address all of these issues if the function \(f\) satisfies the following two conditions: (i) the function \(f\) is designed in such a way that for any input vector \(\vec {y}\) in the search space, there exists a region \(A\) which contains \(\vec {y}\) and \( f\!\left( {\vec {y}} \right) \) can be located anywhere in \(A\) , and (ii) \(f\) is invariant under any affine transformation. An example of function \(f\) is provided that satisfies these conditions and its performance is examined through some experiments. The experiments confirm that the proposed algorithm (with an appropriate function \(f)\) can effectively address all of these issues at the same time. Also, comparisons with earlier methods show that the overall ability of the proposed method for solving benchmark functions is significantly better.     

Head impact accelerations for brain strain-related responses in contact sports: a model-based investigation

Both linear \((\mathbf{a}_{\mathrm{lin}})\) and rotational \((\mathbf{a}_{\mathrm{rot}} )\) accelerations contribute to head impacts on the field in contact sports; however, they are often isolated in injury studies. It is critical to evaluate the feasibility of estimating brain responses using isolated instead of full degrees-of-freedom (DOFs) accelerations. In this study, we investigated the sensitivities of regional brain strain-related responses to resultant \(\mathbf{a}_{\mathrm{lin}}\) and \(\mathbf{a}_{\mathrm{rot}}\) as well as the relative contributions of these acceleration components to the responses via random sampling and linear regression using parameterized, triangulated head impacts with kinematic variable values based on on-field measurements. Two independently established and validated finite element models of the human head were employed to evaluate model-consistency and dependency in results: the Dartmouth Head Injury Model and Simulated Injury Monitor. For the majority of the brain, volume-weighted regional peak strain, strain rate, and von Mises stress accumulated from the simulation significantly correlated with the product of the magnitude and duration of \(\mathbf{a}_{\mathrm{rot}}\) , or effectively, the rotational velocity, but not to \(\mathbf{a}_{\mathrm{lin}}\) . Responses from \(\mathbf{a}_{\mathrm{rot}}\) -only were comparable to the full-DOF counterparts especially when normalized by injury-causing thresholds (e.g., volume fractions of large differences virtually diminished (i.e., \(<\) 1 %) at typical difference percentage levels of 1–4 % on average). These model-consistent results support the inclusion of both rotational acceleration magnitude and duration into kinematics-based injury metrics and demonstrate the feasibility of estimating strain-related responses from isolated \(\mathbf{a}_{\mathrm{rot}}\) for analyses of strain-induced injury relevant to contact sports without significant loss of accuracy, especially for the cerebrum.     

Oxygen removal from water versus arterial oxygen delivery: calibrating the Fick equation in Pacific salmon

While it is well known that O₂ is directly removed from the water by skin and gill tissues of fish, the mismatch between O₂ removal from water (O₂ uptake; \(\dot{V}{\text{O}}_{ 2}\) ) and the O₂ delivered to tissues by the primary circulation (O₂ consumption; \(\dot{V}{\text{aO}}_{ 2}\) ) has never been measured directly. Using data from four recent studies that simultaneously measured \(\dot{V}{\text{O}}_{ 2}\) and \(\dot{V}{\text{aO}}_{ 2}\) in 2–5 kg Pacific salmon, our analysis revealed that sockeye salmon can remove an additional 12–48 % more O₂ from the water than the primary circulation delivers to the systemic tissues. This percentage did not change significantly during swimming activity, a result that contradicts an earlier prediction that the difference should decrease when \(\dot{V}{\text{O}}_{ 2}\) increases during exercise. In resting Chinook salmon, a similar percentage difference in simultaneously measured \(\dot{V}{\text{O}}_{ 2}\) and \(\dot{V}{\text{O}}_{ 2}\) was observed, yet the difference tended to disappear during acute heat stress to a near lethal temperature. These results emphasize that caution should be exercised when using the Fick equation to estimate cardiac output because the overestimate of cardiac output that results from using the Fick equation in Pacific salmon is not small, may not be fixed and may exist in other teleosts.     

Principal trabecular structural orientation predicted by quantitative ultrasound is strongly correlated with \upmu FEA determined anisotropic apparent stiffness

The microarchitecture and alignment of trabecular bone adapts to the particular mechanical milieu applied to it. Due to this anisotropic mechanical property, measurement orientation has to be taken into consideration when assessing trabecular bone quality and fracture risk prediction. Quantitative ultrasound (QUS) has demonstrated the ability in predicting the principal structural orientation (PSO) of trabecular bone. Although the QUS prediction for PSO is very close to that of \(\upmu \) CT, certain angle differences still exist. It remains unknown whether this angle difference can induce significant differences in mechanical properties or not. The objective of this study was to evaluate the mechanical properties in different PSOs predicted using different methods, QUS and \(\upmu \) CT, thus to investigate the ability of QUS as a means to predict the PSO of trabecular bone noninvasively. By validating the ability of QUS to predict the PSO of trabecular bone, it is beneficial for future QUS applications because QUS measurements in the PSO can provide information more correlated with the mechanical properties than with other orientations. In this study, seven trabecular bone balls from distal bovine femurs were used to generate finite element models based on the 3-dimensional \(\upmu \) CT images. Uniaxial compressive loading was performed on the bone ball models in the finite element analysis (FEA) in six different orientations (three anatomical orientations, two PSOs predicted by QUS and the longest vector of mean intercept length (MIL) tensor calculated by \(\upmu \) CT). The stiffness was calculated based on the reaction force of the bone balls under loading, and the von Mises stress results showed that both the mechanical properties in the PSOs predicted by QUS are significantly higher than the anatomical orientations and comparatively close to the longest vector of MIL tensor. The stiffness in the PSOs predicted by QUS is also highly correlated with the stiffness in the MIL tensor orientation (ATTmax vs. MIL, \(R^{2}\)  = 0.98, \(p<001\) ; UVmax vs. MIL, \(R^{2}\)  = 0.92, \(p<001\) ). These results were validated by in vitro mechanical testing on the bone ball samples. This study demonstrates that the PSO of trabecular bone predicted by QUS has an equally strong apparent stiffness with the orientation predicted by \(\upmu \) CT.     

N-site Phosphorylation Systems with 2N-1 Steady States

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of \(n\) -site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29–52, 2008), it is shown that models of \(n\) -site sequential distributive phosphorylation admit at most \(2n-1\) steady states. Wang and Sontag furthermore conjecture that for odd \(n\) , there are at most \(n\) and that, for even \(n\) , there are at most \(n+1\) steady states. This, however, is not true: building on earlier work in Holstein et al. (Bull Math Biol 75(11):2028–2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a \(3\) -site system has \(5\) steady states and parameter values where a \(4\) -site system has \(7\) steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in \(n\) -site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.     

Counting glycans revisited

We present an algorithm for counting glycan topologies of order \(n\) that improves on previously described algorithms by a factor \(n\) in both time and space. More generally, we provide such an algorithm for counting rooted or unrooted \(d\) -ary trees with labels or masses assigned to the vertices, and we give a “recipe” to estimate the asymptotic growth of the resulting sequences. We provide constants for the asymptotic growth of \(d\) -ary trees and labeled quaternary trees (glycan topologies). Finally, we show how a classical result from enumeration theory can be used to count glycan structures where edges are labeled by bond types. Our method also improves time bounds for counting alkanes.     

Basic Features of a Cell Electroporation Model: Illustrative Behavior for Two Very Different Pulses

Science increasingly involves complex modeling. Here we describe a model for cell electroporation in which membrane properties are dynamically modified by poration. Spatial scales range from cell membrane thickness (5 nm) to a typical mammalian cell radius (10  \(\upmu\) m), and can be used with idealized and experimental pulse waveforms. The model consists of traditional passive components and additional active components representing nonequilibrium processes. Model responses include measurable quantities: transmembrane voltage, membrane electrical conductance, and solute transport rates and amounts for the representative “long” and “short” pulses. The long pulse—1.5 kV/cm, 100  \(\upmu\) s—evolves two pore subpopulations with a valley at \({\sim}\) 5 nm, which separates the subpopulations that have peaks at \({\sim}\) 1.5 and \({\sim}\) 12 nm radius. Such pulses are widely used in biological research, biotechnology, and medicine, including cancer therapy by drug delivery and nonthermal physical tumor ablation by causing necrosis. The short pulse—40 kV/cm, 10 ns—creates 80-fold more pores, all small ( \(<\) 3 nm; \(\sim\) 1 nm peak). These nanosecond pulses ablate tumors by apoptosis. We demonstrate the model’s responses by illustrative electrical and poration behavior, and transport of calcein and propidium. We then identify extensions for expanding modeling capability. Structure-function results from MD can allow extrapolations that bring response specificity to cell membranes based on their lipid composition. After a pulse, changes in pore energy landscape can be included over seconds to minutes, by mechanisms such as cell swelling and pulse-induced chemical reactions that slowly alter pore behavior.     

Approximation of sojourn-times via maximal couplings: motif frequency distributions

Sojourn-times provide a versatile framework to assess the statistical significance of motifs in genome-wide searches even under non-Markovian background models. However, the large state spaces encountered in genomic sequence analyses make the exact calculation of sojourn-time distributions computationally intractable in long sequences. Here, we use coupling and analytic combinatoric techniques to approximate these distributions in the general setting of Polish state spaces, which encompass discrete state spaces. Our approximations are accompanied with explicit, easy to compute, error bounds for total variation distance. Broadly speaking, if \({\mathsf{T}}_n\) is the random number of times a Markov chain visits a certain subset \({\mathsf{T}}\) of states in its first \(n\) transitions, then we can usually approximate the distribution of \({\mathsf{T}}_n\) for \(n\) of order \((1-\alpha )^{-m}\) , where \(m\) is the largest integer for which the exact distribution of \({\mathsf{T}}_m\) is accessible and \(0\le \alpha \le 1\) is an ergodicity coefficient associated with the probability transition kernel of the chain. This gives access to approximations of sojourn-times in the intermediate regime where \(n\) is perhaps too large for exact calculations, but too small to rely on Normal approximations or stationarity assumptions underlying Poisson and compound Poisson approximations. As proof of concept, we approximate the distribution of the number of matches with a motif in promoter regions of C. elegans. Mathematical properties of the proposed ergodicity coefficients and connections with additive functionals of homogeneous Markov chains as well as ergodicity of non-homogeneous Markov chains are also explored.     

Interdependency and hierarchy of exact and approximate epidemic models on networks

Over the years numerous models of \(SIS\) (susceptible \(\rightarrow \) infected \(\rightarrow \) susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure.     

Effect of the cosolutes trehalose and methanol on the equilibrium and phase-transition properties of glycerol-monopalmitate lipid bilayers investigated using molecular dynamics simulations

The influence of the cosolutes trehalose and methanol on the structural, dynamic and thermodynamic properties of a glycerol-1-monopalmitate (GMP) bilayer and on its main transition temperature \(T_m\) is investigated using atomistic molecular dynamics simulations (600 ns) of a GMP bilayer patch (2 × 8 × 8 lipids) at different temperatures in the range of 302 to 338 K and considering three different cosolute concentrations. Depending on the environment and temperature, these simulations present no or a single GL \(\rightarrow \) LC, LC \(\rightarrow \) GL or LC \(\rightarrow \) ID transition, where LC, GL and ID are the liquid crystal, gel and interdigitated phases, respectively. The trehalose molecules form a coating layer at the bilayer surface, promote the hydrogen-bonded bridging of the lipid headgroups, preserve the interaction of the headgroups with trapped water and induce a slight lateral expansion of the bilayer in the LC phase, observations that may have implications for the phenomenon of anhydrobiosis. However, this cosolute does not affect \(T_m\) and its dependence on hydration in the concentration range considered. On the other hand, methanol molecules intercalate between the lipid headgroups, promote a lateral expansion of the bilayer in the LC phase and induce a concentration dependent decrease of \(T_m\) , observations that may have implications for the phenomenon of anesthesia. The occurrence of an ID phase in the presence of this cosolute may be viewed as an extreme consequence of lateral expansion. The analysis of the simulations also suggests the existence of two basic conservation principles: (1) the hydrogen-bond saturation principle rests on the observation that for all species present in the different systems, the total numbers of hydrogen-bonds per molecule is essentially constant, the only factor of variability being their distribution among different partners; (2) the densest packing principle rests on the observation that the effective volume per methylene group in the interior of the bilayer is only weakly sensitive to the environment, with values comparable to those for liquid (LC) and solid (ID) alkanes, or intermediate (GL).