Revolutionaries and spies: Spy-good and spy-bad graphs

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if $m$ of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever. Let $\sigma(G,m,r)$ denote the minimum number of spies needed to win. To avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the lower bound is sharp when $G$ has a rooted spanning tree $T$ such that every edge of $G$ not in $T$ joins two vertices having the same parent in $T$. As a consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where $\gamma(G)$ is the domination number; this bound is nearly sharp when $\gamma(G)\le m$. For the random graph with constant edge-probability $p$, we obtain constants $c$ and $c'$ (depending on $m$ and $p$) such that $\sigma(G,m,r)$ is near the trivial upper bound when $r<c\ln n$ and at most $c'$ times the trivial lower bound when $r>c'\ln n$. For the hypercube $Q_d$ with $d\ge r$, we have $\sigma(G,m,r)=r-m+1$ when $m=2$, and for $m\ge 3$ at least $r-39m$ spies are needed. For complete $k$-partite graphs with partite sets of size at least $2r$, the leading term in $\sigma(G,m,r)$ is approximately $\frac{k}{k-1}\frac{r}{m}$ when $k\ge m$. For $k=2$, we have \sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and \sigma(G,3,r)=\floor{r/2}, and in general $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.Comment: 34 pages, 2 figures. The most important changes in this revision are improvements of the results on hypercubes and random graphs. The proof of the previous hypercube result has been deleted, but the statement remains because it is stronger for m<52. In the random graph section we added a spy-strategy resul

Prediction of the Translocation Kinetics of a Protein from Its Mechanical Properties

AbstractProteins are actively unfolded to pass through narrow channels in macromolecular complexes that catalyze protein translocation and degradation. Catalyzed unfolding shares many features that characterize the mechanical unfolding of proteins using the atomic force microscope (AFM). However, simulations of unfolding induced by the AFM and when a protein is translocated through a pore suggest that each process occurs by distinct pathways. The link, if any, between each type of unfolding, therefore, is not known. We show that the mechanical unfolding energy landscape of a protein, obtained using an atomistic molecular model, can be used to predict both the relative mechanical strength of proteins when unfolded using the AFM and when unfolded by translocation into a pore. We thus link the two processes and show that the import rate through a pore not only depends on the location of the initiation tag but also on the mechanical properties of the protein when averaged over all the possible geometries that are relevant for a given translocation initiation site

A randomised controlled study of high intensity exercise as a dishabituating stimulus to improve hypoglycaemia awareness in people with type 1 diabetes:a proof of concept study

Aims/hypothesis Approximately 25% of people with type 1 diabetes have suppressed counterregulatory hormonal and symptomatic responses to insulin-induced hypoglycaemia, which renders them at increased risk of severe, disabling hypoglycaemia. This is called impaired awareness of hypoglycaemia (IAH), the cause of which is unknown. We recently proposed that IAH develops through habituation, a form of adaptive memory to preceding hypoglycaemia. Consistent with this hypothesis, we demonstrated restoration of defective counterregulatory hormonal responses to hypoglycaemia (referred to as dishabituation) in a rodent model of IAH following introduction of a novel stress stimulus (high intensity training [HIT]). In this proof-of-concept study we sought to further test this hypothesis by examining whether a single episode of HIT would amplify counterregulatory responses to subsequent hypoglycaemia in people with type 1 diabetes who had IAH (assessed by Gold score ≥4, modified Clarke score ≥4 or Dose Adjustment For Normal Eating (DAFNE) hypoglycaemia awareness rating 2 or 3). The primary outcome was the difference in adrenaline response to hypoglycaemia following both a single episode of HIT and rest. Methods In this randomised, crossover study 12 participants aged between 18 and 55 years with type 1 diabetes for ≥5 years and an HbA1c < 75 mmol/mol (9%) were recruited. Individuals were randomised using computer generated block randomisation to start with one episode of HIT (4 × 30 s cycle sprints [2 min recovery] at 150% of maximum wattage achieved during V˙O2peak assessment) or rest (control). The following day they underwent a 90 min hyperinsulinaemic–hypoglycaemic clamp study at 2.5 mmol/l with measurement of hormonal counterregulatory response, symptom scores and cognitive testing (four-choice reaction time and digit symbol substitution test). Each intervention and subsequent clamp study was separated by at least 2 weeks. The participants and investigators were not blinded to the intervention or measurements during the study. The investigators were blinded to the primary outcome and blood analysis results. Results All participants (six male and six female, age 19–54 years, median [IQR] duration of type 1 diabetes 24.5 [17.3–29.0] years, mean [SEM] HbA1c 56 [3.67] mmol/mol; 7.3% [0.34%]) completed the study (both interventions and two clamps). In comparison with the rest study, a single episode of HIT led to a 29% increase in the adrenaline (epinephrine) response (mean [SEM]) (2286.5 [343.1] vs 2953.8 [384.9] pmol/l); a significant increase in total symptom scores (Edinburgh Hypoglycaemia Symptom Scale: 24.25 [2.960 vs 27.5 [3.9]; p < 0.05), and a significant prolongation of four-choice reaction time (591.8 [22.5] vs 659.9 [39.86] ms; p < 0.01] during equivalent hypoglycaemia induced the following day. Conclusions/interpretation These findings are consistent with the hypothesis that IAH develops in people with type 1 diabetes as a habituated response and that introduction of a novel stressor can restore, at least partially, the adapted counterregulatory hormonal, symptomatic and cognitive responses to hypoglycaemia.Output Status: Forthcoming/Available Onlin

The natural history and ecology of melanism in red wolf and coyote populations of the southeastern United States – evidence for Gloger’s rule

Background: Gloger’s rule postulates that animals should be darker colored in warm and humid regions where dense vegetation and dark environments are common. Although rare in Canis populations, melanism in wolves is more common in North America than other regions globally and is believed to follow Gloger’s rule. In the temperate forests of the southeastern United States, historical records of red wolf (Canis rufus) and coyote (Canis latrans) populations document a consistent presence of melanism. Today, the melanistic phenotype is extinct in red wolves while occurring in coyotes and red wolf-coyote hybrids who occupy the red wolf’s historical range. To assess if Gloger’s rule could explain the occurrence and maintenance of melanistic phenotypes in Canis taxa, we investigated differences in morphology, habitat selection, and survival associated with pelage color using body measurements, GPS tracking data, and long-term capture-mark-recapture and radio-telemetry data collected on coyotes and hybrids across the southeastern United States. Results: We found no correlation between morphometrics and pelage color for Canis taxa. However, we observed that melanistic coyotes and hybrids experienced greater annual survival than did their gray conspecifics. Furthermore, we observed that melanistic coyotes maintained larger home ranges and exhibited greater selection for areas with dense canopy cover and wetlands than did gray coyotes. Conclusions: In the southeastern United States, pelage color influenced habitat selection by coyotes and annual survival of coyotes and hybrids providing evidence that Gloger’s rule is applicable to canids inhabiting regions with dense canopy cover and wetlands. Greater annual survival rates observed in melanistic Canis may be attributed to better concealment in areas with dense canopy cover such as coastal bottomland forests. We suggest that the larger home range sizes of melanistic coyotes may reflect the trade-off of reduced foraging efficiency in lower quality wetland habitat for improved survival. Larger home ranges and differential use of land cover by melanistic coyotes may facilitate weak assortative mating in eastern coyote populations, in which melanistic animals may have lower success of finding compatible mates in comparison to gray conspecifics. We offer that our observations provide a partial explanation for why melanism is relatively low (\u3c 10%) but consistent within coyote populations throughout southeastern parts of their range

Robustness Generalizations of the Shortest Feasible Path Problem for Electric Vehicles

Electric Vehicle routing is often modeled as a Shortest Feasible Path Problem (SFPP), which minimizes total travel time while maintaining a non-zero State of Charge (SoC) along the route. However, the problem assumes perfect information about energy consumption and charging stations, which are difficult to even estimate in practice. Further, drivers might have varying risk tolerances for different trips. To overcome these limitations, we propose two generalizations to the SFPP; they compute the shortest feasible path for any initial SoC and, respectively, for every possible minimum SoC threshold. We present algorithmic solutions for each problem, and provide two constructs: Starting Charge Maps and Buffer Maps, which represent the tradeoffs between robustness of feasible routes and their travel times. The two constructs are useful in many ways, including presenting alternate routes or providing charging prompts to users. We evaluate the performance of our algorithms on realistic input instances