Correspondence between geometrical and differential definitions of the sine and cosine functions and connection with kinematics

In classical physics, the familiar sine and cosine functions appear in two forms: (1) geometrical, in the treatment of vectors such as forces and velocities, and (2) differential, as solutions of oscillation and wave equations. These two forms correspond to two different definitions of trigonometric functions, one geometrical using right triangles and unit circles, and the other employing differential equations. Although the two definitions must be equivalent, this equivalence is not demonstrated in textbooks. In this manuscript, the equivalence between the geometrical and the differential definition is presented assuming no a priori knowledge of the properties of sine and cosine functions. We start with the usual length projections on the unit circle and use elementary geometry and elementary calculus to arrive to harmonic differential equations. This more general and abstract treatment not only reveals the equivalence of the two definitions but also provides an instructive perspective on circular and harmonic motion as studied in kinematics. This exercise can help develop an appreciation of abstract thinking in physics.Comment: 6 pages including 1 figur

Exact phase diagram of quasispecies model with mutation rate modifier

We consider an infinite asexual population with a mutator allele which can elevate mutation rates. With probability $f$, a transition from nonmutator to mutator state occurs but the reverse transition is forbidden. We find that at $f=0$, the population is in the state with minimum mutation rate and at $f=f_c$, a phase transition occurs between a mixed phase with both nonmutators and mutators and a pure mutator phase. We calculate the critical probability $f_c$ and the total mutator fraction $Q$ in the mixed phase exactly. Our predictions for $Q$ are in agreement with those seen in microbial populations in static environments.Comment: Revised versio

Point Interaction in two and three dimensional Riemannian Manifolds

We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac delta interactions on two and three dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator. In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for general class of manifolds, e.g., for compact and Cartan-Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by Perron-Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the beta-function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by S. G. Rajeev.Comment: 43 page

Transcriptional Infidelity Promotes Heritable Phenotypic Change in a Bistable Gene Network

Bistable epigenetic switches are fundamental for cell fate determination in unicellular and multicellular organisms. Regulatory proteins associated with bistable switches are often present in low numbers and subject to molecular noise. It is becoming clear that noise in gene expression can influence cell fate. Although the origins and consequences of noise have been studied, the stochastic and transient nature of RNA errors during transcription has not been considered in the origin or modeling of noise nor has the capacity for such transient errors in information transfer to generate heritable phenotypic change been discussed. We used a classic bistable memory module to monitor and capture transient RNA errors: the lac operon of Escherichia coli comprises an autocatalytic positive feedback loop producing a heritable all-or-none epigenetic switch that is sensitive to molecular noise. Using single-cell analysis, we show that the frequency of epigenetic switching from one expression state to the other is increased when the fidelity of RNA transcription is decreased due to error-prone RNA polymerases or to the absence of auxiliary RNA fidelity factors GreA and GreB (functional analogues of eukaryotic TFIIS). Therefore, transcription infidelity contributes to molecular noise and can effect heritable phenotypic change in genetically identical cells in the same environment. Whereas DNA errors allow genetic space to be explored, RNA errors may allow epigenetic or expression space to be sampled. Thus, RNA infidelity should also be considered in the heritable origin of altered or aberrant cell behaviour

A high-throughput synthetic platform enables the discovery of proteomimetic cell penetrating peptides and bioportides

Collectively, cell penetrating peptide (CPP) vectors and intrinsically active bioportides possess tremendous potential for drug delivery applications and the discrete modulation of intracellular targets including the sites of protein–protein interactions (PPIs). Such sequences are usually relatively short (< 25 AA), polycationic in nature and able to access the various intracellular compartments of eukaryotic cells without detrimental influences upon cellular biology. The high-throughput platform for bioportide discovery described herein exploits the discovery that many human proteins are an abundant source of potential CPP sequences which are reliably predicted using QSAR algorithms or other methods. Subsequently, microwave-enhanced solid phase peptides synthesis provides a high-throughput source of novel proteomimetic CPPs for screening purposes. By focussing upon cationic helical domains, often located within the molecular interfaces that facilitate PPIs, bioportides which act by a dominant-negative mechanism at such sites can be reliably identified within small number libraries of CPPs. Protocols that employ fluorescent peptides, routinely prepared by N-terminal acylation with carboxytetramethylrhodamine, further enable both the quantification of cellular uptake kinetics and the identification of specific site(s) of intracellular accretion. Chemical modifications of linear peptides, including strategies to promote and stabilise helicity, are compatible with the synthesis of second-generation bioportides with improved drug-like properties to further exploit the inherent selectivity of biologics

A translocation motif in relaxase TrwC specifically affects recruitment by its conjugative type IV secretion system

Type IV secretion system (T4SS) substrates are recruited through a translocation signal that is poorly defined for conjugative relaxases. The relaxase TrwC of plasmid R388 is translocated by its cognate conjugative T4SS, and it can also be translocated by the VirB/D4 T4SS of Bartonella henselae, causing DNA transfer to human cells. In this work, we constructed a series of TrwC variants and assayed them for DNA transfer to bacteria and human cells to compare recruitment requirements by both T4SSs. Comparison with other reported relaxase translocation signals allowed us to determine two putative translocation sequence (TS) motifs, TS1 and TS2. Mutations affecting TS1 drastically affected conjugation frequencies, while mutations affecting either motif had only a mild effect on DNA transfer rates through the VirB/D4 T4SS of B. henselae. These results indicate that a single substrate can be recruited by two different T4SSs through different signals. The C terminus affected DNA transfer rates through both T4SSs tested, but no specific sequence requirement was detected. The addition of a Bartonella intracellular delivery (BID) domain, the translocation signal for the Bartonella VirB/D4 T4SS, increased DNA transfer up to 4% of infected human cells, providing an excellent tool for DNA delivery to specific cell types. We show that the R388 coupling protein TrwB is also required for this high-efficiency TrwC-BID translocation. Other elements apart from the coupling protein may also be involved in substrate recognition by T4SSs

Aptamer-based multiplexed proteomic technology for biomarker discovery

Interrogation of the human proteome in a highly multiplexed and efficient manner remains a coveted and challenging goal in biology. We present a new aptamer-based proteomic technology for biomarker discovery capable of simultaneously measuring thousands of proteins from small sample volumes (15 [mu]L of serum or plasma). Our current assay allows us to measure ~800 proteins with very low limits of detection (1 pM average), 7 logs of overall dynamic range, and 5% average coefficient of variation. This technology is enabled by a new generation of aptamers that contain chemically modified nucleotides, which greatly expand the physicochemical diversity of the large randomized nucleic acid libraries from which the aptamers are selected. Proteins in complex matrices such as plasma are measured with a process that transforms a signature of protein concentrations into a corresponding DNA aptamer concentration signature, which is then quantified with a DNA microarray. In essence, our assay takes advantage of the dual nature of aptamers as both folded binding entities with defined shapes and unique sequences recognizable by specific hybridization probes. To demonstrate the utility of our proteomics biomarker discovery technology, we applied it to a clinical study of chronic kidney disease (CKD). We identified two well known CKD biomarkers as well as an additional 58 potential CKD biomarkers. These results demonstrate the potential utility of our technology to discover unique protein signatures characteristic of various disease states. More generally, we describe a versatile and powerful tool that allows large-scale comparison of proteome profiles among discrete populations. This unbiased and highly multiplexed search engine will enable the discovery of novel biomarkers in a manner that is unencumbered by our incomplete knowledge of biology, thereby helping to advance the next generation of evidence-based medicine