A Tutorial on Mathematical Modeling of Biological Signaling Pathways

Mathematical models have been widely used in the studies of biological signaling pathways. Among these studies, two systems biology approaches have been applied: top-down and bottom-up systems biology. The former approach focuses on X-omics researches involving the measurement of experimental data in a large scale, for example proteomics, metabolomics, or fluxomics and transcriptomics. In contrast, the bottom-up approach studies the interaction of the network components and employs mathematical models to gain some insights about the mechanisms and dynamics of biological systems. This chapter introduces how to use the bottom-up approach to establish mathematical models for cell signaling studies

Band structure renormalization and weak pseudogap behavior in Na_{0.33}CoO_2: Fluctuation exchange study based on a single band model

Based on a single band Hubbard model and the fluctuation exchange approximation, the effective mass and the energy band renormalization in Na$_{0.33}$CoO$_2$ is elaborated. The renormalization is observed to exhibit certain kind of anisotropy, which agrees qualitatively with the angle-resolved photoemission spectroscopy (ARPES) measurements. Moreover, the spectral function and density of states (DOS) in the normal state are calculated, with a weak pseudogap behavior being seen, which is explained as a result of the strong Coulomb correlations. Our results suggest that the large Fermi surface (FS) associated with the $a_{1g}$ band plays likely a central role in the charge dynamics.Comment: 5 pages, 5 figure

Model Wavefunctions for the Collective Modes and the Magneto-roton Theory of the Fractional Quantum Hall Effect

We construct model wavefunctions for the collective modes of fractional quantum Hall systems. The wavefunctions are expressed in terms of symmetric polynomials characterized by a root partition and a "squeezed" basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, the model wavefunctions reduce to those predicted by the single-mode approximation, and remain accurate at energies above the continuum of roton pairs.Comment: 4 pages, 3 figures, minor changes for the final prl versio

Orbital-transverse density-wave instabilities in iron-based superconductors

Besides the conventional spin-density-wave (SDW) state, a new kind of orbital-transverse density-wave (OTDW) state is shown to exist generally in multi-orbital systems. We demonstrate that the orbital character of Fermi surface nesting plays an important role in density responses. The relationship between antiferromagnetism and structural phase transition in LaFeAsO (1111) and BaFe$_2$As$_2$ (122) compounds of iron-based superconductors may be understood in terms of the interplay between the SDW and OTDW with a five-orbital Hamiltonian. We propose that the essential difference between 1111 and 122 compounds is crucially determined by the presence of the two-dimensional $d_{xy}$-like Fermi surface around (0,0) being only in 1111 parent compounds.Comment: several parts were rewritten for clarity. 6 pages, 3 figures, 1 tabl

Absolute Quantification of TGF-β Signaling Proteins Using Quantitative Western Blot

Cell signaling governs the basic functions of cells by molecular interactions that involve of many proteins. The abundance of signaling proteins can directly influence cellular responses to external signal, contributing to cellular heterogeneity. Absolute quantification of proteins is important for modeling and understanding the complex signaling network. Here, we introduce how to measure the amount of TGF-β signaling proteins using quantitative immunoblotting. In addition, we discuss how to convert the measurements of protein abundance to the quantities of absolute molecules per cell. This method is generally applicable to the absolute quantification of other proteins

Measuring TGF-β Ligand Dynamics in Culture Medium

TGF-β plays an important role in a myriad of cell activities including differentiation, proliferation, and growth arrest. These effects are influenced by the concentration of TGF-β in the surrounding milieu, which is interpreted by mammalian cells and subsequently translated into meaningful signals that guide their proliferation, survival, or death. To predict cellular responses to TGF-ß signaling based on molecular mechanisms, it is important to consider how cells respond to different ligand doses and how variations in ligand exposure impact Smad signaling dynamics and subsequent gene expression. Here we describe methods to measure TGF-β concentration in the environment and approaches to perturb cellular TGF-β exposure to gain a quantitative understanding of signaling dynamics of this pathway

Inferring cellular regulatory networks with Bayesian model averaging for linear regression (BMALR)

Bayesian network and linear regression methods have been widely applied to reconstruct cellular regulatory networks. In this work, we propose a Bayesian model averaging for linear regression (BMALR) method to infer molecular interactions in biological systems. This method uses a new closed form solution to compute the posterior probabilities of the edges from regulators to the target gene within a hybrid framework of Bayesian model averaging and linear regression methods. We have assessed the performance of BMALR by benchmarking on both in silico DREAM datasets and real experimental datasets. The results show that BMALR achieves both high prediction accuracy and high computational efficiency across different benchmarks. A pre-processing of the datasets with the log transformation can further improve the performance of BMALR, leading to a new top overall performance. In addition, BMALR can achieve robust high performance in community predictions when it is combined with other competing methods. The proposed method BMALR is competitive compared to the existing network inference methods. Therefore, BMALR will be useful to infer regulatory interactions in biological networks. A free open source software tool for the BMALR algorithm is available at https://sites.google.com/site/bmalr4netinfer/

Optogenetic Control of TGF-β Signaling

Cells employ signaling pathways to make decisions in response to changes in their immediate environment. The Transforming Growth Factor β (TGF-β) signaling pathway plays pivotal roles in regulating many cellular processes, including cell proliferation, differentiation, and migrations. In order to manipulate and explore the dynamic behavior of TGF-β signaling at high spatiotemporal resolution, we developed an optogenetic system (the optoTGFBRs system), in which light is used to control TGF-β signaling precisely in time and space. Here, we describe about experimental details of how to build the optoTGFBRs system and utilize it to manipulate TGF-β signaling in a single cell or a cell population using microscope or LED array, respectively

Lattice QCD calculation of ππ\pi\pi scattering length

We study s-wave pion-pion ($\pi\pi$) scattering length in lattice QCD for pion masses ranging from 330 MeV to 466 MeV. In the "Asqtad" improved staggered fermion formulation, we calculate the $\pi\pi$ four-point functions for isospin I=0 and 2 channels, and use chiral perturbation theory at next-to-leading order to extrapolate our simulation results. Extrapolating to the physical pion mass gives the scattering lengths as $m_\pi a_0^{I=2} = -0.0416(2)$ and $m_\pi a_0^{I=0} = 0.186(2)$ for isospin I=2 and 0 channels, respectively. Our lattice simulation for $\pi\pi$ scattering length in the I=0 channel is an exploratory study, where we include the disconnected contribution, and our preliminary result is near to its experimental value. These simulations are performed with MILC 2+1 flavor gauge configurations at lattice spacing $a \approx 0.15$ fm.Comment: Remove some typo

The Universal Edge Physics in Fractional Quantum Hall Liquids

The chiral Luttinger liquid theory for fractional quantum Hall edge transport predicts universal power-law behavior in the current-voltage ($I$-$V$) characteristics for electrons tunneling into the edge. However, it has not been unambiguously observed in experiments in two-dimensional electron gases based on GaAs/GaAlAs heterostructures or quantum wells. One plausible cause is the fractional quantum Hall edge reconstruction, which introduces non-chiral edge modes. The coupling between counterpropagating edge modes can modify the exponent of the $I$-$V$ characteristics. By comparing the $\nu=1/3$ fractional quantum Hall states in modulation-doped semiconductor devices and in graphene devices, we show that the graphene-based systems have an experimental accessible parameter region to avoid the edge reconstruction, which is suitable for the exploration of the universal edge tunneling exponent predicted by the chiral Luttinger liquid theory.Comment: 7 pages, 6 figure