beta-1 integrin

Examples of beta-1 integrin in the following topics:

• Mammary Glands

  • The binding of laminin and collagen in the myoepithelial basement membrane with beta-1 integrin on the epithelial surface insures correct placement of prolactin receptors on basal lateral side of alveoli cells and directional secretion of milk into lactiferous ducts.
  • Cross-section of the mammary-gland. 1.

• Beta Decay

  • Beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus.
  • There are two types of beta decay.
  • Beta minus (β) leads to an electron emission (e−); beta plus (β+) leads to a positron emission (e+).
  • A typical Q is around 1 MeV, but it can range from a few keV to a several tens of MeV.
  • One example is the odd-proton odd-neutron nuclide 40 K, which undergoes both types of beta decay with a half-life of 1.277 ·109 years.

• Basement Membranes and Diseases

  • The electron-dense lamina densa membrane is about 30–70 nanometers in thickness, and consists of an underlying network of reticular collagen (type IV) fibrils (fibroblast precursors) which average 30 nanometers in diameter and 0.1–2 micrometers in thickness.
  • This is an inherited connective tissue disease causing blisters in the skin and mucosal membranes, with an incidence of 1/50,000.
  • It is caused by a mutation in the integrin α6β4 cell-adhesion molecule on either the alpha or beta subunit.

• The Fields

  • $\displaystyle \beta \equiv \frac{\bf u}{c},~\text{so}~ \kappa = 1 - {\bf n} \cdot \beta$
  • $\displaystyle {\bf E}(r,t) = \kern-2mm q \left [ \frac{({\bf n} - \beta)(1-\beta^2)}{\kappa^3 R^2} \right ]_\mathrm{ret}\!
  • The first part is proportional to $1/R^2$ and it is simply a generalization of the field for a stationary charge.
  • The second terms are proportional to $1/R$.
  • $\displaystyle {\bf S} = {\bf n} \frac{q^2}{4\pi c \kappa^6 R^2} \left | {\bf n} \times \left \{ \left ( {\bf n} - \beta \right ) \times {\dot{\beta}} \right \} \right |^2$

• Inverse Compton Spectra - Single Scattering

  • $\displaystyle I'(E',\mu') = F_0 \left (\frac{E'}{E}\right)^2 \delta (E-E_0)\\ \displaystyle = F_0 \left (\frac{E'}{E_0}\right)^2 \delta (\gamma E' (1+\beta\mu') -E_0) \\ \displaystyle = \frac{F_0}{\gamma\beta E'} \left (\frac{E'}{E_0}\right)^2 \delta \left (\mu' - \frac{E_0-\gamma E'}{\gamma\beta E'} \right )$
  • $\displaystyle j'(E_f') = \frac{N' \sigma_T E_f' F_0}{2 E_0^2 \gamma \beta}~\text{ if }~ \frac{E_0}{\gamma (1+\beta)} < E_f' < \frac{E_0}{\gamma(1-\beta)}$
  • $\displaystyle j(E_f,\mu_f) = \frac{E_f}{E_f'} j'(E_f') \\ \displaystyle = \frac{N \sigma_T E_f F_0}{2 E_0^2 \gamma^2 \beta} \\ \displaystyle ~~\text{ if }~ \frac{E_0}{\gamma (1+\beta)(1-\beta \mu_f)} < E_f < \frac{E_0}{\gamma(1-\beta)(1-\beta \mu_f)} \nonumber$
  • $\displaystyle \frac{1}{\beta} \left [ 1 - \frac{E_0}{E_f} \left ( 1 + \beta \right ) \right ] < \mu_f < \frac{1}{\beta} \left [ 1 - \frac{E_0}{E_f} \left ( 1 - \beta \right ) \right ].$
  • $\displaystyle j(E_f) = \frac{N \sigma_T F_0}{4 E_0 \gamma^2 \beta^2} \left \{ \begin{array}{lc} (1+\beta) \frac{E_f}{E_0} - (1 -\beta ), \frac{1-\beta}{1+\beta} < \frac{E_f}{E_0} < 1 \\ \displaystyle (1+\beta) - \frac{E_f}{E_0} (1 -\beta ), 1 < \frac{E_f}{E_0} < \frac{1+\beta}{1-\beta} \\ \displaystyle 0, \text{ otherwise } \end{array} \right .$

• Angle Addition and Subtraction Formulae

  • $\begin{aligned} \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta \end{aligned}$
  • $\begin{aligned} \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \end{aligned}$
  • $\displaystyle{ \begin{aligned} \tan(\alpha + \beta) &= \frac{ \tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \\ \tan(\alpha - \beta) &= \frac{ \tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \end{aligned} }$
  • Apply the formula $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$:
  • $\displaystyle{ \sin{\left(45^{\circ} - 30^{\circ}\right)} = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) }$

• Fluid Mosaic Model

  • For comparison, human red blood cells, visible via light microscopy, are approximately 8 µm wide, or approximately 1,000 times wider than a plasma membrane.
  • A phospholipid molecule consists of a three-carbon glycerol backbone with two fatty acid molecules attached to carbons 1 and 2, and a phosphate-containing group attached to the third carbon.
  • Integral proteins (some specialized types are called integrins) are, as their name suggests, integrated completely into the membrane structure, and their hydrophobic membrane-spanning regions interact with the hydrophobic region of the the phospholipid bilayer .
  • Integral membrane proteins may have one or more alpha-helices that span the membrane (examples 1 and 2), or they may have beta-sheets that span the membrane (example 3).

• Distribution in Frequency and Angle

  • $\displaystyle {\hat E} (\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} E(t) e^{i\omega t} d t.$
  • $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^3} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ]_\mathrm{ret} e^{i\omega t} d t.$
  • $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^2} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ] e^{i\omega (t'+R(t')/c)} d t'.$
  • $\displaystyle \frac{d W}{d\Omega d\omega} = \frac{q^2}{4\pi^2 c} \left | \int_{-\infty}^{\infty} \frac{{\bf n} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]}{\left ( 1-\beta\cdot {\bf n} \right )^2} e^{i\omega (t'-{\bf n}\cdot {\bf r}(t')/c)} d t' \right |.$
  • $\displaystyle \frac{{\bf n} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]}{\left ( 1-\beta\cdot {\bf n} \right )^2} = \frac{d}{d t'} \left [ \frac{{\bf n} \times ({\bf n} \times \beta ) }{1-\beta\cdot {\bf n}} \right ].$

• Beta Coefficient for Portfolios

  • If we think of the S&P 500 as the index, a portfolio that fluctuates identically to the market has a Beta of 1.
  • What would the following portfolios have for Beta values?
  • Thus, the portfolio would have a Beta value of 3.
  • Every time the market is up 1%, the portfolio is down half a percent.
  • If you invested equal amounts in each portfolio, it would leave you over-exposed to the market because it would have a Beta of 1.5.

• How about relativistic particles?

  • $\displaystyle \frac{dP(t')}{d\Omega} = \frac{q^2}{4\pi c} \frac{ \left | {\bf n} \times \left \{ \left ( {\bf n} - \beta \right ) \times {\dot{\beta}} \right \} \right |^2}{\left ( 1 - {\bf n} \cdot \beta \right )^5} $
  • Let's start by assuming the $\beta$ is parallel to ${\dot{\beta}}$, so $\beta \times {\dot{\beta}}=0$.
  • $\displaystyle P = 2\pi \frac{q^2 {\dot u}^2}{4\pi c^2} \int_{-1}^1 \frac{1-\mu^2}{\left ( 1 - \beta \mu \right )^5} d\mu = \frac{2}{3} \frac{q^2 {\dot u}^2}{c^2} \gamma^6$
  • Let's take a second look at the angular distribution of power for small angles and $\beta \approx 1$ .
  • $\displaystyle \frac{dP(t')}{d\Omega} = \frac{q^2 {\dot u}^2}{4\pi c^3} \frac{ 1 }{\left ( 1 - \beta \cos\Theta \right )^3} \left [ 1 - \frac{ \sin^2 \Theta \cos^2\phi }{\gamma^2 \left ( 1 - \beta \cos\Theta \right )^2} \right ]$