Z-line

Examples of Z-line in the following topics:

• Sliding Filament Model of Contraction

  • A sarcomere is defined as the segment between two neighbouring, parallel Z-lines.
  • Actin myofilaments attach directly to the Z-lines, whereas myosin myofilaments attach via titin molecules.
  • Surrounding the Z-line is the I-band, the region where actin myofilaments are not superimposed by myosin myofilaments.
  • The I-band is spanned by the titin molecule connecting the Z-line with a myosin filament.
  • Titin molecules connect the Z-line with the M-line and provide a scaffold for myosin myofilaments.

• Fts Proteins and Cell Division

  • FtsZ has been named after "Filamenting temperature-sensitive mutant Z".
  • This ring is called the Z-ring.
  • The Z-ring forms from smaller subunits of FtsZ filaments.
  • Lines of FtsZ would line up together parallel and pull on each other creating a "cord" of many strings that tightens itself.
  • The Z-ring forms from smaller subunits of FtsZ filaments.

• Equations of Lines and Planes

  • A line is described by a point on the line and its angle of inclination, or slope.
  • Every line lies in a plane which is determined by both the direction and slope of the line.
  • A line in three dimensional space is given by a point, $P_0 (x_o,y_o,z_o)$, or a plane, $M$, and its direction.
  • $x = x_0 + at \\ y=y_0 +bt \\ z = z_0 +ct$
  • $z> \ = \ z_0 + ct>$

• Solving Systems of Equations in Three Variables

  • The introduction of the variable z means that the graphed functions now represent planes, rather than lines.
  • $\left\{\begin{matrix} 3x+2y-z=6\\ -2x+2y+z=3\\ x+y+z=4\\ \end{matrix}\right.$ Since the coefficient of z is already 1 in the first equation, solve for z to get
  • $z=3x+2y-6$.
  • Working up again, plug $y=2$ and $x=1$ into the first substituted equation, $z=3x+2y-6$, which simplifies to $z=1$.
  • Now subtract two times the first equation from the third equation to get $2x+2y+z-2(x+y+z)=3-2(2)$, which simplifies to $z=1$.

• Tangent Planes and Linear Approximations

  • The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • For a surface given by a differentiable multivariable function $z=f(x,y)$, the equation of the tangent plane at $(x_0,y_0,z_0)$ is given as:
  • where $(x_0,y_0,z_0)$ is a point on the surface.
  • Note the similarity of the equations for tangent line and tangent plane.
  • The approximation works well as long as the point $(x,y,z) $ under consideration is close enough to $(x_0,y_0,z_0)$, where the tangent plane touches the surface.

• Complex Numbers in Polar Coordinates

  • $r$ is how far the point is from the origin, which is usually denoted $r=\sqrt{a^2+b^2}=\abs{z}.$ The other parameter is the angle $\phi$ which the line from the origin to the point makes with the horizontal, measured in radians.
  • It turns out, due to a theorem of the great mathematician Euler that we can write $z$ as the complex expression $z=re^{i\phi}$.
  • So if $z=re^{i\phi}$ and $w=se^{i\theta}$ are complex numbers, then the product of $z$ and $w$ is $zw=rse^{i(\phi+\theta)}$, which comes from simply multiplying as usual for exponential functions.
  • From this we can see that the product of $z$ and $w$ is the complex numbers whose distance from the origin is the product of the distances from the origin of $z$ and $w$, and whose angle with the horizontal is the sum of the angles of $z$ and $w$ with the origin.
  • For example, consider the complex number $z=\sqrt2e^{i\pi/4}=1+i$ and $w=\sqrt2e^{3i\pi/4} = -1+i$.

• Complex Conjugates

  • The symbol for the complex conjugate of $z$ is $\overline{z}$.
  • The number $a^2+b^2$ is the square of the length of the line segment from the origin to the number $a+bi$.
  • The symbol for the modulus of $z$ is $\abs{z}$.
  • Thus we can write $z\overline{z} = (a+bi)(a-bi)=a^2+b^2=\abs{z}^2$, or in other words, $\abs{z}=\sqrt{z\overline{z}}.$
  • The length of the line segment from the origin to the point $a+bi$ is $\sqrt{a^2+b^2}$.  

• Normal probability examples

  • Z = 0.92.
  • Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
  • Z = 0.92.
  • Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
  • Erik's height is in the 40th percentile, marked with a black line.

• Partial Derivatives

  • For instance, $z = f(x, y) = x^2 + xy + y^2$.
  • To every point on this surface, there is an infinite number of tangent lines.
  • Partial differentiation is the act of choosing one of these lines and finding its slope.
  • By finding the derivative of the equation while assuming that $y$ is a constant, the slope of $f$ at the point $(x, y, z)$ is found to be:
  • That is to say, the partial derivative of $z$ with respect to $x$ at $(1, 1, 3)$ is $3$.

• Volumes

  • One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in three-dimensional space.
  • It can also mean a triple integral within a region $D$ in $R^3$ of a function $f(x,y,z)$, and is usually written as:
  • of the function $z = f(x, y) = 5$ calculated in the region $D$ in the $xy$-plane, which is the base of the cuboid.