gradient

Examples of gradient in the following topics:

• Secondary Active Transport

  • In secondary active transport, a molecule is moved down its electrochemical gradient as another is moved up its concentration gradient.
  • Instead, another molecule is moved up its concentration gradient, which generates an electrochemical gradient.
  • The molecule of interest is then transported down the electrochemical gradient.
  • As sodium ion concentrations build outside the plasma membrane because of the action of the primary active transport process, an electrochemical gradient is created.
  • An electrochemical gradient, created by primary active transport, can move other substances against their concentration gradients, a process called co-transport or secondary active transport.

• Electrochemical Gradient

  • Simple concentration gradients are differential concentrations of a substance across a space or a membrane, but in living systems, gradients are more complex.
  • The electrical gradient of K+, a positive ion, also tends to drive it into the cell, but the concentration gradient of K+ tends to drive K+ out of the cell.
  • The combined gradient of concentration and electrical charge that affects an ion is called its electrochemical gradient .
  • Electrochemical gradients arise from the combined effects of concentration gradients and electrical gradients.
  • Define an electrochemical gradient and describe how a cell moves substances against this gradient

• Chemiosmosis and Oxidative Phosphorylation

  • Chemiosmosis is the movement of ions across a selectively permeable membrane, down their electrochemical gradient.
  • The uneven distribution of H+ ions across the membrane establishes both concentration and electrical gradients (thus, an electrochemical gradient) owing to the hydrogen ions' positive charge and their aggregation on one side of the membrane.
  • This protein acts as a tiny generator turned by the force of the hydrogen ions diffusing through it, down their electrochemical gradient.
  • In oxidative phosphorylation, the hydrogen ion gradient formed by the electron transport chain is used by ATP synthase to form ATP.
  • ATP synthase is a complex, molecular machine that uses a proton (H+) gradient to form ATP from ADP and inorganic phosphate (Pi).

• Fundamental Theorem for Line Integrals

  • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
  • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
  • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
  • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

• Proton Reduction

  • Anaerobic respiration utilizes highly reduced species - such as a proton gradient - to establish electrochemical membrane gradients.
  • An electrochemical gradient has two components.
  • Cellular respiration (both aerobic and anaerobic) utilizes highly reduced species such as NADH and FADH2 to establish an electrochemical gradient (often a proton gradient) across a membrane, resulting in an electrical potential or ion concentration difference across the membrane.
  • Proton reduction is important for setting up electrochemical gradients for anaerobic respiration.
  • In contrast, fermentation does not utilize an electrochemical gradient.

• Diffusion

  • On the contrary, concentration gradients are a form of potential energy, dissipated as the gradient is eliminated.
  • Each separate substance in a medium, such as the extracellular fluid, has its own concentration gradient independent of the concentration gradients of other materials.
  • In addition, each substance will diffuse according to that gradient.
  • This lack of a concentration gradient in which there is no net movement of a substance is known as dynamic equilibrium.
  • Extent of the concentration gradient: The greater the difference in concentration, the more rapid the diffusion.

• Transport of Electrolytes across Cell Membranes

  • Water passes through semi-permeable membranes by passive diffusion, moving along a concentration gradient and equalizing the concentration on either side of the membrane.
  • Active transport requires energy in the form of ATP conversion, carrier proteins, or pumps in order to move ions against the concentration gradient.
  • Passive transport, such as diffusion, requires no energy as particles move along their gradient.
  • Active transport requires additional energy as particles move against their gradient.

• Gas Exchange across the Alveoli

  • Since this pressure gradient exists, oxygen can diffuse down its pressure gradient, moving out of the alveoli and entering the blood of the capillaries where O2 binds to hemoglobin.
  • Due to this gradient, CO2 diffuses down its pressure gradient, moving out of the capillaries and entering the alveoli.
  • Oxygen and carbon dioxide move independently of each other; they diffuse down their own pressure gradients.
  • This pressure gradient drives the diffusion of oxygen out of the capillaries and into the tissue cells.
  • The pressure gradient drives CO2 out of tissue cells and into the capillaries.

• Directional Derivatives and the Gradient Vector

  • If the function $f$ is differentiable at $\mathbf x$, then the directional derivative exists along any vector $\mathbf v$, and one has $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$, where the $\nabla f(\mathbf{x})$ is the gradient vector and $\cdot$ is the dot product.
  • The gradient of the function $f(x,y) = −\left((\cos x)^2 + (\cos y)^2\right)$ depicted as a projected vector field on the bottom plane.

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Here $\nabla\varphi$ denotes the gradient of $\varphi$.
  • If $\mathbf{v}=\nabla\varphi$ is a conservative vector field, then the gradient theorem states that $\int_P \mathbf{v}\cdot d\mathbf{r}=\varphi(B)-\varphi(A)$.