In a study of proteins mediating cell membrane transport, biologists measure current versus time through the cell membranes of oocytes (nearly mature egg cells) taken from the African clawed frog, Xenopus. The measured current versus time is given approximately by \(I=60 t+200 t^{2}+4.0 t^{3}\) with \(t\) in seconds and I in nA. Find the total charge that flows through the cell membrane in the interval from \(t=0\) to \(t=5.0 \mathrm{s}.\)

Biophysics is a discipline at the intersection of biology and physics, which includes the study of biological systems using physical methods and concepts. It plays a pivotal role in understanding cell membrane transport mechanisms. The cell membrane is a critical biological structure that controls the passage of substances into and out of cells.

In biophysics, the transport of molecules across cell membranes is often quantified by measuring the flow of ions, which creates an electrical current. This is of particular interest in the study of cell physiology, for example in nerve and muscle cells, where ion flow underlies the transmission of signals.

The problem in question involves a study of proteins that mediate cell membrane transport in oocyte cells. The flow of ions, in this case, is represented by an electrical current measured in nanoamperes (nA). Understanding how this current changes over time is essential for interpreting the active biological processes in the cell membrane and characterizing the behavior of the proteins involved.

In the realm of biology, the concept of electrical current is closely associated with the movement of charged particles or ions across membranes. Biological membranes are semi-permeable barriers that separate the inside of the cell from the extracellular environment. Ion channels and transporters in these membranes can move ions, which carry electric charges, thus creating an electric current.

The equation given in the exercise, \(I=60t+200t^2+4.0t^3\), represents the current flowing through the membrane as a function of time, with \(I\) representing the current in nanoamperes and \(t\) the time in seconds. We can infer that this current results from a combination of time-dependent processes within or across the cell membrane, possibly reflecting different stages of activity of the transport proteins.

Understanding how these currents work not only sheds light on fundamental biological processes but also paves the way for innovations in medical technology, diagnostics, and treatments related to electrical signaling in the body, such as in the case of cardiac pacemakers and neurostimulation devices.

Charge integration is a fundamental concept in biophysics, especially when it comes to electrical current in biology. In the context of the exercise, the total charge transported through the cell membrane over a specific period is obtained by integrating the current with respect to time.

Mathematically, the total charge \(Q\) is the area under the current versus time curve from \(t=0\) to \(t=5.0s\), which is calculated using the definite integral: \[Q=\int_0^{5.0}I dt\]. By performing the integration, we find the mathematical expression for the total charge, which in this case involves adding up contributions from each term in the current function individually.

After integrating and evaluating the definite integral, we end up with the expression: \[Q=\left[30t^2+66.67t^3+t^4\right]_0^{5.0}=30(5.0)^2+66.67(5.0)^3+(5.0)^4\], which represents the total charge in nanoCoulombs (nC). This quantifies the total amount of electric charge that has moved through the membrane in the given time frame, reflecting the activity of the transport proteins in the oocyte cells.