Chemistry of Neruons and Neurotoxins in Biology
- Page ID
- 418897
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Topics Covered
- Action potential
- Nernst equation
- Example problem: concentration cell
- Neurotoxins
- Electrochemical Gradient
- Challenge problem
- Neurotoxin challenge problem
Topics from the ACS Examinations Institute for General Chemistry--ACCM
- VI. Energy and Thermodynamics:
- 3. There are a wide variety of energy units, so care must be taken to use consistent units when considering energy changes quantitatively.
- E 1. d. Nonstandard situations are addressed via the Nernst equation.
- H. 2. Gibbs free energy is a state function that simultaneously calculates entropy for the system and surroundings, and is useful for determining whether or not a process occurs spontaneously.
- VIII Equilibrium:
- F. 2. For electrochemical systems, the cell potential is also related to the change in free energy.
Introduction to Neurons: Action Potential
The nervous system operates through a series of chain reactions between neurons as the cells fire one after another, transmitting an electrochemical signal throughout the body. Neurons release chemical signals, called neurotransmitters, into the synapse, which are then received by the following neuron. These neurotransmitters cause the next neuron to fire, which then releases its own set of neurotransmitters causing the next sequential neuron to fire.
Each neuron has a resting electrochemical potential of around -70mV due to the concentration of ions inside and outside the cell.1 The neuron receives a chemical signal when the neurotransmitters released from another neuron bind to its receptors. This triggers the opening of ligand-gated ion channels along the cell membrane of the neuron. Positive ions diffuse across the membrane, and if the voltage increase due to this initial excitatory reaction reaches the threshold potential, the voltage-gated sodium channels open leading to the rapid depolarization of the neuron as ions flood through the channels.1 The firing of a neuron is an all-or-nothing event– the neuron either fires at full force or it does not. After depolarization, the potassium ion pumps on the cell membrane begin to restore the resting potential. In the process, the voltage-gated sodium ion channels close. The efflux of potassium combined with the closed sodium ion channels lead to the resting potential being undershot. Eventually, the resting potential is restored via diffusion through potassium leakage channels. During this reset period, the neuron cannot fire.2
The concentration half cells of the neuron can be calculated, predicting which direction the ions are likely to flow.
The cell potential is called Ecell, with the potential at standard or starting conditions being Eocell
A positive Ecell correlates to a spontaneous reaction (and a negative ΔG) so ions flow into the cell. Likewise, a negative Ecell correlates to ions flowing out of the cell and the reaction being non-spontaneous in the forward direction.
At standard conditions, Eocell can be determined using3:
\[\Delta G^o = -nFE^o_{cell}\]
Likewise, at nonstandard conditions:
\[\Delta G = -nFE_{cell}\]
Where ΔGo is the change in free energy in J. n is the integer charge. The Faraday constant (F) is 9.649x104, the energy of one mole of electrons with units Coulombs (C) per mole. And Eocell is the cell potential with units J/C (Volts) at standard conditions 1M | 1atm.
The Gibbs Free energy at nonstandard conditions can be calculated with3:
\[\Delta G = \Delta G^o + RT \ln Q \]
Where ΔG is the free energy in J, ΔGo is the free energy at standard conditions. n is the integer charge. R is the universal gas constant 8.314 \(\tfrac{J}{mol K}\). T is Temperature in Kelvin and Q is the reaction quotient (molar concentrations at the given time).
Substituting the definition of ΔGo from eq. 1 and 2 into eq. 3 we get3:
\[-nFE_{cell} = -nFE^o_{cell} + RT \ln Q \nonumber \]
\[E_{cell} = E^o_{cell}- \frac{RT}{nF} \ln Q \]
Determine the Ecell of the neuron at the given concentrations.
Givens:
Na+ concentration inside: 14mM | Na+ concentration outside: 145mM
Body temperature = 37oC
Solution
37oC = 310K
\(E_{cell}(\tfrac{J}{C})= E^o_{cell}(\tfrac{J}{C})- \dfrac{R(\tfrac{J}{molK})310(K)}{(1)F(\tfrac{C}{mol})} \ln \frac{14}{145} \)
because this is a concentration gradient problem and both half-cells are the same, Eo = 0. Therefore,
\(E_{cell} = - \dfrac{(8.314)(310)}{(1)9.649·10^4} \ln \tfrac{14}{145} = 0.0624V \)
Note that since the ions flow into the cell during an action potential, this reaction must be spontaneous and Ecell will be greater than 0.
Neurotoxins in Context:
In cases of chronic and hypersensitive pain, neurons firing in excess can be harmful to the body. It is important to note that chronic pain is not equivalent to long lasting acute pain. Acute pain serves as a reactionary mechanism by the body to warn the brain of some impending danger and usually subsides quickly as the body heals. In contrast, chronic pain typically stems from inflammation due to disease or damage to the surrounding nervous system of a region. Chronic pain often has no benefits for the body and instead leaves the patient suffering4.Hypersensitive pain can also be a detriment, causing great distress to the patient over relatively little biological need.
Some of the most popular and widely used pain inhibitors are NSAID’s, or nonsteroidal anti-inflammatory drugs. The most common of which are ibuprofen and acetaminophen. These drugs are non-selective and often need to be taken in high doses to be effective which can have harmful long term effects on the liver and kidney.4 Because of this, highly potent, highly selective drugs are always of great interest to the medical community.
While undoubtedly dangerous in the wrong dosage, neurotoxins, such as the cone snail’s conotoxin or pufferfish tetrodotoxin, possess both of the required criteria mentioned above. Conotoxin specifically has recently garnered mass attention from the medical community for its widespread diversity and incredible potency for the inhibition of calcium-gated ion channels.
Neurotoxins function by attacking voltage-gated ion channels and interrupting their regular processes. In doing so, they prevent a signal from being properly perpetuated across neuronal links. A few examples of this are ω-conotoxin, which physically binds to the subunits of the ion channel and blocks the pores allowing ions to transfer in and out.4 Another type of inhibitor functions by preventing ion channels from closing. So rather than stopping the perpetuation of action potential by preventing the transfer of ions, these types of neurotoxins force a continued stream of charged particles, meaning that the cells are unable to return to a base state of resting potential. Most signals from the brain take approximately 1 millisecond to perpetuate, any longer than that, and the neuronal chain reaction is disrupted.5
Extension: Electrochemical Potential
electrochemical potential \(\mu = \mu^o + RT\ln [A_x] + nF(V)\) (Feher, 2012)
where µ is the free energy per mole, an intensive property of free energy affected by the conditions (Note: Gibbs ΔG is extensive and determined by the amount of substance). R is the universal gas constant 8.314 \(\tfrac{J}{mol K}\), T in Kelvin, [Ai] and [Ao] are the concentrations of ions inside and outside the cell, respectively. n is the number of moles of charge (for Na+ it would be 1 and for Mg2+ it would be 2). 9.649x104 is the energy of one mole of electrons, also called the Faraday constant (F). ΔV is the cell potential in volts \(\tfrac{J}{C}\).
From the equation above, we can determine the electrochemical potential difference by doing the potential inside minus that of the outside.
\[\Delta\mu = \mu_i - \mu_o\]
Inserting the equation from above, we get:
\[\Delta\mu = \mu^o + RT\ln [A_i] + nF(V_i) - \mu^o - RT\ln [A_o] - nF(A_o)\]
\[\Delta\mu = RT\ln( \tfrac{[A_i]}{[A_o]} ) + nF\Delta V\]
\[\Delta\mu\tfrac{J}{mol} = R\tfrac{J}{mol K}T(K)\ln( \tfrac{[A_i]}{[A_o]} ) + nF\tfrac{C}{mol}\Delta V\tfrac{J}{C}\]
Is the transport of Na+ into the cell spontaneous?
Support your conclusion by calculating Δµ.
Givens:
Na+ concentration inside: 14mM | Na+ concentration outside: 145mM
Body temperature = 37oC | ΔV = -70mV
Solution
37oC = 310K
\(\Delta\mu = 8.314\tfrac{J}{mol K}310 K \ln(\tfrac{14}{145})+ 1 · (9.649·10^4\tfrac{C}{mol})(-.070\tfrac{J}{C})\)
\(\Delta\mu =-12779\tfrac{J}{mol}= -12.779\tfrac{kJ}{mol}\)
\(\Delta\mu < 0 \) therefore the transport of Na+ into the cell is spontaneous
Is the transport of K+ into the cell spontaneous?
Support your conclusion by calculating Δµ.
K+ concentration inside: 150 mM | Na+ concentration outside: 4 mM
Body temperature = 37oC , 310K | ΔV = -70mV
Solution
\(\Delta\mu = 8.314\tfrac{J}{mol K}310 K \ln \tfrac{150}{4} + (1)(9.649 ·10^4\tfrac{C}{mol}(-.070\tfrac{J}{C})\)
\(\Delta\mu = 2,587\tfrac{J}{mol}= 2.587\tfrac{kJ}{mol}\)
\(\Delta\mu > 0 \) therefore the transport of K+ into the cell is non-spontaneous
The K+ ions will flow in the reverse direction, out of the cell.
Neurotoxin Challenge Problem
Neurotoxins can work in several different ways, most of which involve the targeting of the voltage-gated ion channels. By disrupting the flow of electrons in and out of a cell, the neurotoxin can disrupt the overall electrical signal sent throughout the nervous system, impairing function in a multitude of ways. Example 3 below outlines what would happen if a neurotoxin which prevented the ion-channels from closing appropriately entered into the nervous system.
Given the following information, calculate the percent change in Na+ concentration in a cell if action potential is extended for 2.5 ms.
- A 100-mV change occurs within a cell during action potential
- The intracellular concentration of Na+ is 0.01M
- The cell has a surface are of 314 µM2 and a volume of 524 µM3
- The specific membrane capacitance of the cell is 10-6 \(\frac{F}{cm^2}\)
- The charge of ONE sodium ion is 1.6 x 10-19 Coulombs
The best way to approach this problem is to split it up into parts, first finding the concentration of sodium in the cell and then determining how much it changes.
Solution
Step 1: Calculate the change in charge/cm2 of the cell membrane during action potential
- 1 Faraday = 1 Coulomb / 1 Volt
- Charge = \(Q = 1·10^{-6}\tfrac{F}{cm^2} (100mV)(\frac{1V}{1000mV}) = 1·10^{-7}\tfrac{C}{cm^2}\)
Step 2: Calculate the number of sodium ions required to see this change in charge
- \((1·10^{-7} \tfrac{C}{cm^2})( \frac{Na^+}{1.6·10^{-19}C})\)= 6.25x1011 Na+ ions/cm2
Step 3: Calculate the total number of Na+ ions that pass through the entire cell membrane
- \((314 \mu M^2 )(\dfrac{1cm^2}{10^8\mu M^2}) (6.25·10^{11}\tfrac{Na^+ ions}{cm^2})\) = 1.963 x 106 Na+ ions transferred through the membrane/action potential
Step 4: Calculate the initial concentration of Na+ in the cell
- \(524 \mu M^3 \tfrac{1cm^3}{10^{12}\mu M^3}) (\frac{1L}{10^3 cm^3})\) = 5.24 x 10-13 L
- \((5.24·10^{-13}\tfrac{L}{cell})(.01 \tfrac{mol}{L})\)= 5.24 x 10-15 moles of Na+ per cell
- \((5.24·10^{-15} \tfrac{mol}{cell})(\frac{6.022·10^{23})}{mol})\) = 3.1555 x 109 Na+ ions per cell
Step 5: Calculate the amount of Na+ that enters the cell after 2.5 ms if action potential usually lasts for 1 ms
- \((2.5ms) (1.963·10^6 \tfrac{Na^+ions}{1ms})\) = 4.908 x 106 Na+ ions
Step 6: Calculate the new concentration of Na+ ions in the cell and find the percent increase
- \(3.1555·10^9 + 4.908·10^9\) = 3.160 x 109 Na+ ions
- \(\frac{3.160·109-3.1555·10^9}{3.1555·10^9}\)= 0.156% increase
Note that the extension problems above go beyond the scope of Chem 110.
References
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Byrne, J. Resting Potentials and Action Potentials (Section 1, Chapter 1) Neuroscience Online: An Electronic Textbook for the Neurosciences | Department of Neurobiology and Anatomy - The University of Texas Medical School at Houston /neuroscience/m/s1/chapter01.html.
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Katzman, Shoshana D. & Alessandra L. Barrera, et. alt. “Chapter 8 - The Electrochemical Gradient” in “Fundamentals of Cell Biology” on OpenALG https://alg.manifoldapp.org/read/fundamentals-of-cell-biology/section/f4d10886-6492-41bb-9ab0-bbbc67bb0292.
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17.2: The Gibbs Free Energy and Cell Voltage https://chem.libretexts.org/Bookshel...d_Cell_Voltage.
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McGivern, J. G. Ziconotide: A Review of Its Pharmacology and Use in the Treatment of Pain. Neuropsychiatric Disease and Treatment 2007, 3 (1), 69–85. https://doi.org/10.2147/nedt.2007.3.1.69.
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Stevens, M.; Peigneur, S.; Tytgat, J. Neurotoxins and Their Binding Areas on Voltage-Gated Sodium Channels. Frontiers in Pharmacology 2011, 2. https://doi.org/10.3389/fphar.2011.00071.
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Feher, J. Active Transport. Quantitative Human Physiology 2012, 134–140. https://doi.org/10.1016/b978-0-12-382163-8.00016-5.