My talk on the paper: A Bipedal DNA Brownian Motor with Coordinated Legs

Hello everyone

The paper I gave my presentation on was titled: A Bipedal DNA Brownian Motor with Coordinated Legs
By Tosan Omabegho, Ruojie Sha, Nadrian C. Seeman

A copy of it can be accessed at the websites below.

Abstract:
http://www.sciencemag.org/cgi/content/abstract/324/5923/67

Paper:
http://www.sciencemag.org.ezproxy.library.uq.edu.au/cgi/reprint/sci;324/5923/67?maxtoshow=&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&resourcetype=HWCIT.pdf

Practice final exam

Here is the final exam from last year.

Structure of the torque ring of the flagellar motor and the molecular basis for rotational switching

Hi all,

I will be presenting the paper titled "Structure of the torque ring of the flagellar motor and the molecular basis for rotational switching" by Lee et al on Wednesday. Have a look at some of the supplementary videos on this site - the torque ring in particular.

http://www.nature.com/nature/journal/v466/n7309/full/nature09300.html

Cheers,

Calvin

Our Video!

How do neurons perform calculations?

If the brain is analogous to a computer is it possible to describe a small number of neurons as similar to a logic gate? What would this analogy leave out?

Paper presentations

Before you presentation please post a link to the paper you will be discussing.
After the presentation (or even better before) post a copy of your presentation.

When axons get on your nerves

Thanks to the work of Camillo Golgi and Santiago Ramon y Cajal, we have been able to understand the functioning of neurons. Between each neuron is a synapse. On either side of this synapse is the presynaptic (axon) and the postsynaptic (dendrite) components. As it goes, information passes from the pre- to the post-synaptic side.


Axon Terminals are capable of releasing various neurotransmitters, which can be used to alter the membrane potential in another dendritic tree.
Arrivign action potentials at the presynaptic cleft can cause depolarization or hyperpolarization of the postsynaptic, based around the released neurotransmitters, and on the postsynaptics ion channels.
The synapse can be considered excitatory or inhibitory based on whether it is depolarizing of hyperpolarizing.Often, single action potentials just aren't good enough to do anything, soi cells incorporate integrate-and-fire model of neuron activity.

Electrophysiology

Also known as the axon's electrical response, there are several features that are key to the functioning of an action potential.

1. It's all or nothing - either you're going to have an action potential, or you're not. The membrane depolarization must reach a certain threshold (commonly quote in textbooks as -50mV). If subthreshold, electrotonus occurs (no response far from the point of stimulation), but if above the threshold, a wave of excitations will occur, with each peak potential non-determined by the input stimulus.

2. action potentials move at a constant speed of 0.1 to 120 m/s.

3. Action potential peaks are independent of distance, though the decaying behaviour of hyperpolarization or subthreshold stimuli are. This is to say, one input of stimulus can send an action potential along the longest axon (very important, think of your leg)

4. action potential shape remains constant over time.

5. afterhyperpolarization is where the membrane potential overshoots, become more negative than at rest.

6. The neuron is harder to stimulate during what is known as the refractory period.

Proof that ion channels exist?

The experiment that produced the graphs in figure 12.13 is described as the most remarkable in the book. I suppose it is the most remarkable because showing that the action potential of an intact axon is identical to that of an axon with replaced internal contents proves that: a) the internal contents of the axon do not matter and b) there must be some way for the ions thought to be responsible for the action potential to enter the axon. Could that way be the ion channels?

Threshold stimulus effect on crab axon

Figure 12.2 shows the effect of depolarizing (upper traces) and hyperpolarizing (lower traces). It shows a quick dying off (electrotonus) behaviour below a threshold but large propagating potential once this threshold is reached.

Movement of an Action Potential

Initiation of an action potential.
Most voltage-sensing channels have voltage-sensing ‘paddles’. These paddles consist of charged helices that can move according to the charges of its surroundings. Movements of these voltage-sensors create conformational changes in the channel that leads to their opening/closing.



Action potentials move in one direction.
Na+ channels can also be inactivated. After activation Na+ channels are inactivated for a time (refractory period). In the ‘ball and chain’ model the channel is blocked by a ball-structure which binds in the channel pore. This ensures that the action potential does not propagate backwards. The channel will remain inactivated until the membrane has reached its resting potential.



Repolarisation is also delayed by the hyperpolarisation that occurs from K+ channel flux. This is seen as a further dip in Figure 12.8a in Nelson. K+ voltage gated channels are opened once the action potential depolarisation has occurred. They then remain open until the membrane potential returns to its resting state.



Information taken from: Molecular Cell Biology, Lodish, 6th edn, Chapter 23.
Information and Channel figure taken from: The Voltage Sensor in Voltage-Dependent Ion Channels, F. Bezanilla, 2000, Pysiological Reviews.
1st polarisation figure taken from: http://www.ncbi.nlm.nih.gov/bookshelf/br.fcgi?book=mcb&part=A6125

Myelination allows the Action Potential to jump down the axon.

The myelin sheath provides insulation for the neuron and increases the speed of the action potential transmission. For neurons with no sheath the speed of the action potential is proportional to the axon diameter (you may remember this from Krassen’s lectures in Biph2000).

The region of the axon covered in the sheath does not have any channels present in the membrane, they are instead localised to the nodes between the sheaths. In this way the potential can jump to each node with practically no loss of signal. This type of transmission is called “salutatory conduction”.

Information taken from: Molecular Cell Biology, Lodish, 6th edn, Chapter 23.
Image taken from: http://www.arts.uwaterloo.ca/~bfleming/psych261/lec4se21.htm

The Hodgkin and Huxley Equations

In the biological maths course I took last semester, we looked at the Hodgkin and Huxley model for action potential. I hoped that this chapter would mention it, but it didn’t go into many of the details. So I thought I’d share a bit of what I learnt here.

They modelled the rate of change of the potential as a function of the current injected into a cell minus the change in voltage caused by the change in conductance of each ion (Ex is the equilibrium Nernst potential of the ion x).

They had to find a relation to model the conductances of each ion as function of membrane potenial. They chose the equations below. The variables n and m measure (respectively) the probability of the potassium or sodium voltage gated ion channels being open and thus 0 <= n <=1. The term for the sodium conductance has an extra variable, h, which measures the probability of the sodium gated ion channel being inactivated. As section 12.3.2 on page 540 of Nelson points out, it’s ok to use a probabilistic model for the activation of ion channels, as there are many of them in a single cell.

A model must then be found for these probabilities. Each of the variables n, m and h were assumed to have first order kinetics, where there was one rate constant governing the opening of the channel, and one rate constant governing the closing. These rates are α and β in the equations below, and are assumed to be the same for each variable m, n and h. It is these variables, α and β, which are membrane potential dependant.

This model might seem complicated, and it is, but it produces quite realistic results. One of our assignments for this course was to solve these equations numerically. This is an example of the action potentials produced by this model.

I think this chapter would have been better off discussing this model rather than the model in section 12.2.

The Goldman Equation

This chapter derived some interesting models for the way action potentials are generated. But I noticed they let out an equation which I thought was quite important when considering the membrane potential. This equation is the Goldman-Hodgkin-Katz equation. This equation is effectively just the Nernst equation, but for multiple ionic species. But there is one important difference. The Nernst equation uses relative concentrations on each side of the cell membrane to determine the potential, whereas the Goldman equation goes a step further, and includes a permeability term for each ion.

Px represents the permeability of ionic species x, F is Faraday’s constant, and R and T have their usual meaning, the gas constant and temperature.

The permeability term allows this equation to remain valid during the entire action potential, unlike the Ohmic hypothesis used in Nelson. To convince yourself this equation will still give an accurate measurement of the membrane potential during an action potential, consider what happens to V as the permeability of sodium (PNa) increases. [Na+out]>[Na+in], so the numerator in the natural log increases, increasing the membrane potential V.

A realistic ratio of the permeabilities of these ions in the resting state is PNa:PK:PCl = 1 : 0.03 : 0.1. Using these pemeabilities and the values in table 11.1 on page 477 of Nelson, I calculated a resting potential of -63.8mV, which is very close to the measured value of 60mV. The ratio of permeabilities for the excited state is PNa:PK:PCl = 1 : 15 : 0.1, which yields a membrane potential of +43.7mV, which is again close to the accepted value of the peak potential of an action potential.

If you don’t want to repeat these calculations again, but would like to play around with values, I suggest going to this website: http://www.nernstgoldman.physiology.arizona.edu/. It allows you to vary the permeabilities of the ions and observe the effect on the membrane potential using the Goldman equation.

Meeting Summary 13th of October

As today was the first time we had seen the sun in a few days, we decided to have this meeting outside. On our way we discussed aspects of the biophysics course, such as the prerequisites, the textbook, etc. We came back to this topic briefly a number of times during the meeting, but I didn’t think it was necessary to record these digressions.

The main topic of chapter 11, and thus our discussions, was ion channels. Much of the maths derived for topics that we have studied in previous chapters, which deal with bulk solutions and continuums, breaks down when applied to ion channel physics. Before the structures of the ion channels were known, many models were suggested, but were not able to reflect reality. This is because ion channels often allow transmission of single molecules at a time, and statistical assumptions like those made for the Boltzmann distribution break down at this level.

We briefly discussed voltage gated ion channels, and how a potential can be stored in a cell membrane. This discussion moved on to how electric eels generate charge, and how the charge hurts/kills prey. They have thousands of cells in series called electroplaques which generate and store potential, like a capacitor.

The sodium anomaly was discussed, which lead to the concept of the anion gap. This is where a coma patient’s electrolyte count is measured, to indicate how the coma was induced. Brain injury patients are at risk of low electrolyte concentrations, as brain swelling increases the amount of water in the brain.

The unusual features of the chlorine channels were mentioned. These channels come in pairs, and have inward curving channel walls, and have a glutamate residue gating the channel, which only allows chlorine ions to pass through in one direction. KcsA channels were also brought up and the way that the protein mimics the hydration sphere of the potassium ion, to allow only potassium ions to pass through, rather than sodium ions, which as smaller. The concept of the potential mean force on an ion as it passes thorough a channel was introduced. Ligand gated ion channels were also mentioned.

The nature of torque was discussed, and how it relates to the ATP synthase motor experiment in section 11.3.4. The mitochondrial role in glycolysis was touched upon, as well as the molecule pyruvate, but most of us were happy with this area.

How some toxins work as ion channel blockers, and how lethal they can be was also discussed. We then moved into Megan’s lab to view some videos illustrating how detergents can affect crystallised protein structures and how ions travel through ion channels.

How to Make the Most of Your Chemiosmotic Coupling

So the body needs ion pumping across membranes as a necessity. It can be used for segregating macromolecules inside cellular components (often they need to be in a certain environment for optimal activity), to give macromolecules an overall net negative charge (to prevent clumping), and to maintain osmotic balance, or to osmoregulate.

So from pg 497, here are some you can make use (or more appropriate, cells force an organism to make use) of chemiosmotic coupling.

Proton pumping in chloroplasts and bacteria.

Chloroplasts are ATP-generating organelles, which use the free energy of the sunlight they absorb to pump protons across their membrane, and this proton gradient drives the "CF0CF1" complex, similar to the mitochondria complex "F0F1"

Bacteria also find their proton gradient and contain F0F1 synthases.

Flagellar Motor.

The flagella motor converts the electrochemical potential jump of protons into a mechanical torque. (to quote from pg 497 of the text)

Other Pumps.

Pumps such as the calcium ATPase are powered by ATP. There are others known as symports, and antiports.

How now to blow your cells up 101

Unfortunately, unlike plant, algal, fungal and bacterial cells, human cells just don't have a lot of strength behind them. They are weak, flexible and easily broken by the osmotic pressure occuring during Donnan Equilibrium. So, to maintain this equilibrium, the body has developed a method of being able to unsure that our cells don't swell up to the point that they burst, nor remove solution away to the point that the cells shrivel up like an old person. This solution - allow both the cell to swell and shrivel at the same time in such an equilibrium that neither burstage nor complete shrivelling occur. The cells allow for continuous pumping of the sodium ions by using metabolic energy. This creates a nonequilibrium system, but a steady state.

The text provides another example of this as the water fountain. Allow the fountain to keep running without a pump will cause it to eventually stop, but by using a pump, one can move the water back to the water source, and the cycle of fountain flow can continue.

Finally, the text eludes to some genetic defects which can interfere with osmoregulation. If one has spherocytosis, then their red blood cells become more permeable to sodium. Because the cells have to work a lot harder to remove the sodium, then it results in some cells surcoming to the eventual swelling and bursting. As the text says, "Entropic forces can kill"

Finally (this time it actually is finally) I was going to elude to action potentials. These are an example of continuous pumping of ions in and out of the cell but which can perform a useful action. Once a certain energy threshold has been achieved in the cells, a cascade of ions can be pumped in and out of the cell, allowing the muscle to perform useful work.

Evidence for the chemiosmotic mechanism: generation and utilization

ATP synthase is independent of the electrochemical potential generating proteins. In the words of the text: generation and utilization of chemiosmotic mechanism are independent. An experiment confirming this was done by E. Racker and W. Stoeckenius in which an artificial system resembling an ATP factory was constructed. Artificial lipid bilayers with the light-driven proton pump, bacteriorhodopsin from a bacterium, was confirmed as generating a pH gradient in response to light. A foreign ATP synthase from beef heart was added. With these two elements ATP was synthesized when the experiment was exposed to light. Because of the very different nature and source of the two proteins it was concluded that the two events are independent.

Figure 11.9 from page 493 shows an example of oxidative phosphorylation and the independence of proton (black dots) gradient generation and ATP synthasis (far right).

Meeting summary 06.10.10 CH10 Enzymes & Molecular Machines

Sorry about the late posting of this summary!
Last week we discussed chapter 10 of Nelson which was on enzymes and molecular machines. A definition of molecular devices can be found on pages 402-403 where there is a distinction between catalysts and 1-shot machines. An example of a 1-shot machine is translocation of a protein across a membrane.

We discussed where the motive force used by Listeria fits in to there definitions of molecular devices. At the tail end of Listeria an actin nucleation site where actin polymerizes is found. This polymerization causes listeria to be pushed forward. Should the Listeria motive force be considered a one-shot machine or a cyclic motor? p402 definition of a cyclic motor suggests that by using a renewable fuel (ATP) this it should be considered cyclic.

Mitch wondered where the data in fig 10.8 comes from.

Megan mentioned that the fact that these molecular machines do work on both local and global random fluctuations was not emphasized in this chapter. They tended to be represented by a smooth switching function when they would be better represented by an energy funnel.

Gilbert & Sullivan keep interrupting the chapter.

Figure 10.26 compares rachets to kinesin. The question was raised: what role does ADP or ATP play in this mechanism? It must be coupled to free energy difference but it is not clear how this happens. p463 mentions assumptions hold for large deltaG.

ATP, ADP provides large energy that allows weak, strong binding.

Figure 10.24 load is moved on upper right, hydrolysis on lower left. motion is not coupled to fuel burning. Are the events connected irreversibly?

A nucleotide sandwich dimer was mentioned where ATP is between myosin & kinesin.

The Smoluchowski equation (10.4) was discussed. Which describes diffusion with drift with ohmic/linear response/damped/dissipitive where velocity is linear in force - this basically means low reynolds number.

We discussed section 10.4 on Michaelis-Menten kinetics and discussed non- vs un-competitive binding. See Matt & Mitch's blog posts from last for further information.

We discussed the lack of statistics/probability knowledge in science - this seems to be a common theme.

Brief discussion on potential surfaces and their usefulness.

S-rachet was presented as protein translocation through membrane. When considering molecular machines even the best once go both ways due to thermal motion KbT.

We discussed molecular models of ATP & ADP + Pi and how it is difficult to model both as a single system.

Alan posed a question: What is the cause of catalysis? Could proteins have evolved to fluctuate randomly in one degree of freedom & can this enhance the rate of reaction. Is it possible to tell the difference between the different proposals for how enzymes work? Thermodynamically they all appear the same.

Your Turn 11F

As this is a fairly short chapter, and I wanted to leave things for other people to blog about, I’m taking a leaf out of Heather’s book and making my second post my solution to Your Turn 11F.

a) Torque, τ is force, f × (cross product) distance from the pivot to the point the force is applied, r. The force described in equation 11.17 is for the force acting perpendicular to the long axis of the rod, so the angle between the force vector and the position vector is 90o. So the cross product can be treated as a regular product. I will assume the force applied to the rod is uniform along the rod, so the average position of the point where the force is being applied is just the halfway down the rod. Thus, τ= f ×r≈3.0ηLvr=1.5ηL^2v.
The velocity of a point rotating at ω rad/s is the angular velocity times the distance that point is from the pivot. The average distance a point on the rod is from the pivot is again just halfway down the rod. So the torque becomes τ≈0.75ηL^3ω.

For a rod of length 1μm and an angular velocity of 6 rev/s=12π/s, the torque is 2.83x10^-20Nm.

b) The work done by torque is just the torque times the angle moved. In this case, the angle moved is one third of a revolution, which is ⅔π. So the work done by the F1 motor every third of a revolution is 5.92x10^-20J. Thermal energy at room temperature is 4.1x10^-21J, which is less than the value calculated, which is a good sign.

Differences between normal cells and neurons

After reading this chapter I was confused about which cells had sodium pumps, which cells had voltage drops across their membrane, and which had voltage gated ligand channels. The giant axon of the squid is used when introducing this topic, and is the cell in which the values in table 11.1 come from. Because a nerve cell was used as the example, I wasn’t sure if the topics discussed are specific to a nerve cell or to cells in general. So I took to the internet.

First of all, all cells have sodium-potassium pumps. This makes sense, as all cells need to osmoregulate. According to Wikipedia, the sodium-potassium pump in a normal cell expends a third of the cell’s energy.

All cells do have a membrane potential, around that of the resting potential of a neuron. It is used to power some of the molecular devices in the membrane. However, not all cells can use it for signalling like neurons.

Voltage-gated ion channels are necessary in neurons, but again, according to Wikipedia, can be found in many kinds of cell. I had trouble finding examples other than muscle and neuronal cells, but I found one example. The Transient Receptor Potential Channels are voltage triggered. An example of a protein in this family is the capsaicin receptor, which responds to the chemicals which make chilli taste hot. Voltage gated ion channel are not found in every cell.

So everything that I found out here is pretty much what I expected. But I am glad I found out for sure, and I hope this helps confirm things for you too.

P.S. I can’t remember if I was taught this last year in BIPH2000, but in case I wasn’t, I thought I’d share: The research done on squid giant axons was not performed on the axons of a giant squid, but the giant axon of a regular sized squid. You probably already know this, but I didn’t realise this until it was pointed out to me.

Your Turn 11B

Consider a fictitious membrane permeable to Cl- but not to K+:

Assuming only the negative ions can permeate the membrane, and then the negatively charged chloride ions will leak. The concentration of ions inside the membrane is higher than the outside, c2 > c1, therefore the ions will move outwards. This will increase the negative charge just outside the membrane and decrease it just inside. This is the reverse of the situation shown in Figure 11.2 a.
The electrostatic potential across the membrane will also be opposite. Visualise a positively charged test object. As the object moves from the outside of the membrane in its potential energy will increase; as it moves from a complementary-charged negative region to a repulsive positively-charged region. Thus the potential curve decreases moving from the inside to the outside of the membrane. This again is the reverse of the situation shown in the text, Figure 11.2 b.

Some extra info on glycolysis...

Here’s a simplified Biologists look at glycolysis, the citric acid cycle and oxidative phosphorylation. Three processes involved in ATP production.

Watch these:
http://www.youtube.com/watch?v=x-stLxqPt6E
http://www.youtube.com/watch?v=aCypoN3X7KQ&feature=related
http://www.youtube.com/watch?v=Idy2XAlZIVA&feature=related
http://www.youtube.com/watch?v=H0RchC0FbYU&feature=related

Glycolysis involves two stages. All the reactions are catalysed by specific enzymes. First glucose is prepared for catabolism by adding phosphates to its structure. This actually uses some ATP energy. The glucose is then processed and broken down to produce pyruvate.
The pyruvate is passed into the citric acid cycle (also named the Krebs cycle) it undergoes a number of changes involving a different enzymes and metabolites.
Two of these reactions are described by Nelson. Equations: 11.14; 11.15.
These reactions produce reduced NADH and FADH2 molecules. NADH is then used to generate a hydrogen ion gradient in the mitochondrion. The chemiosmotic gradient is generated using four protein complexes. Complex 1 is a NADH dehydrogenase; it catalyses NADH -> NAD+ and pushes 4 protons across the membrane. The electrons captured in this reaction are passed through to the other complexes in succession. Complex 2 is a succinate dehydrogenase; it doesn’t directly transfer hydrogen ions but instead donates further electrons by converting succinate to fumarate. Complex 3 couples the transfer of the electrons (from 1 and 2) with the transport of 4 protons. Complex 4 then reacts the electrons with oxygen to create water and transports a further 2 protons across the membrane. These successive reactions work to fully capture the energy generated from glucose. Without these incremental energy-capturing steps most of this energy would be lost to heat.

Landscaping

From figure 10.6, figures (a) and (b) both follow one-dimensional landscapes.

For (a) there is a negative slope (lower dashed line) related to the Umotor = -t(theta), the coiled spring contribution, while the positive slope (upper dashed line) relates to the external load contribution. This results in an overall downward slopes. the total potential energy (equal to (w1R-t)(theta)) decreases as time increases (theta is constant).

For (b) the graph is rather similar to (a) except is related to an imperfect shaft. What can be noticed is that due to the set up of (b) the contributation by the spring follows a sine-wave like curve, which is then reflected in the total potential energy line.

From figure 10.6, figure (c) contains a 2D landscape graph, since it depends on both the angles alpha and beta. Figure 10.8 displays the landscape graph for an unloaded machine, containing both valleys and barriers. Travelling along a valley is ideal but impossible to do indefinitely, so movements over the barriers are necessary. For a loaded machine, figure 10.9 shows the landscape graph, explaining that it is simply a tilted version of figure 10.8

Week 9 Session Notes - sorry for late upload

Point 9.1

Using a set number of degrees of freedom – we don’t overanalyse, we save ourselves a huge amount of mathematics etc

Elasticity comes from entropy. Entropy determines the elasticity of an object

Rubber band force pulling it back becomes less at high temperature

Elastic energy cost increases with temperature

Shape of curve determines elastic energy (equation 9.4 assumes circular curve)

Approximate a curve as a bunch of little circular segments

Figure 9.4

C) still a coil

d) unwinds and becomes straight

slope of A = entropic elasticity

figure 9.5

gives 4 different models to describe 9.4’s regime A and part of B

talked about reasons why we’d stretch out dna and other things

1D chain model – chain containing segments that pointed either left or right. Only two parameters occurring are temperature (which is fixed), and the persistence length (length of segment)

3D chain – segments can point in any direction. Adds two degrees of freedom (the two extra dimensions)

Elastic rod model – 3dfjc model plus an energy cost for every bend. Each segment cares about what’s going on with its neighbours

Figure 9.7

Find it weird that the structure become more ordered at higher temperatures, as opposed to denaturing. Does show however that at low temperatures it is denatured, and for all we know, it may denature again at a higher temperature.

Cold denaturing – loss of electrostatics, lowering of disulfide bonds

Long protein can be approximated as an infinitely long coil

Short protein – ends can’t be stabilised as easily, due to fraction of residues that can’t become part of the helix

Figure 9.8

Rotation determines fraction of helices

Double conc = double rotation

One helical rotation turns light to the left, the handedness of rotation turns light to the right.

CD

Figure 9.9

Effects of changing degree of cooperativity

In long chains, hard to tell if cooperativity matter

In short chains though, makes a hell of a difference

Cooperativity allows for sharp transitions

Figure 9.10

HbO2 -><- Hb + O2

Keq = [Hb][O2]/[HbO2]

Y = [HbO2]/([Hb] + [HbO2] = fraction bound

Bad to have myoglobin.....20% oxygenation difference in tissue

Haemoglobin has 4 binding sites....effective value of 3 due to its binding nature

Small fraction have 1, 2 or 3 bound

Response to Mitch's Post on Enzyme Inhibition

So when inhibitors are involved with our simple enzyme reaction system, there are two main ways in which this inhibitor can incorporate itself.

The first way in which it can be involved is to bind to the enzyme while not bound to the substrate. By forming the EI complex, the inhibitor can prevent the substrate from binding and forming the ES complex, thereby prohibiting the formation of a product.

The second way in which an inhibitor can be involved is by binding to the ES complex to form the ESI complex. Again, by binding to this, the production of the product in prevented. In this case, it is because the binding of the inhibitor usually distorts the shape of the enzyme thereby making it lose its fitting bind to the substrate disabling the catalytic effects that come from a fitted binding of the ES.

I have heard that there can be some situations where there is the step EI + S <-> ESI, but I cannot name any examples, and I personally find it hard to imagine that happening, as surely the EI complex would straight away prevent it from binding the substrate due to either being bound in the same pocket that the substrate would be binding it, or it would distort the conformation of the enzyme, and as such the binding pocket of the substrate.

Michaelis-Menten and simple kinetics

Let's start with the general procession of an enzymatic reaction (which is unaffected by inhibitors). We have:

E + S <-> ES<-> EP <-> E + P

In the case where the is none, or next to no product present initially, the chemical potential is a large negative number, resulting in a steep downhill slope for the third step, allowing us to write EP -> E + P

We now have:

E + S <-> ES <-> EP -> E + P

We shall also assume that E+S, ES, and E+P are separated by large barriers, allowing each transition to be treated independently. The transition involving binding of substrates from solution is characterised by a first order rate, proportional to the substrate concentration.

Throwing a single enzyme into a vat of substrate at initial concentration cs, and negligible product, will spend a fraction of time Pe unoccupied, and the rest (1-Pe) bound to substrate. These two times can be said to be nearly constant in time, so the enzyme converts the substrate at a constant rate.

We are now left with the final reaction of:

E + S <-> ES -> E + P

Forward rate of the first step = csk1

Reverse rate of the first step = k-1

Forward rate of the second step = k2

Michaelis constant = (k-1 + k2)/k1

vmax = k2Ce

Michaelis-Menten rule

v = vmax(cs / (Km + cs))

Ratchet?

I have to say Nelsons use of analogies in this chapter seems overly complicated, such as his introduction of the G and S-ratchets.



So in order to understand his analogy I propose to consider a different analogy. Instead of a rod moving through a membrane driven by thermal fluctuations; take a simple chemical reaction.

A + B <-> C + D

The reaction can move either left or right. There is only a small difference in energies which can easily be supplied by thermal fluctuations.
The equilibrium can become biased when the activation energy for the products becomes large. In this way the amount of product will increase and reactant decrease (like the sliding of the rod).
By simply removing amounts of the product (Le Chatelier’s Principle) the reaction will keep forming products and use up the available reactants (like applying a force on the rod).

Molecular Machine Example

One amazing molecular machine is ATP synthase.



It operates in a rotational catalysis mechanism. The β subunits perform the catalysis of ADP + Pi -> ATP. The γ subunit is responsible for rotating the β subunits and is driven by the influx of Hydrogen ions.

As the β subunits are rotated their conformation changes depending on their contacts with the γ subunit. Firstly ADP and Pi are bound loosely within one β subunit. The rotational conformation change then produces ATP by binding the reactants tightly and stabilizing the ATP form. In the next rotation, the ATP molecule is only loosely bound by the β subunit and the molecule can dissociate. There are many videos of this process available on Youtube.

This synthase is a great example of both a cyclic machine (10.1.1) and an enzyme catalyses a reaction by binding to the transition state (10.3.3). Can you think of any other examples?

(Information taken from textbook: Lehninger- Principles of Biochemistry
Image from article: Molecular Motors: Turning the ATP motor, Richard L. Cross, Nature 427, 407-408(29 January 2004), doi:10.1038/427407b)

Enzyme Inhibition

The Lineweaver-Burk equation

 

As my first post this week was quite long, I’ll make my second a short one and save you some reading time. I’m pretty sure we’ve all done CHEM2002, and have learnt about enzyme competition before. This chapter mentions competitive inhibition and noncompetitive inhibition. To refresh our memories, competitive inhibition is where the inhibitor molecule binds to the same site as the enzyme’s substrate. Thus they are in direct competition with each other at the binding site.

Noncompetitive inhibition is where the enzyme has two binding sites, which are effectively mutually exclusive. When the inhibitor binds to its binding site, the enzyme cannot bind the substrate any more, and thus the reaction again is inhibited. We learnt a third kind of inhibition, uncompetitive inhibition. This is where the inhibitor binds to the enzyme substrate complex.

These three kinds of inhibition have different effects on the Lineweaver-Burk plot as the inhibitor concentration is increased. If the inhibition is competitive, the y-intercept of the Lineweaver-Burk plot is constant, but the gradient changes. If the inhibition is noncompetitive, the x-intercept of the plot is constant, but the slope changes. If the inhibition is uncompetitive, the gradient is constant, but the y-intercepts changes (it increases). This means that competitive inhibition does not affect the vmax of the reaction and noncompetitive inhibition does not affect Km. Uncompetitive inhibition lowers both the Km and vmax.

Kinesin Motion

I thought a key topic in this chapter was the motion of kinesin. I was confused at first about how this molecule works, but I think I understand now. Please correct me if I am wrong anywhere. Each head of the kinesin molecule has two sites, one which binds one ADP molecule and one which binds to the microtubule. Both these sites bind strongly to their respective substrates. However, the head cannot bind strongly to both substrates at once. This is an example of non-competitive inhibition.

Because these sites bind strongly, and there is an abundance of ADP and microtubule binding sites in the cell compared to kinesin molecules, at least one of these sites is bound at any time. So if the molecule is not bound to the microtubule, then both these heads of the dimer have ADP bound. When the molecule is near a microtubule, eventually one of the heads will lose its ADP and bind to the microtubule. I suppose that a backward step could be taken at this point, and the head bound to the microtubule releases the microtubule and rebinds ADP. However, as the concentration of ATP in the cell is much higher than the concentration of APT, an ATP molecule is more likely to bind to the vacant site on the kinesin head bound to the microtubule.

Unlike ADP, kinesin is able to bind ATP and the microtubule strongly. When the ATP molecule binds, the head does not let go of the microtubule. The neck linker of the bound head then attaches to the head. This state is a local energy minimum. The other head is free to diffuse around; however, the position of the neck linker biases this diffusion. This is the asymmetry which is necessary for directed molecular motion.

Eventually the other head diffuses close enough to the next microtubule binding site for it to bind. However, this head still has ADP bound, so it can only bind weakly to the next site. It is likely that it binds and unbinds several times. However, eventually the head will spontaneously release the ADP, and will be free to bind the microtubule.

The protein chain connecting the two heads is only just long enough to reach the next microtubule binding site, so when both heads are bound, the molecule is strained due to stretching. Since the protein chain is only just long enough to reach the next microtubule binding site, even with the bias in the neck linker, it is nearly impossible for the unbound head to accidentally diffuse close enough to the previous site ad take a step backwards, because the neck linker will always point in the forward direction.

Perhaps this added strain on the molecule deforms the ATP binding site on the initially bound head, which causes it to hydrolyse the ATP bound to it. The head only weakly binds the resulting ADP and the microtubule, so the molecule will likely release one of these substrates. Due to the extra strain the molecule is under in its stretched state, the head preferentially releases the microtubule, to cancel this stress.

The energy released by ATP hydrolysis detaches the neck linker from the head, and the molecule is left with one head bound to the microtubule and one head bound to an ADP molecule. This is the state it began in, thus the cycle is available to repeat, providing the substrate concentrations remain the same. I presume this process will repeat until an ADP binds to the head bound to the microtubule instead of an ATP molecule, and the kinesin molecule detaches from the microtubule altogether.