How to Solve It: Kinesin Turnover Rate

Problem:
If an individual kinesin molecule can move at a rate of 800 nm/sec in an in vitro motility assay, what is the maximal turnover rate (ATP molecules hydrolyzed per second) by one of the molecule’s motor domains?

I strongly recommend whenever you try to solve any sort of quantitative problem, you should start by trying to draw a diagram of what the problem is describing.

In this case, we have a microtubule (draw) and a kinesin walking along it - remember the kinesin has two motor heads that take turns stepping (draw a kinesin with two heads).     You'll note that I've drawn the two heads in two different colors, so we can keep them straight.

Each head is an ATPase, and every time one head hydrolyses an ATP, the head will move forward in the hand-over-hand motion we learned about in this chapter.  Let's try to draw this.   It's a chemical reaction, so starting with the initial state shown here (point to diagram), we draw an arrow (draw arrow) to the final state (draw the MT with the formerly trailing head now the leading head).  

For a step like this, one of the heads has hydrolyzed one ATP, so we're going to draw an ATP coming in and getting converted to ADP.   (add arrows bringing in ATP and bringing away ADP).

OK so now the challenge is to relate the chemical rate of this equation to the mechanical speed that it produces.   We know that every time the motor steps, it moves ahead by 8 nm, the length of one alpha beta tubulin dimer.   So, we can say that the motor moves by 8 nm per ATP hydrolysed, since as this diagram shows, ATP hydrolysis is directly coupled to stepping.   You can't do one without doing the other.  

Let's try to write an equation to describe what's going on.  We need to make up some variable names.   You can call your variables whatever you want but you can choose simple ones that will help you to keep track of what's going on.   For the motor movement, we can use "v" to describe its velocity.   The units of v are  nanometers per second.  ( write  v, and then in parentheses, nm/s).     For the rate of ATP hydrolysis, let's use r, its units are molecules per second  (write r, then in parentheses, molecules/s).   

The problem is asking us to solve for r, the ATPase rate, given v, the speed of the motor.   Unfortunately, these two variables use two different units so we can't directly compare them.   Somehow, we need to relate these two units, nm/s and molecules/s.   How do we do that?   The trick is that we know the step size of kinesin is 8 nm per ATP molecule hydrolysed.   So, we can convert molecules of atp hydrolysed per second into distance moved per second, like this:   (write   8 nm/molecule x   molecules/s).   So now we can get an equation that relates r and v.  {write    v (nm/s)  = 8 nm/molecule * r (molecules/s).  } 

Now we can plug in our numbers.   We have v = 800 nm/s, so we get {write it out:} 800 nm/s = 8 nm/molecule x r molecule/s.      

Now we can solve for r  {write out   r =   800 nm/s /  8 nm/molecule } which is {r = 100 molecules/s}

So it looks like we have our answer right?   Wait, hold on a second, this is kind of a trick question actually.   The ATPase rate we've calculated here is for the whole kinesin molecule, which has two heads.  Those two heads take turns hydrolyzing ATP, so actually if the total rate of ATP hydrolysis is 100 molecules per second, each individual head is only hydrolysing half as many ATPs, so the rate per head is going to be 50 molecules per second.    This is why it is super important to always draw out your diagram and keep looking at it while you solve problems.