calcium diffusion in different shells

Extending NEURON to handle reaction-diffusion problems.

Moderators: hines, wwlytton, ramcdougal

Post Reply
ylzang
Posts: 27
Joined: Sun Mar 08, 2015 3:38 am

calcium diffusion in different shells

Post by ylzang » Wed Dec 30, 2015 9:22 pm

In the Neuron Book, there is a cdp model used as Example 9.8 in Chapter 9. I have a question about this model. In this model, the calcium concentration in the outermost shell is regarded as the model output Cai. I think this value is much higher than the average calcium concentration. In some recent publications, they also used this value as the cai signal. However, do you think calcium in the outermost shell can represent the intracellular calcium concentration? Thanks.

ramcdougal
Posts: 205
Joined: Fri Nov 28, 2008 3:38 pm
Location: Yale School of Public Health

Re: calcium diffusion in different shells

Post by ramcdougal » Wed Dec 30, 2015 10:40 pm

You are right that cai as defined in those models is not the average calcium concentration. That is by design.

cai and cao are the two calcium concentrations made available to ion channel, pump, etc mechanisms defined in NMODL via USEION. As such, they are the concentrations that the mechanism can "detect." Note this is not the average intracellular and extracellular concentration, but rather the local concentration in a region just inside the membrane and just outside the membrane near the mechanism.

If you really want to know the average intracellular calcium concentration, you can compute that as a weighted average of the shell concentrations, but beware: deciding to use shells is deciding that the average alone does not explain the dynamics.

ted
Site Admin
Posts: 5724
Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
Contact:

Re: calcium diffusion in different shells

Post by ted » Thu Dec 31, 2015 10:29 am

ylzang wrote:In the Neuron Book, there is a cdp model used as Example 9.8 in Chapter 9. I have a question about this model. In this model, the calcium concentration in the outermost shell is regarded as the model output Cai. I think this value is much higher than the average calcium concentration. In some recent publications, they also used this value as the cai signal. However, do you think calcium in the outermost shell can represent the intracellular calcium concentration?
ramcdougal is quite correct in his answer to your question, but some readers may benefit from a more explicit discussion.

The starting point for all discussions of the "correctness" or "appropriateness" of any computational model must be the underlying conceptual model, that is, the hypothesis that serves as the basis for the computational model. If your hypothesis includes the effects of transmembrane calcium flux on intracellular calcium concentration, and the consequences of those effects, you will be most interested in the intracellular calcium concentration at particular locations in the cell. At the very least, you need to know the concentration adjacent to the inner surface of the membrane. Why? This is the concentration that determines the transmembrane electrochemical gradient for calcium--it's the concentration that you need to use to calculate eca, or the flux through channels that are described by the Goldman-Hodgkin-Katz formalism--and it's also the concentration that is "seen" by calcium-gated potassium channels.

If your hypothesis includes the effects of calcium concentration on BK channels, you may need to take into account the fact that BK channels tend to cluster around voltage-gated calcium channels. This special anatomical arrangement means that a one-dimensional (radial) or two-dimensional (radial and longitudinal) diffusional geometry based on concentric cylindrical shells may not be sufficient--instead, for some purposes it may be necessary include three-dimensional diffusion in the near neighborhood of calcium channels.

ylzang
Posts: 27
Joined: Sun Mar 08, 2015 3:38 am

Re: calcium diffusion in different shells

Post by ylzang » Thu Dec 31, 2015 8:55 pm

Thanks for ramcdougal and Ted's explanation. I hope to compare the model computed calcium transients including the amplitude, rise and decay dynamics with experimental observations. I think experimentally measured calcium-bounded dye signal should represent the average concentration. Is this correct? Then in the model it should also be corresponding to the average concentration of calcium bounded dye concentration instead of that in the outermost shell. Is my understanding correct? Specifically speaking, in experiments, the peak of the calcium transient after dendritic spikes is about several uM after calibration. What does this signal really correspond to the model components? I guess it should not be the calcium in the outermost shell. Thanks.

ted
Site Admin
Posts: 5724
Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
Contact:

Re: calcium diffusion in different shells

Post by ted » Fri Jan 01, 2016 3:06 pm

ylzang wrote:I hope to compare the model computed calcium transients including the amplitude, rise and decay dynamics with experimental observations. I think experimentally measured calcium-bounded dye signal should represent the average concentration.
Sure, but the average of what?
1. Is all of the dye in the cytoplasm, or is some of it in the cell's membrane?
2. Your calcium accumulation mechanism needs to take buffering into account--not only the cell's endogenous buffer(s), but also the effect of the dye. What fraction of intracellular ca does the dye bind, and how fast is the binding reaction?
3. Given the answers to 1 and 2, and the diffusional geometry, only algebra is needed to calculate the average optical signal from the shape and location of the imaged region of interest.
in experiments, the peak of the calcium transient after dendritic spikes is about several uM after calibration. What does this signal really correspond to the model components?
I don't understand the meaning of this statement
"the peak of the calcium transient after dendritic spikes is about several uM after calibration"
because I don't know what you mean by "calibration" and I have always seen uM used as a unit of concentration (micromolar).

Am I right to guess that "calibration" is a process in which you evoke a spike and observe the maximum dye signal that it elicits? And did you mean to say that the peak dye signal occurs several milliseconds after the action potential? That's an old observation, dates back to the early 1980s if not earlier, when people were first using aequorin in experiments on neurons in Aplysia and other invertebrates. The delay reflects the fact that most calcium influx occurs well after the spike peak. The same is true for sodium influx elicited by an action potential. The explanation is that driving force for Na and Ca entry is greatest after the spike, not during the spike, and opened channels take many ms to close.

ylzang
Posts: 27
Joined: Sun Mar 08, 2015 3:38 am

Re: calcium diffusion in different shells

Post by ylzang » Mon Jan 04, 2016 2:35 am

I hope to compare the model computed calcium transients including the amplitude, rise and decay dynamics with experimental observations. I think experimentally measured calcium-bounded dye signal should represent the average concentration.Sure, but the average of what?
1. Is all of the dye in the cytoplasm, or is some of it in the cell's membrane?
2. Your calcium accumulation mechanism needs to take buffering into account--not only the cell's endogenous buffer(s), but also the effect of the dye. What fraction of intracellular ca does the dye bind, and how fast is the binding reaction?
3. Given the answers to 1 and 2, and the diffusional geometry, only algebra is needed to calculate the average optical signal from the shape and location of the imaged region of interest.
First thanks for the reply. I am sorry that I did not make it clear. In experiments, calcium imaging data are expressed as normalised relative fluorescence increase and then converted to free intracellular calcium concentration based on a cuvette calibration. Here the fluorescence increase should correspond to the average calcium-bounded dye concentration instead of the calcium-bounded dye concentration in the outermost shell. Is this correct? Therefore The calibrated calcium concentration should also be the average free calcium concentration on the condition without dye.
To simulate the naive or in vivo situation, dye is not included in the model. Then to compare the calcium signal generated in the model with the experimentally calibrated free calcium concentration by above mentioned methods, the average calcium concentration should be the model output? Is this correct? I know in fact, calcium in the outermost shell may be more functionally important. But here I am not sure whether calcium in the outermost shell or the average calcium concentration should be used to compare the fluorescence increase.

in experiments, the peak of the calcium transient after dendritic spikes is about several uM after calibration. What does this signal really correspond to the model components?
I don't understand the meaning of this statement
"the peak of the calcium transient after dendritic spikes is about several uM after calibration"
because I don't know what you mean by "calibration" and I have always seen uM used as a unit of concentration (micromolar).

Am I right to guess that "calibration" is a process in which you evoke a spike and observe the maximum dye signal that it elicits? And did you mean to say that the peak dye signal occurs several milliseconds after the action potential? That's an old observation, dates back to the early 1980s if not earlier, when people were first using aequorin in experiments on neurons in Aplysia and other invertebrates. The delay reflects the fact that most calcium influx occurs well after the spike peak. The same is true for sodium influx elicited by an action potential. The explanation is that driving force for Na and Ca entry is greatest after the spike, not during the spike, and opened channels take many ms to close.
[/quote]

Here again, I did not make it clear. What I tried to say is in my model, if I used the calcium concentration in the outermost shell as the output ( no dye in the model), then the peak of calcium concentration is on the order of tens of uM (micro molar). However, in experiments, the calibrated (as mentioned in the beginning) value is on the order of 1 or 2 uM (micro molar). There are two possibilities, either too many calcium channels distributed on the membrane or the calcium concentration in the outermost shell should not be compared with the experimentally calibrated value. Instead the average calcium concentration in the dendrite is the right object.
Thanks and Hope this time I make it clear.

ted
Site Admin
Posts: 5724
Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
Contact:

Re: calcium diffusion in different shells

Post by ted » Tue Jan 05, 2016 5:44 pm

ylzang wrote:In experiments, calcium imaging data are expressed as normalised relative fluorescence increase and then converted to free intracellular calcium concentration based on a cuvette calibration. Here the fluorescence increase should correspond to the average calcium-bounded dye concentration instead of the calcium-bounded dye concentration in the outermost shell. Is this correct?
The methods I'm familiar with for calibration for the purpose of measuring intracellular calcium are similar to those described in
Neher, E. Quantitative aspects of calcium fluorimetry. Cold Spring Harb Protoc 2013, 918-924
and
Helmchen, F. Calibration of fluorescent calcium indicators. Cold Spring Harb Protoc 2011, 923-930
The calibrated calcium concentration should also be the average free calcium concentration on the condition without dye.
I don't know anything about the dye you're interested in, or whether your experimental methods are such that the dye will or will not affect either the apparent amplitude or the time course of the calcium transient. If you're the experimentalist who generated the data, you'll know whether you're dealing with a high- or low-affinity dye, and whether it will affect free calcium amplitude or time course. If not, I assume you are collaborating with an experimentalist who can advise you about these issues.
To simulate the naive or in vivo situation, dye is not included in the model. Then to compare the calcium signal generated in the model with the experimentally calibrated free calcium concentration by above mentioned methods, the average calcium concentration should be the model output? Is this correct?
Yes, if
1. you know that the dye does not affect the amplitude or time course of the experimentally evokde calcium transient
2. you know what is the region of interest from which the fluorescence signal was measured
3. you know the distribution of dye in whatever membrane or cytoplasm is contained in the region of interest.

The calculation will be easiest if
1. the region of interest spans the entire diameter of the neurite from which measurements are made
and
2. the dye is present only in the cytoplasm, and is distributed uniformly throughout the cytoplasm
Then you can calculate the weighted sum of calcium concentrations over all radial shells.

In theory the weights would be the relative volumes of the shells. For any compartment with normalized inner and outer radii ri and ro, the relative volume is
ro^2 - ri^2

In the models of calcium accumulation with radial diffusion that are described in the NEURON book, the concentration in each radial compartment is a STATE variable, which means that it is automatically a RANGE variable that is visible to hoc. You will probably want to identify the segment in your model that corresponds to the location at which experimental measurements were made, and use Vector.record to capture the time course of calcium in each of the radial compartments. For example, if you're interested in the middle segment of dendx, these hoc statements will set up Vector recording for you.

Code: Select all

// $1 is number of radial shells
// $2 is location of segment
// call with section of interest as the currently accessed section
obfunc mkcaveclist() { local i  localobj tmpobj, tmplist
  tmplist = new List()
  for i=0,$1-1 {
    tmpobj = new Vector()
    tmpobj.record(&ca[i]_cadifus($2))
    tmplist.append(tmpobj)
  }
  return tmplist
}

objref cavecs, tvec
tvec.record(&t)
tvec = new Vector()
// numshells = number of cylindrical shells in cadifus
dendx cavecs = mkcaveclist(numshells, 0.5)
Then after a simulation, you can calculate the time course of the weighted sum of the concentrations

Code: Select all

// $o1 is a List that holds the Vectors to which the shell ca concentrations were recorded
// $o2 is a Vector that holds the weights
obfunc wtsum() { local i  localobj tmpsum, tobj
  tmpsum = new Vector($o1.o(0).size, 0) // same size as recorded vecs, filled with 0
  for i=0,$o2.size()-1 {
    tobj=$o1.o.(i).c // copy of ith Vector in $o1
    tobj.mul($o2.x[i]) // scaled by ith element in $o2
    tmpsum.add.tobj
  }
  return tmpsum
}

// assumes weights is a vector with numshells elements
// whose values are the relative volumes of each compartment
// or whatever other weighting scheme turns out to work best for you
objref meanca // with contain the weighted sum of the shell concentrations
meanca = new Vector()
meanca = wtsum(cavecs, weights)

ylzang
Posts: 27
Joined: Sun Mar 08, 2015 3:38 am

Re: calcium diffusion in different shells

Post by ylzang » Wed Jan 06, 2016 1:45 am

Thanks for the reply. Yes, I am collaborating with other experimentalist and I will contact him to get more details. Thanks again.

ted
Site Admin
Posts: 5724
Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
Contact:

Re: calcium diffusion in different shells

Post by ted » Wed Jan 06, 2016 11:51 am

Two more comments.

First, free ca concentration is a proxy for concentration of the ca*dye complex only if the reaction
ca + dye <-> ca*dye
is "instantaneous" (well, at least fast enough that ca*dye tracks free ca concentration closely).

Second, the relationship between the concentrations of ca and ca*dye will be linear only if the reaction does not significantly affect either the free ca concentration or the free dye concentration. In other words, the dye must have low affinity for ca. If this condition is not met, the relationship will be nonlinear. Nonlinearity means that the volume-weighted average of calcium concentration in a segment will be meaningless.

Instead you have to devise a function f that calculates the final concentration of ca*dye complex from the initial values of free ca and free dye in a system that initially has no ca*dye complex--that is, assume the gedankenexperiment
"At t = 0, the concentrations of calcium, dye, and ca*dye are all 0.
An instant later, the concentrations of calcium and dye change suddenly to cai0 and dye0.
When the binding reaction reaches steady state, you measure the concentration of the ca*dye complex."
This function will be easiest to use if it has only one argument: the initial free calcium concentration; for the sake of convenience, the initial free dye concentration can be a hoc global variable.

Then you apply f to a copy of each vector recording of calcium concentration to get the time course of the ca*dye complex in each cylindrical compartment.

Code: Select all

// assumes that f is a function that,
// given initial concentrations of ca and dye,
// returns the final concentration of the ca*dye complex
// $o1 is a List that contains the recordings of free ca
// $2 is the desired value of initial dye concentration

dye0 = 0 // create a global variable that will be used by f

obfunc xformcai() { local i  localobj tmpvec, tmplist
  dye0 = $2
  tmpvec = new Vector()
  tmplist = new List()
  for i = 0,$o1.count()-1 {
    tmpvec = $o1.o(i).c
    tmpvec.apply("f")
    tmplist.append(tmpvec)
  }
  return tmplist
}

objref cadyevecs
cadyevecs = new List()
cadyevecs = xformcai(cavecs, 0.15) // for the sake of this example,
  // I assume that initial dye concentration has a numerical value of 0.15
Finally, you use obfunc wtsum() to calculate the volume weighted averages of the ca*dye complex.

Code: Select all

// assumes weights is a vector with numshells elements
// whose values are the relative volumes of each compartment
// or whatever other weighting scheme turns out to work best for you
objref meancadye // will contain the weighted sum of the cadye concentrations
meancadye = new Vector()
meancadye = wtsum(cavecs, weights)

Post Reply