Timescales for calcium indicators in cells

What determines timescales of calcium indicator signals in cells? Is it intrinsic to the kinetics of the sensor molecule (e.g. the of GCaMP)? Or does it depend on other factors like cell size and expression level?

(Note that I prepared this for my own understanding and in the hopes that it might be informative to others. The literature on this topic is vast and I apologize in advance if I've failed to cite relevant literature. Despite this literature on the biophysical influences of calcium signals, I think that with the explosion of use of GCaMPs, there's an occasional tendency to interpret calcium signals as simply 'neural activity.' This post is in some ways a reminder to me to try not to do this!).

Calcium indicators provide insight into the workings of neurons, by measuring changes in calcium concentration in the cell. Calcium influx is influenced by membrane voltage, and calcium indicator signals typically rise when neurons fire action potentials, providing an easy way to measure when neurons fire.

However, the mere presence of a calcium indicator alters normal calcium dynamics, and thus calcium indicators provide a warped view of normal calcium in/outflux (discussed in many papers, for example Helmchen et al 1996, Neher 1995, Neher 1998). Here, I want to show a simple model that underscores how the size of the cell and the concentration of the calcium indicator affects the measured indicator signal. Understanding the relationship between calcium indicator signals and underlying calcium dynamics is important for interpreting our measurements! Quantitatively, this is important for deconvolution of calcium indicator signals, and the ability to identify multiple closely spaced action potentials.

The thing to remember is that that calcium indicator dynamics are not solely dependent on the binding rates ( and ) of the indicator. As cells get larger, their fluorescent decay times get longer. And as expression of GCaMP increases, it prolongs calcium decays. Calcium transients in somas should be slower than in processes (dendrites, axons), and that calcium transients should be slower in higher-expressing cells (e.g. many months post-transfection) than in higher-expressing cells (shortly following transfection).

Why aren't fluorescent rise/decay rates just and ?

One might start by imagining that calcium indicator dynamics (rise times and decay times) are dictated by the binding rates of the indicator. When a neuron fires an action potential or dendritic spike, calcium channels open and calcium rushes into the cell. The indicator then binds the calcium and starts to fluoresce, informing us of calcium influx into the cell. After the action potential, calcium is rapidly pumped out of the cell. However, first, the indicator has to release it. If this release step is rate-limiting, then the fluorescence decay rate will be equal to the unbinding rate, ).

However, in a real neuron, when calcium is released from the indicator, it has a nonzero chance of rebinding another indicator molecule before it is reaches the cell membrane and is pumped out of the cell! The calcium indicator keeps the calcium in the cell longer than it would be without the calcium indicator. And here is where you can start to imagine that cell geometry and indicator expression level might affect the timescale at which cellular gcamp signals rise/decay.


A simple model shows that cell size and indicator concentration strongly affect fluorescence decay times

I'm going to make a simple model of a cell. It has the indicator GCaMP in it, some calcium, and it's divided into a number of discrete 'shells' (Fig 1). Calcium, GCaMP, and calcium bound to GCaMP can all diffuse from one shell to the next, except that they initially can't diffuse out of the cell. They can however diffuse, and bind/unbind. In my model, I'll then suddenly make the outer wall perfecly absorbing of free calcium (any free calcium that hits the outer wall is pumped straight out). Finally, I'll sum up the observed fluorescence of the cell so that we can see what we'd expect to see experimentally from the indicator.

How does calcium outflow depend on cell size?

First, how much does cell size affect the rates at which calcium exits, in the absence of a calcium indicator? It must take longer for calcium to diffuse out of cells, simply because it has farther to go to reach the membrane. This sets a fundamental limit on how fast the response would be for a perfect calcium indicator.

But this appears to be an order of magnitude or more faster than any sensor currently in use, decaying with a time constant of 17 milliseconds in a cell with a 10 m radius. As a footnote, I've fit these decays to exponential functions to estimate 'effective' decay rates. However, it's worth note that they are not simply exponential functions, and therefore the rates I'm listing are not exact (though I've checked that they are reasonable approximations of the decay).

If we express GCaMP in these cells, we don't observe the same dynamics. I'll now simulate the cells with a GCaMP calcium indicator expressed in them. I don't know the typical intracellular concentration of GCaMP in just about any context (if you find it, email me and tell me!), but it's probably in the range of 1 to 100 M. Below, I've simulated both GCaMP6f (Chen et al. 2013) and GCaMP6fu, an ultrafast variant (Helassa et al. 2016) across this range of concentrations. Note that I'm plotting what one would actually measure - the amount of GCaMP bound to calcium (which is proportional to fluorescence), relative to the beginning of the decay.

To simplify this data, I fit each curve to an exponential decay (note that the curves are not actually exponential decays, but are reasonably approximated as such). This yields a single decay rate for each condition.



From this simple model, it was easy to see the following results:

  1. The dynamics of calcium indicator signals depend not only on the molecular kinetics of the indicator ( and ), but also on the concentration of the indicator and the size of the cell.
  2. Dynamics get slower as indicator concentration increases.
  3. Dynamics get slower as cells get bigger.
  4. An indicator with a 25x-faster molecular off rate (e.g. of GCaMP6fu vs GCaMP6f) will only have 25x-faster decays for very small compartments. For a soma, the improvement may be less than 2x!
  5. Thus, the benefits of indicators with faster off-rates will come predominantly in imaging small compartments like dendrites.


In my simulation, the parameters are as follows:

  • for GCaMP6f, = 10/s and = 3.5e7/M/s  (Helassa et al. 2016, Badura et al 2014, Chen et al 2013).
  • for GCaMP6fu, = 250/s and = 7.4e8/M/s (Helassa et al. 2016)
  • The diffusion coefficient of calcium is 5e-10 (Donahue and Abercrombie 1987).
  • The diffusion coefficient of GCaMP (both variants) is 2.5e-11 square meters per second [based on GFP estimate from (Gura-Sadovsky et al. 2017).
  • Shells are 0.1 m thick.
  • Note that I ignored cooperative binding (GCaMPs show Hill coefficients of 2-3 [Badura et al 2014, Helassa et al 2016]).

Transport from one shell to the next is given by Fick's law of diffusion - the net flow of calcium from shell to shell is , where is the surface area of the boundary between shell and shell , and is the diffusion coefficient of free calcium. Diffusive transport of free and bound GCaMP follows a similar equation.

The full equations for each shell are thus as follows.


Badura, Aleksandra, Xiaonan Richard Sun, Andrea Giovannucci, Laura A. Lynch, and Samuel S.-H. Wang. 2014. “Fast Calcium Sensor Proteins for Monitoring Neural Activity.” Neurophotonics 1 (2): 025008. doi:10.1117/1.NPh.1.2.025008.

Chen, Tsai-Wen, Trevor J. Wardill, Yi Sun, Stefan R. Pulver, Sabine L. Renninger, Amy Baohan, Eric R. Schreiter, et al. 2013. “Ultrasensitive Fluorescent Proteins for Imaging Neuronal Activity.” Nature 499 (7458): 295–300. doi:10.1038/nature12354.

Donahue, B. S., and R. F. Abercrombie. 1987. “Free Diffusion Coefficient of Ionic Calcium in Cytoplasm.” Cell Calcium 8 (6): 437–48.

Gura Sadovsky, Rotem, Shlomi Brielle, Daniel Kaganovich, and Jeremy L. England. 2017. “Measurement of Rapid Protein Diffusion in the Cytoplasm by Photo-Converted Intensity Profile Expansion.” Cell Reports 18 (11): 2795–2806. doi:10.1016/j.celrep.2017.02.063.

Helassa, Nordine, Borbala Podor, Alan Fine, and Katalin Török. 2016. “Design and Mechanistic Insight into Ultrafast Calcium Indicators for Monitoring Intracellular Calcium Dynamics.” Scientific Reports 6: 38276. doi:10.1038/srep38276.

Helmchen, F., K. Imoto, and B. Sakmann. 1996. “Ca2+ Buffering and Action Potential-Evoked ca2+ Signaling in Dendrites of Pyramidal Neurons.” Biophysical Journal 70 (2): 1069–81. doi:10.1016/S0006-3495(96)79653-4.

Neher, E. 1995. “The Use of Fura-2 for Estimating ca Buffers and ca Fluxes.” Neuropharmacology 34 (11): 1423–42. doi:10.1016/0028-3908(95)00144-U.

Neher, Erwin. 1998. “Usefulness and Limitations of Linear Approximations to the Understanding of ca++ Signals.” Cell Calcium 24 (5-6): 345–57. doi:10.1016/S0143-4160(98)90058-6.

Three ways to do non-negative least squares in R

Since I find myself doing this all the time, perhaps this would be of interests to others.

# generate some fake data
x = matrix(rnorm(1000),ncol=10)
y = matrix(rnorm(100), ncol=1)

# method 1 (generally seems slowest)
mod1 = nnls(x,y)

# method 2
mod2 = glmnet(x,y, lambda=0, lower.limits=0, intercept=FALSE)

# method 3 - fastest with multiple y vectors, but occasionally throws a bug I can't figure out
mod3 = .fcnnls(x,y)

# compare results
data.frame(nnls=coef(mod1), glmnet=as.matrix(coef(mod2)[-1]), fcnnls=mod3$coef)

Things I would like to remember if I become a PI

I wrote this post two years ago, and thought perhaps I'd add some updates. It feels perhaps less likely I'll be a PI these days, but perhaps this could be good advice for others.

My original post included the following points:

  • I should offer my students the option to be corresponding authors on their papers. Similarly, I should seriously consider what my role was in the project and therefore whether I should be on the paper in the first place. I should strive to feel like I always deserve that spot and was doing more than contributing money and lab space.
  • If I have favorites, I should not treat them any differently than anyone else in the lab. I should celebrate all accomplishments as much as reasonably possible.
  • I should meet regularly with students and try to understand their goals and aspirations beyond their time in the lab.
  • I should give students an opportunity to provide honest feedback on how I am doing. This is not easy, but some PIs have had success with 6-month reviews.
  • I should encourage all my students to think about how to work on projects with the maximal possible impact, subject to the time constraints they have.
  • I should remind students that publications are not the only currency of value, and indeed are a very volatile and unreliable currency. However, unreported results are of no value to no one.

I think it's worth potentially adding the following points:

  • Students should be my top priority.
  • I should always keep meetings with students. Poor time management is inexcusable. A shared calendar so that students can see my schedule may be a good idea.
  • I should ask my students about the degree to which they would like me to be involved in their work. Some students may need a lot of interaction, others may prefer to be independent, and student needs may change over time.
  • I should be as encouraging as possible.

My new go-to for cheap PCBs: Seeed studio

Full disclosure: Seeed studio emailed me and said I'd get a $30 coupon if I wrote a review of my experience ordering PCBs from them. Edit: May 3, 2017 - they have since delivered on this promise, so I've got $30 to spend on new boards! Sweet. Definitely my new source for pcb manufacturing.

First off, I usually order 2-layer boards, and I'm not pushing the fab limits at all (usually my minimum clearance and trace widths are both >12 mil). My bottom line is price, since I pay for the stuff myself, and I'm a postdoc (hence I'm not rich).

In the past, I've ordered PCBs from Sunstone and Advanced Circuits (4PCB). Mostly, I care about price. At advanced circuits, for $33 I can get a seemingly arbitrarily large 2-layer board (only available to students, otherwise the minimum order quantity is 4), but if I tile a small PCB so that I can get 10 small PCBs for the price of one, then they slap on a $50 array fee. And then there's shipping (which for some reason is always really expensive, like >$20) so the minimum cost is about $50 for a single PCB. Sunstone used to be significantly more expensive, but I recently ordered some boards there for around $50 each.  So, similarly priced.

Recently, a friend of mine told me about Seeed studio. They are fantastically priced - I bought ten 2.2-square-inch boards (2-layer) for $15 there (plus an $18 shipping fee).

I received the board 9 days after ordering - comparable to sunstone and advanced circuits. Quality-wise they're clearly a little bit below sunstone and advanced circuits, as there are minor misalignments in the silkscreen and the soldermask. But so far as I can tell with my first batch, totally functional. Definitely will order again (and update this post if I find any further issues!)


New goal

For every dollar I spend on alcohol, I'm going to donate that same amount to a charity.

Are bacterial mutation rates higher in space?

Apparently Nanobiosym, a biotech company in the area, has decided to send MRSA up to the ISS on a spacex rocket. This is headline-grabbing stuff. From the NASA page:

Proof-of-Concept for Gene-RADAR® Predictive Pathogen Mutation Study (Nanobiosym Genes) evaluates the feasibility of one day using this device from Nanobiosym® to identify bacterial mutations in space. The X Prize-winning device can accurately detect any disease that has a genetic fingerprint, in real time and at the point-of-care. Microgravity may accelerate the rate of bacterial mutations and this pilot investigation analyzes this process in two strains aboard the International Space Station, which may provide insight into how deadly bacteria become drug-resistant.

(Source: https://www.nasa.gov/mission_pages/station/research/experiments/2420.html)

Seriously, what????? $20 to anyone who can provide a reasonable and nontrivial explanation (e.g. not like, the cultures were less aerated in space because they weren't on a shaker) for why microgravity would accelerate the rate of bacterial mutations.

From a Salon.com article, Anita Goel (founder of Nanobiosym) is quoted as saying “[Low Earth Orbit] acts as an incubator to accelerate the mutations that happen with some  bacteria and viruses”.

That makes no sense at all to me. Can anyone explain?

Correlations in time series are sensitive to timescale

... and it's something that perhaps we don't look at quite often enough!

What am I talking about? Well, let's imagine that we're interested in the relationship between two signals, , and . One of the most basic analyses we might do is ask "are they correlated?" But perhaps the correlation depends on the timescale that we focus on. Could a signal be positively correlated at once timescale and negatively correlated at another scale?

Since I've shown you an example, hopefully you believe that a signal could be positively correlated at one timescale and negatively correlated at another! Here, and are positively correlated on a long timescale but negatively correlated on a short timescale.

Can we characterize this sort of relationship? Yes, and I'll outline one way of characterizing this sort of time-scale dependent correlation below.

Continue reading "Correlations in time series are sensitive to timescale"