Newton, Leibniz or sperm - who was first?

Sperm have only one aim: to find the egg. The egg supports sperm in their quest by releasing attractants that induce changes in the calcium level inside sperm. Calcium determines the beating pattern of the sperm tail which enables sperm to steer. We discovered that sperm only react to changes in calcium concentration and not to the absolute calcium concentration itself. Probably sperm make this calculation to remain capable of maneuvering independent of the cellular calcium concentration.

To increase the chances of fertilization, eggs release chemicals (chemoattractants) that generate a gradient around them (Figure 1 A). When sperm detect such gradients, calcium ion channels on the cell membrane open, and calcium ions flow into the sperm tail. The increase in intracellular calcium modulates the beating of the tail, and thereby alters the swimming path (Figure 1 B).

Figure 1: Navigation of sperm in a gradient. A. The egg is surrounded by a chemical gradient of attractant molecules. B. After detecting the chemoattractant, sperm alternate between symmetric and asymmetric beat to steer their swimming path. Symmetrical beat leads to straight swimming paths, whereas asymmetric beat leads to curved paths.

It was believed for long that at high intracellular calcium concentration, the tail beats asymmetrically like a whip, while at low calcium concentration, the tail beats symmetrically and sperm swim on a straight line. The alternation between high and low calcium concentrations was thought to steer sperm along their swimming path. This view was put into question by recent studies on sperm from marine invertebrates. Although these studies challenge previous concepts, it remained elusive how calcium shapes the beat of the sperm tail.

In a recent publication, we show that sperm do not react to the absolute calcium concentration itself, but to the rate, or dynamics, by which the calcium concentration is changing. In other words, the swimming of sperm is controlled by the time derivative of the calcium concentration (Figure 2).

Figure 2:Swimming of sperm is controlled by the time derivative of the calcium concentration. A. Calcium recording of a sperm cell navigating towards a gradient of chemoattractant (blue). The cell is loaded with a calcium indicator (green), and the intracellular calcium concentration is linear with the brightness of the cell. The average swimming path is shown in dark red. B. Comparison between the intracellular calcium concentration (green) and the curvature of the swimming path (violet). C. Comparison of the time derivative of the calcium concentration (red) and the curvature of the swimming path (violet).

But why do sperm carry out this complicated calculus that we first encounter at the upper secondary school level? The concentration of the attractant and, therefore, also the calcium concentration in sperm is very high near the egg. The mathematical trick enables sperm to react an maneuver even in the presence of such high calcium concentrations. How? Simply by ignoring how much calcium is inside the tail and reacting only to the small changes on top of the high calcium level. Moreover, the swimming path of sperm is composed of curved and straight segments (turns and runs, respectively). Turns allow sperm to reorient in the gradient, whereas runs carry sperm up the gradient (where the egg awaits). The calcium signals elicited in sperm when exposed to a chemical gradient rise and fall repeatedly (Figure 2 B, in green). The rise in calcium evokes a turn (positive time derivative), whereas the decrease in calcium produces a run (negative time derivative).

The mechanism by which sperm compute the time derivative is a mystery, but we suspect that sperm detect calcium ions with the help of two proteins coupled to a motor protein. Calcium (Ca2+) binds to one protein (P1) fast and to the other (P2) slow. By comparing the amount of calcium bound on both proteins, sperm can compute a "chemical derivative". We call our model the "chemical differentiator" (Figure 3).

Figure 3: The "chemical differentiator" model. A. Inside the sperm tail, two proteins bind to calcium. One reaction is fast (binding to P1), and the other reaction is slow (binding to P2). Both reactions promote the bending of the tail in opposite directions. The net bending of the tail is given by the difference of P1 and P2 occupancy. B. When calcium enters inside the cell, the tail first bends in one direction due to the fast reaction of calcium with P1. This binding is counterbalanced by the slow reaction with P2 that favors the bending of the tail on the opposite direction. Thus, the cell adapts to the calcium concentration and only responds to changes in intracellular calcium.

In a gradient, sea urchin sperm swim on periodic paths. The paths can vary from slowly drifting circles to looping trajectories with tight "turns" and wide "arcs" (Figure 4 A). We showed that, from prototypical calcium signals generated in the computer (Figure 4 B), we can calculate the corresponding time derivative and predict the resulting swimming path (Figure 4 C). Using such a modeling approach, we can reproduce the wealth of swimming paths observed in nature.

Figure 4: Numerical simulations reproduce the wealth of swimming paths found in sperm from marine invertebrates. A. Three swimming paths recorded from single sperm cells. The chosen paths displaying stereotypical swimming paths found in sperm. B. Calcium signals used for the numerical reconstruction showed in C. C. Swimming paths derived from the calcium signals displayed on B. The swimming paths obtained are very similar to those encountered in nature.

As an anectode, the father of differential calculus, thus the derivative, is still a question of debate. It is unclear whether Isaac Newton or Gottfried Leibniz discovered it first in the XVII. century. Our finding indicates that, even without a mathematical formal description, sperm started using differential calculus at least 400 million years ago. Thus, they were the first!


Alvarez, L., Dai, L., Friedrich, B. M., Kashikar, N., Gregor, I., Pascal, R. & Kaupp, U. B. (2012) "The rate of change in Ca2+ concentration controls sperm chemotaxis" J. Cell. Biol. 196, 653-663