Final answer:
Mathematical challenges in counting ticks of a molecular clock relate to the variable rates of genetic change and the assumption of a steady average mutation rate. Precise atomic clocks are essential for calibration, but uncertainties in mutation rates present significant complexities in calculating evolutionary divergences accurately.
Step-by-step explanation:
Counting the ticks of a molecular clock involves mathematical challenges, primarily due to the premise that genetic change through mutations occurs at a steady average rate, which is now contested. The concept of a molecular clock is used to estimate the time since related species diverged from a common ancestor using DNA or protein differences. The mathematical difficulties arise because genetic changes can occur for various reasons and at various rates, making the assumption of a fixed rate historically variable. Moreover, calibrating these molecular clocks requires precise and sophisticated technology like atomic clocks, which use oscillations of the resonant frequency of cesium atoms. The second, used to measure time in this context, was defined in 1967 as the duration of 9,192,631,770 such oscillations.
The extensive scale of atomic and molecular counts, such as Avogadro's number, exemplifies the impracticality of counting at the molecular level, highlighting why such precise instruments and assumptions are necessary for the functionality of molecular clocks. Calculating genetic divergence accurately is imperative for evolutionary studies, but because the molecular clock hypothesis is an approximation, scientists contend with significant uncertainties and complexities in their estimations.