Question

The following problem arises in molecular biology. The surface of a bacterium is supposed to consist of several sites at which a foreign molecule may become attached if it is of the right composition. A molecule of this composition will be called acceptable. We consider a particular site and postulate that molecules arrive at the site according to a Poisson process with parameter \(\mu .\) Among these molecules a proportion \(\beta\) is acceptable. Unacceptable molecules stay at the site for a length of time which is exponentially distributed with parameter \(\lambda .\) While at the site they prevent further attachments there. An aceeptable molecule "fixes" the site preventing any further attachments. What is the probability that the site in question has not been fixed by time \(t ?\)

Short answer

The probability that the site in question has not been fixed by an acceptable molecule by time \(t\) can be expressed as: \(P(t) = \sum_{n=0}^{\infty}\bigg(\frac{(\mu t)^{n}}{n!} e^{-\mu t} \bigg) (1-\beta)^{n} \big(1 - (1 - e^{-\lambda t})\big)^n\)

Step-by-step solution

Compute the probability of no arrival by time \(t\).

For a Poisson process with parameter \(\mu\), the probability of no arrival in the interval [0, t] is given by: \(P_{0}(t) = e^{-\mu t}\)

Compute the probability of arrivals and all being unacceptable until time t.

Conditional on the number of arrivals, we need to compute the probabilities that all are unacceptable and don't stay until time \(t\). Denoting the number of arrivals as n, we can write the probability as: \(P_{n}(t)= \sum_{n=1}^{\infty}\bigg(\frac{(\mu t)^{n}}{n!} e^{-\mu t} \bigg) (1-\beta)^{n} P_{\lambda}(n, t)\) where \(P_{\lambda}(n, t)\) is the probability that none of the \(n\) molecules stay at the site until time \(t\).

Express the probability referring to exponential distribution.

Since the time length each molecule stays at the site follows an exponential distribution with parameter \(\lambda\), the probability of a single unacceptable molecule staying at the site until time t is given by: \(P_{\lambda}(t) = 1 - (1 - e^{-\lambda t})\) To express \(P_{\lambda}(n, t)\), we consider the probability that all of the \(n\) molecules do not stay until time \(t\), which we can express as the product of their individual probabilities: \(P_{\lambda}(n, t) = \big(1 - (1 - e^{-\lambda t})\big)^n\)

Sum up all the probabilities.

Now we can express the probability that the site has not been fixed by time \(t\) as the sum of probabilities of no arrival and arrivals with all unacceptable molecules that don't stay until time t: \(P(t) = P_{0}(t) + \sum_{n=1}^{\infty}\bigg(\frac{(\mu t)^{n}}{n!} e^{-\mu t} \bigg) (1-\beta)^{n} \big(1 - (1 - e^{-\lambda t})\big)^n\)

Simplify the expression.

Now we have the expression for the probability \(P(t)\), but it seems quite complex. However, we can simplify it by observing that the first term, \(P_{0}(t)\), can be included in the sum by setting \(n=0\). In that case, \(\frac{(\mu t)^{n}}{n!} e^{-\mu t} = e^{-\mu t}\) and \((1-\beta)^{n} \cdot \big(1 - (1 - e^{-\lambda t})\big)^n = 1\). Therefore, the simplified probability formula is: \(P(t) = \sum_{n=0}^{\infty}\bigg(\frac{(\mu t)^{n}}{n!} e^{-\mu t} \bigg) (1-\beta)^{n} \big(1 - (1 - e^{-\lambda t})\big)^n\) This is the final expression for the probability that the site in question hasn't been fixed by an acceptable molecule by time \(t\).

Key concepts

Poisson Process

Understanding the Poisson process is crucial when addressing problems in molecular biology relating to random events. It is a mathematical model used to describe a series of events that occur randomly and independently of each other over time.

The key parameter in a Poisson process is the rate \(\mu\), which signifies the average number of events happening per unit time. For instance, when analyzing the arrival of molecules to a bacterium's site, a Poisson process can be used to model this random occurrence. The essential property of a Poisson process is that the number of events occurring in a fixed interval of time follows a Poisson distribution.

The probability of witnessing no events (in this case, no molecular arrivals) over a period of time \(t\) can be expressed as \(P_0(t) = e^{-\mu t}\), which is essential for further calculations in the problem at hand. The Poisson process is a foundational concept for understanding the randomness and distribution of molecular attachments in biological contexts.

Exponential Distribution

The Exponential distribution is often used to model the time between events in a continuous-time process, such as the period molecules stay attached at a site. It has a single parameter \(\lambda\), which is the rate parameter, and the mean time between events is \(\frac{1}{\lambda}\).

In the context of our molecular biology problem, each unacceptable molecule remains attached for a duration that follows an exponential distribution. Importantly, the probability that an unacceptable molecule has not left the site by time \(t\) is \(P_\lambda(t) = 1 - (1 - e^{-\lambda t})\).

This property illustrates the \'memoryless\' nature of the exponential distribution—the probability of an event occurring in the next instant is independent of how much time has already passed. It simplifies calculations in stochastic processes where the duration of attachments must be considered. Including this concept in the solution helps to illuminate the stochastic nature and temporal behavior of molecular attachments, contributing to a more profound understanding of biological interactions.

Molecular Attachment Probability

The molecular attachment probability in our problem reflects the chance of a particular site on the bacterium’s surface being unoccupied by an acceptable molecule by time \(t\).

We derive this probability by considering two scenarios: first, the probability of zero arrivals at the site by time \(t\), denoted by \(P_0(t)\), and second, the probability of one or more arrivals by time \(t\), with all of them being unacceptable and not remaining attached at the site, denoted by \(\sum_{n=1}^{\infty}(\frac{(\mu t)^{n}}{n!} e^{-\mu t}) (1-\beta)^{n} P_\lambda(n, t)\).

The term \(1-\beta\) represents the proportion of unacceptable molecules, and \(P_\lambda(n, t)\) signifies the probability that none of the \(n\) unacceptable molecules stay until time \(t\). By summing this series, we account for all possible numbers of arrivals and their respective probabilities.

This concept allows us to understand the dynamics of molecular interactions at the site and can help explain more complex biological phenomena. It also can inform experimental designs and interventions in microbiology and bioengineering, demonstrating the vast applications of probability theory in science.