Biological Processes as Exploratory Dynamics

Abstract

Many biological processes can be thought of as the result of an underlying dynamics in which the system repeatedly undergoes distinct and abortive trajectories with the dynamical process only ending when some specific process, purpose, structure or function is achieved. A classic example is the way in which microtubules attach to kinetochores as a prerequisite for chromosome segregation and cell division. In this example, the dynamics is characterized by apparently futile time histories in which microtubules repeatedly grow and shrink without chromosomal attachment. We hypothesize that for biological processes for which it is not the initial conditions that matter, but rather the final state, this kind of exploratory dynamics is biology’s unique and necessary solution to achieving these functions with high fidelity. This kind of cause and effect relationship can be contrasted to examples from physics and chemistry where the initial conditions determine the outcome. In this paper, we examine the similarities of many biological processes that depend upon random trajectories starting from the initial state and the selection of subsets of these trajectories to achieve some desired functional final state. We begin by reviewing the long history of the principles of dynamics, first in the context of physics, and then in the context of the study of life. These ideas are then stacked up against the broad categories of biological phenomenology that exhibit exploratory dynamics. We then build on earlier work by making a quantitative examination of a succession of increasingly sophisticated models for exploratory dynamics, all of which share the common feature of being a series of repeated trials that ultimately end in a “winning” trajectory. We also explore the ways in which microscopic parameters can be tuned to alter exploratory dynamics as well as the energetic burden of performing such processes.

It is a great privilege to take part in this special volume dedicated to the life and work of Prof. Erich Sackmann (1934–2024). For one of us (RP), at the time of making a switch from traditional condensed matter physics to a life engaged in the study of life, he went to a meeting near Munich which completely opened his eyes to the ways in which the approach of physics could be brought to bear on the study of the living. Sackmann’s work was an inspiring presence at that meeting. One of the hallmarks of his work was a principled approach to dissecting biological processes over a range of scales and phenomena. One common thread to much of his work was that it acknowledged the dynamical character of living organisms. The present paper attempts to follow in the tradition of Sackmann’s studies of dynamics by suggesting a new way of looking at many biological processes all through the unifying perspective of what we will call exploratory dynamics.

1. Theories of Inanimate and Animate Dynamics

Whatever level of phenomena we look at, small or large spatial scales, short or long time scales, low or high energies, these phenomena are usually dynamic. The quest to understand the dynamics of broader and broader classes of phenomena has repeatedly over the long arc of the history of science resulted in “new physics” [1, 2, 3, 4]. Indeed, as shown in Figure 1, there is a long and justly celebrated history of different approaches to working out the dynamics of processes in the world around us. The template for this subject both mathematically and physically was achieved by Newton standing on the shoulders of Galileo and others [5]. Newton saw that $\mathbf{F}=m\mathbf{a}$ provided an update rule that allowed us, using what we now know as calculus, to find the current state of the system by time stepping from previous states [6]. Though most of us learn about Newton’s analysis of the planets in polar coordinates culminating in elliptical orbits, Newton’s treatment of the problem was as a series of straight line steps punctuated by impulses due to the force of gravitation resulting in an orbit that was a polygon [6]. We might summarize this and most subsequent dynamical laws as having the form

$$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}(t+\mathrm{\Delta }t)=\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\left(t\right)+\mathrm{u}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}\left(t\right)\mathrm{\Delta }t,$$ (1)

the simplest embodiment of the forward-Euler method for solving differential equations, but which we think of differently as being the discrete representation of most versions of deterministic, initial-condition dictated, dynamics. The state now is determined by the state a time $\mathrm{\Delta }t$ earlier plus some update term.

Figure 1.

Gallery of dynamical equations. The defining example of dynamics in physics is the $F=ma$ dynamics of Newton that allows us to solve problems such as falling bodies or planetary motion. The dynamics of chemical reactions is another well established example of initial-condition driven dynamics, in this case showing the dynamics of ligand (red)-receptor (green) binding. The heat equation is a deterministic dynamical equation that tells us how an initial temperature distribution will evolve in space and time. There are a host of different dynamical equations describing the dynamics of populations, here we show the logistic equation. Maxwell’s theory of is a quintessential example of a field theory which vastly expands the repertoire of classical physics. Langevin-like equations even allow us to work out the time evolution of stochastic chemical dynamics.

In the centuries that followed Newton’s creation of the “System of the World [6],” $F=ma$ dynamics was brilliantly generalized to continuous media with examples such as hydrodynamics and elastodynamics to describe the mechanics of fluids and solids [7, 8]. But other phenomena called for something beyond the mechanical world view [9]. As shown in Figure 1, new dynamical laws were articulated that made it possible to work out the time evolution of a myriad of processes. For example, the study of chemical transformations led to the emergence of the law of mass action and the rate equations of chemical kinetics, allowing for the appearance and disappearance of different chemical species [10, 11]. Another class of problems highlighted in Figure 1 arose as Fourier and others tackled the dynamics of temperature fields and heat flow in the form of a partial differential equation (the heat equation) showcased in the figure [12, 2]. In this case, a material subjected to some initial temperature distribution will evolve to some new temperature distribution [13, 14]. The dynamical equations describing the diffusion of matter and heat had and have a huge reach including to important problems such as the age of the Earth [15, 16, 17]. While physicists were busy working out the laws of inanimate matter, other kinds of scientists and mathematicians were also engaged in thinking about the dynamics of populations as exemplified by the logistic equation shown in the figure [18, 19, 20, 21]. In the middle of the 19th century, laws of dynamical evolution received a great boost in the form of Maxwell’s theory of the electromagnetic field to describe the spacetime evolution of electromagnetic fields [1, 3, 4]. In addition to these exercises in deterministic dynamics, the discovery of Brownian motion inched scientists and mathematicians towards the consideration of random processes such as the jiggling motions of pollen grains observed by Brown, in that case using the Langevin equation for example [22, 23]. Many of these dynamical descriptions share the common feature that there is some initial state of the system, and the dynamical laws tell us how to compute the subsequent evolution of the system in time.

But the dynamical laws described above left out one of the most important categories of dynamics in the natural world, namely, the strange dynamics in and of living organisms, processes including the long chains of energy-driven biochemical reactions, structure formation from macromolecular complexes to organelles, embryonic development and a myriad of other fascinating topics. It is fashionable to minimize the brilliant efforts of our predecessors when we later come to realize how “wrong” their ideas might have been. Within physics the 19th century idea of heat as a fluid, the famed caloric, or of the luminiferous ether, the medium within which electromagnetic disturbances were thought to propagate, are sometimes viewed as quaint and misguided. However, a more fair and nuanced view sees them instead as brilliant and creative examples of the continuing refinement that attends most scientific progress [1, 3, 4]. In the study of evolution, Lamarck’s ideas are routinely dismissed as similarly misguided, though a more scholarly judgment is that like many others, Lamarck was engaged in thoughtful hypothesis making that was later learned to be inconsistent with observations and measurements. Perhaps no discredited topic draws such contempt as does that of vitalism. However, even here, a more generous interpretation of the 19th century vitalists is to acknowledge the obvious: living organisms seem different than their inanimate counterparts and clearly, the incomplete dynamical repertoire of classical physics might have something to gain by coming to terms with the purposeful dynamics of living organisms.

One way in which the study of the living might hold the secret to new principles is in the sense that the paradigm of Laplace’s Demon [24], the strict unfolding of initial conditions through deterministic evolution, is often not respected by processes within living cells. Au contraire, cells often require a different kind of dynamics that is driven by the final conditions rather than the initial conditions and it is to this kind of processes that the remainder of our paper is dedicated. In particular, living organisms have developed ways of converting goals or functions into goal-oriented dynamical processes. In many everyday human processes characterized by some purpose, goal or function, the standard paradigm is the construction of machines. We build machines to pump water, to cool materials [25], to move objects and the list goes on and on. By way of contrast, living organisms have used stochasticity in a novel way to project the future into the past, to enable random trajectories to “find” the goal or target in a process which we call exploratory dynamics [26]. Andrew Murray suggested to us an everyday occurrence that might help frame the uniqueness and weirdness of exploratory dynamics as shown in Figure 2. He asks us to imagine building a railroad line between New York City and Boston. We all know how this goes if done according to the strictures of modern civil engineering. There is a goal, there is a plan, the plan is implemented to achieve the goal. Surveyors do their work, steel is ordered, worked are deployed and paid to lay down track. However, if done according to the principles of biological exploratory dynamics, instead, construction workers will start in the two cities, randomly choosing a direction to start laying down track. If after a certain time, they have not encountered their counterparts coming from the other city, they remove the track and try again. That’s all. For most of us, our immediate reaction is that this is woefully inefficient, perhaps even absurd. However, as seen in the examples of Figure 3, this kind of dynamics describes important biological processes over a dizzying array of spatial and temporal scales. Though this idea of exploratory dynamics was first clearly articulated in the context of the search and capture mechanism of chromosome segregation [27], in the time since a growing list of examples of this kind of dynamics has mounted [26]. As a result, we think we have reached a moment of timely rather than premature abstraction in which experience with these specific examples suggests that instead of being a particular mechanistic feature of chromosome search, exploratory dynamics is a much more general phenomenon, calling for its own abstractions, toy models and equations. In this paper, we examine one category of exploratory dynamics which features a series of abortive trajectories followed by a final successful trajectory.

Figure 2.

Comparison of human-scale dynamics and cellular-scale dynamics approaches to building a railroad between New York City and Boston. In both cases, there is a very specific goal. However, the strategy for achieving that goal is completely different in the human engineering context in comparison with the biological strategy of using stochastic variation and “selection” to achieve exploratory dynamics. The concept of the figure is due to a suggestion of Andrew Murray.

Figure 3.

Gallery of exploratory dynamics. Examples from a wide variety of spatial and temporal scales illustrate the way in which the dynamics is characterized by a set of trajectories, all of which repeatedly fail to achieve their function before ultimately succeeding. Each example is explained in more detail in the text.

The organization of the paper is as follows. In section 2 we explain the concept of exploratory dynamics, first by describing biological processes that exemplify it and then by giving a cursory description of the kinds of theoretical ideas that have been put forth that might help explain such processes. Section 3 explains the statistical mechanics of exploratory trajectories using a simple zero-dimensional model. With those preliminaries in hand, section 4 describes the concrete problem of microtubules searching for chromosomes in the language of exploratory dynamics. Section 5 tackles a completely different biological phenomenom, namely, how transcription factors find their target site on DNA, revealing many mathematical similarities to the problem of the microtubule search problem. We note that in both of our case studies, we in no way pretend that we are the first to write down many of the models we explore, nor do we pretend our limited survey of the relevant literature to be comprehensive or encyclopedic. Rather, we spell them out in full detail in the hope of carefully illustrating the statistical mechanics of exploratory trajectories in a few limited examples in the hope that it might encourage analysis of the full suite of examples shown in Figure 3, with the aim of more mathematical generality than the few examples considered here. We finish in section 6 with a high-level summary of the possible broader implications of exploratory dynamics and some final reflections on the career and influence of Prof. Erich Sackmann.

2. Exploratory Dynamics by Variation and Selection

In many cases, when we use or hear the words variation and selection, it is in the context of Darwinian evolution. However, the dynamics of variation and selection has much broader reach than in the context of traditional biological evolution. Perhaps surprisingly, there are many problems in molecular and cellular dynamics that can be couched in this same language. Examples range from the search of transcription factors for their binding sites [28, 29, 30, 31, 32]; microtubules being captured by kinetochores on chromosomes [33, 34, 35, 36, 37]; bacterial cells performing the biased random walks of chemotaxis [38, 39]; the formation of vasculature in developing embryos [40]; the formation of exquisitely precise neuronal connections [41]; the development of antibody repertoires [42, 43, 44]; animals forging for food sources [45, 46, 47, 48, 49, 50] and far beyond. In an earlier work, one of us (MK) in collaboration with John Gerhart wrote a book which considered living organisms in terms of some key principles that might describe broad swathes of biological phenomena [26]. Chapter 4 of that book entitled “The Exploratory Behavior of Biological Systems” described a series of case studies in biological phenomena over a wide range of spatial and temporal scales that could be viewed as being of an exploratory character. Those examples inspired the search for biophysical underpinnings to exploratory dynamics, and this paper is is our attempt to lay out several case studies in some mathematical detail.

Figure 3 provides a gallery of examples of biological processes that we think it might be fruitful to examine quantitatively from the perspective of exploratory dynamics. Though the formal definition of what constitutes exploratory dynamics might still be coming into focus, for the purpose of the present paper, we consider a process as exploratory if it is characterized by a well defined final state which is reached through a series of trials. In particular, we imagine a random process that produces a set of trajectories, a “standing variation” in trajectory space. A selection mechanism then picks out a subset of these trajectories, which are “functional.” One can imagine refinements in which we ask whether the exploratory process requires energy investment such as in the case of chromosome search and capture, or not, as in the case of transcription factor search as considered later in the paper. In addition, one can also imagine situations in which there are cues that reinforce some trajectories that appear to be headed in the “right” direction such as is found in chemotaxis or the growth of vasculature.

To more concretely illustrate what we mean by exploratory dynamics, the first panel of Figure 3 reminds us that the process of protein translation is exploratory in the sense that repeatedly, the “wrong” amino acid is presented to the ribosome and only through this kind of biased trial-and-error dynamics does a protein get translated with high fidelity [51, 52]. In most cases in which a wrong tRNA arrives at the ribosome, there is a reset that restores the system back to its starting state. In this case, the “variation” refers to at least at 20:1 ratio of wrong to right tRNAs arriving at the ribosome, and the “selection” refers to the incorporation of the amino acid of interest into the nascent polypeptide chain. As shown in the second panel, one of the most compelling examples of exploratory dynamics is the dynamics of microtubules which precedes chromosome segregation in dividing cells [33, 34, 35, 36, 37]. Here, microtubules grow and shrink repeatedly from an organizing center known as the centrosome, and those microtubules that find the kinetochore on the chromosomes stay attached to their target. A reframing of the problem as one in exploratory dynamics is schematized in Figure 4. The third example shows the growth of neurons, where connections come and go and are “selected” in some contexts as illustrated in the example of olfactory selection cited here [41]. In the developing fly embryo, though these animals don’t have vasculature in the same sense as mammals, they do have a system of “pipes” that allow for the transport of oxygen to cells and the establishment of this spatial arrangement of pipes is itself exploratory [40]. More generally, the subject of branching morphogenesis provides a broad array of examples of exploratory dynamics [53, 54]. At much larger scales; animal foraging can be thought of as its own kind of exploratory dynamics [45, 46, 47, 48, 49, 50]. The main point of the figure and this discussion is simply to capture the reader’s imagination with the hypothesis that perhaps exploratory dynamics is a new and important part of the dynamics repertoire that has thus far been understudied in general terms from the physics perspective.

Figure 4.

Microtubule search for kinetochores as exploratory dynamics characterized by stochastic variation in the dynamical trajectories of the system coupled with selection and stabilization of those trajectories that culminate in attachment to kinetochores.

Our discussion thus far primarily focused on the biological phenomenology of exploratory dynamics. Interestingly, in parallel, the worlds of both physics and mathematics had developments of their own aimed at solving problems in purposeful dynamics. Though it might seem far afield, the quest of the Allied Forces to search for German U-Boats gave rise to the inspiring and useful field of operations research and search theory [55, 56, 57]. Ironically, several of the 20th century’s most famous biologists were key participants in the development of the mathematical and physical tools to treat search problems [58], including John Kendrew, noted for his resolution of the molecular structure of whale myoglobin [59]. Perhaps more surprisingly is that at the age of 67, CH Waddington, of Waddington landscape (and many other things) fame, wrote a profound text on the use of operations research during the Second World War of which he, along with Kendrew, was a key participant [60]. A parallel avenue of theoretical research that we think is quite relevant to exploratory dynamics has focused on random walks with resets [61, 62]. Yet another set of ideas were introduced that have been dubbed infotaxis [63, 64, 65]. In the remainder of the paper, we focus on a specific approach in which we describe exploratory dynamics as a statistical mechanics of a particular class of trajectories, but we are anxious to see what theorists will have to say about exploratory dynamics in coming years.

3. Zero-Dimensional Exploratory Dynamics

As noted above, one example of exploratory dynamics that is enormously consequential is the way in which microtubules pair up with chromosomes in preparation for the process of chromosome segregation as shown schematically in Figure 4. In this paper, we use this example as a specific case study that illustrates what are perhaps more general ideas of how to formulate exploratory dynamics as a statistical mechanics of trajectories. In this statistical mechanics of exploratory trajectories, all trajectories are characterized by the fact that they end when some “purpose” is fulfilled or some function is achieved. As we show below, in each case considered in this paper, this results in the use of the geometric distribution. We suspect that there are much more general ideas waiting to be discovered that can account for cases in which subsequent trajectories have a memory of those that have come before or in which signals reinforce trajectories that are headed in the right direction.

As a first model, we set up the mathematical language of the statistical mechanics of exploratory trajectories by considering trajectories built up of “wrong” and “right” moves. For the moment, we are noncommittal about the moves themselves since as we will see later in the paper, in different contexts, “move” will refer to the duration of the polymerization process, the direction of polymerization or to changing between one-dimensional and three-dimensional diffusion in the case of transcription factors searching for their binding sites. All the trajectories have the same basic structure. A series of wrong moves followed by a final right move. At each time step, the wrong and right moves are each chosen with probability $f_w$ and $f_r$, respectively, subject to the condition that $f_w+f_r=1$. A given exploratory trajectory is assembled as a collection of $i$ unsuccessful steps before a final successful step. The probability of such a trajectory is given by

$$p_i=\underset{\mathrm{i}}{\underbrace{f_w{f}_w\cdots f_w}}{f}_r=f_w^i{f}_r.$$ (2)

We can rewrite this more intuitively as

$$p_i={\left(1-f_r\right)}^i{f}_r,$$ (3)

a probability distribution known more colloquially as the geometric distribution [66]. This fundamental equation has a structure that we will use over and over in thinking about the statistical mechanics of exploratory trajectories since it highlights the importance of the final state.

Since we will need to manipulate these distributions repeatedly, here we use it to compute the average waiting time until the exploration is complete. We begin by verifying that the distribution is normalized by computing

$$\sum _{i=0}^{\mathrm{\infty }}{p}_i=\sum _{i=0}^{\mathrm{\infty }}{\left(1-f_r\right)}^i{f}_r=f_r\sum _{i=0}^{\mathrm{\infty }}{\left(1-f_r\right)}^i.$$ (4)

We recognize the remaining sum as a geometric series with the property that $\sum _{i=0}^{\mathrm{\infty }}{x}^i=1/(1-x)$ if $0<x<1$. Since we have $0<1-f_r<1$, our case yields

$$\sum _{i=0}^{\mathrm{\infty }}{p}_i=f_r\sum _{i=0}^{\mathrm{\infty }}{\left(1-f_r\right)}^i=\frac{f_r}{\left(1-\left(1-f_r\right)\right)}=1.$$ (5)

As expected, the distribution is normalized.

A more important question is how long does it take on average for the exploratory dynamics to terminate due to success in finding its target? The answer to that question is obtained by averaging over the time spent in all the possible trajectories and given by

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}\underset{\text{failures}}{\underbrace{\left(i{t}_{\text{step}}\right.}}+\underset{\text{success}}{\underbrace{t_{\text{step}}}})p_i,$$ (6)

where we have introduced the quantity $t_{\text{step}}$ as the duration of each instance where the system makes an unsuccessful excursion (e.g. a microtubule polymerizes without finding the chromosome). Later in the paper, we will explicitly calculate $t_{\text{step}}$ on the basis of the microscopic parameters associated with the microtubule polymerization problem or the transcription factor search problem. The second term in the sum immediately yields $t_{\text{step}}$ since the distribution is normalized leaving us with

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}i{t}_{\text{step}}{p}_i+t_{\text{step}}.$$ (7)

Next, we need to evaluate

$$⟨t⟩=t_{\text{step}}\sum _{i=0}^{\mathrm{\infty }}i{p}_i+t_{\text{step}}=t_{\text{step}}{f}_r\sum _{i=0}^{\mathrm{\infty }}i{\left(1-f_r\right)}^i+t_{\text{step}},$$ (8)

which we recognize as having a term of the form

$$\sum _{i=0}^{\mathrm{\infty }}i{x}^i=x\frac{d}{dx}\sum _{i=1}^{\mathrm{\infty }}{x}^i=x\frac{d}{dx}\frac{1}{1-x}=\frac{x}{(1-x{)}^2}.$$ (9)

We now see that we can rewrite eqn. 8 as

$$⟨t⟩=t_{\text{step}}{f}_r\sum _{i=0}^{\mathrm{\infty }}i{\left(1-f_r\right)}^i+t_{\text{step}}=t_{\text{step}}\left(1-x\right)\sum _{i=0}^{\mathrm{\infty }}i{x}^i+t_{\text{step}},$$ (10)

where we have used the definition $x=1-f_r$. We now invoke the derivative trick introduced in eqn. 9 to arrive at

$$⟨t⟩=t_{\text{step}}\left(1-x\right)\frac{x}{(1-x{)}^2}+t_{\text{step}}=t_{\text{step}}\frac{x}{\left(1-x\right)}+t_{\text{step}}.$$ (11)

If we now write this expression in terms of $f_r$, we find the very intuitive result that

$$⟨t⟩=t_{\text{step}}\left(1+\frac{1-f_r}{f_r}\right)=\frac{t_{\text{step}}}{f_r}.$$ (12)

The intuition for this result is best served by concretely imagining that $f_r=1/M$ which means that 1 out of every $M$ trials is successful. This implies in turn that $⟨t⟩=M{t}_{\text{step}}$, stating that on average we need to carry out $M$ trials for the exploration to be a success when the probability of success on a given trajectory is $f_r=1/M$.

Our reason for going through this example is to showcase the structure of the result, since in all of our examples of exploratory dynamics, we will ultimately characterize the exploratory process as a statistical mechanics of trajectories, with all such trajectories sharing the feature that they involve repeated “failures” until they succeed on the final trajectory. Interestingly, for the examples considered throughout the paper, we will see many different realizations of the space of trajectories, but as noted above, they all are of the form $p_i=\underset{\mathrm{i}}{\underbrace{f_w{f}_w\cdots f_w}}{f}_r$.

4. Chromosome Search Via Microtubule Reset Dynamics

4.1. One-Dimensional Chromosome Capture by Exploratory Dynamics

As our next case study, we consider the one-dimensional model summarized in Figure 5(B) where we idealize the growth and catastrophes as taking place in one dimension and ask for the time it takes for the microtubule that is subject to both growth and shrinkage to find its chromosomal target. This analysis largely imitates the beautiful work of Holy and Leibler, especially as developed in their Appendix B [34] as well as more recent work from Wollman et al. [36] and their thorough and insightful Supplementary Information. In the spirit of successively building up models of increasing sophistication, we begin with the unrealistic assumption that every microtubule grows in the right direction and ask only whether it survives long enough to reach the chromosome. One way to think about this model is that the exploratory behavior only exists in the sense that the different exploratory periods have different durations. Later, we will extend this framework to consider the case in which there is also a spatial component to the search process with different microtubules heading off in different directions, lending a second facet to the exploratory nature of the dynamics.

Figure 5.

Schematic of the process of microtubules searching for chromosomes. (A) Schematic showing how microtubules grow out from the centrosome and then suffer catastrophes. (B) One-dimensional idealization of the process of microtubules searching for a chromosome. We imagine that the distance from the centrosome to the kinetochore is $d$ and that the polymerizing microtubule grows with speed $v$. All chromosomes grow in the same direction and the exploratory character is in how long the microtubule survives before suffering a catastrophe. (C) Three-dimensional idealization of the process of microtubules searching for a chromosome. $f_r$ is the probability of the microtubule growing in the right direction and $f_w=1-f_r$ is the probability of growing in the wrong direction. The right direction is defined by a cone of solid angle such that the microtubule will grow in a range of allowed directions that will contact the kinetochore.

Before we can compute the time scale associated with the chromosomal search process, we must first consider the waiting time distribution that tells us how long a microtubule grows before it suffers a catastrophe. To answer that question, we turn once again to the geometric distribution. If the rate constant for a catastrophe is $k_c$, then the probability in a time step $\mathrm{\Delta }t$ that the microtubule will not suffer a catastrophe is $\left(1-k_c\mathrm{\Delta }t\right)$. Hence, if the microtubule is to pass through $N$ time steps before having a catastrophe, the probability of that succession of events is

$$p_N={\left(1-k_c\mathrm{\Delta }t\right)}^N{k}_c\mathrm{\Delta }t.$$ (13)

Because of the beautiful properties of this distribution in the large $N$ limit, namely

$$\underset{N\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left(1-\frac{x}{N}\right)}^N=e^{-x},$$ (14)

and using the fact that the total time elapsed is $t=N\mathrm{\Delta }t$, we see that the probability that the microtubule will suffer a catastrophe in the time interval between $t$ and $t+dt$ is given by

$$p_{\text{survive}}\left(t\right)dt=k_c{e}^{-k_{c}t}dt.$$ (15)

The most naive model of the average time before a microtubule suffers a catastrophe is given by $t_c=1/k_c$ which we find by evaluating the average

$$t_c={\int }_0^{\mathrm{\infty }}t{k}_c{e}^{-k_{c}t}dt=\frac{1}{k_c}.$$ (16)

However, the average time before a catastrophe is actually shorter. Note that if the distance from the initial position of the microtubule to the chromosome is $d$ and it grows at a rate $v$, this implies that if the microtubule grows for a time longer than $\tau =d/v$, the microtubule will have hit the chromosome, thus ending the exploratory set of trajectories. Hence, we need to be more careful in assigning a time scale to the failed trajectories. Instead, we should sum only over those trajectories that last a time shorter than $\tau$. In particular, the average time of the failed trajectories is given by

$$t_f=\frac{{\int }_0^{\tau }{e}^{-k_{c}t}{k}_{c}tdt}{{\int }_0^{\tau }{e}^{-k_{c}t}{k}_{c}dt},$$ (17)

where we only allow those trajectories that are shorter than $\tau$ and we have introduced the notation $t_f$ with subscript $f$ to denote “failed” trajectories. The denominator imposes normalization on the distribution. These integrals can be evaluated to yield

$$t_f=\frac{1}{k_c}\frac{1-\left(k_c\tau +1\right)e^{-k_c\tau }}{1-e^{-k_c\tau }}.$$ (18)

Note that this average time until a catastrophe is shorter than the naive estimate of $1/k_c$ derived above.

Given the survival distribution, the probability that the microtubule will survive long enough to reach the chromosome is given by

$$p_s={\int }_{\tau }^{\mathrm{\infty }}{k}_c{e}^{-k_{c}t}dt=e^{-k_{c}d/v}.$$ (19)

The integral computes the fraction of the probability distribution where the survival of the microtubule lasts longer than the time to reach the chromosome, $\tau =d/v$, where again, $d$ is the distance to the microtubule and $v$ is the velocity of microtubule growth. Our goal is to find the average time it takes for the growing and shrinking microtubule to reach the chromosome. The probability of success after $i$ failures is once again given by the geometric distribution which we now write as

$$p_i={\left(1-p_s\right)}^i{p}_s$$ (20)

which reflects the fact that for $i$ of the growth processes, the microtubule suffers a catastrophe before reaching the chromosome and only on the $i+1^{\mathit{th}}$ try does it reach the chromosome with probability $p_s$. A variety of examples of these kinds of exploratory trajectories are shown in Figure 6. The time taken for this particular class of trajectories is given by $\tau +i{t}_f$ since on average each of the failed growth and catastrophe events takes a time $t_f$ and the final successful growth event takes a time $\tau$.

Figure 6.

Trajectories and weights for one-dimensional model of chromosome search and capture. The index “i” labels the number of times that the microtubule fails to hit the chromosome before its eventual success. The “failed” trajectories are characterized by the fact that the microtubule suffers a catastrophe before reaching the chromosome. The failed trajectories show the maximum length of the microtubule before suffering a catastrophe.

The logic of our calculation of the average “exploration time” is that we need to sum over all trajectories, every possible life history of the growing and shrinking microtubules that ends by ultimately finding the chromosome. In words, this means we have to sum over the case where the microtubule finds the chromosome on the first try, the second try, the third try and so on. Given this procedure, the average time to find the chromosome is given by

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}\left(\tau +i{t}_f\right)p_i.$$ (21)

Using the probability of success on the $(i+1{)}^{\mathit{th}}$ trial introduced in eqn. 20, we can rewrite the average exploration time as

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}\left(\tau +i{t}_f\right){\left(1-p_s\right)}^i{p}_s,$$ (23)

which is helpfully rewritten as

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}\tau {\left(1-p_s\right)}^i{p}_s+\sum _{i=0}^{\mathrm{\infty }}i{t}_f{\left(1-p_s\right)}^i{p}_s.$$ (23)

The first of these sums yields $\tau$ since the geometric distribution is normalized. For the second sum, by exploiting the very useful derivative protocol for evaluating such sums that we already introduced in eqn. 9, we can write the second sum as

$$\sum _{i=0}^{\mathrm{\infty }}i{t}_f{\left(1-p_s\right)}^i{p}_s=t_f{p}_s\frac{\left(1-p_s\right)}{p_s^2}.$$ (24)

Hence, the total search time is given by

$$⟨t⟩=\tau +t_f\frac{\left(1-p_s\right)}{p_s}$$ (25)

If we recall that $p_s=e^{-k_{c}d/v}$ and the result for $t_f$ in eqn. 18, we find

$$⟨t⟩=\tau +\frac{1}{k_c}\frac{1-e^{-k_c\tau }\left(k_c\tau +1\right)}{1-e^{-k_c\tau }}\frac{\left(1-e^{-k_c\tau }\right)}{e^{-k_c\tau }}.$$ (26)

Noting the cancellations of the terms $1-e^{-k_c\tau }$ in the numerator and denominator, this simplifies to the lovely final result

$$⟨t⟩=\frac{1}{k_c}\left(e^{k_c\tau }-1\right).$$ (27)

There are a variety of interesting limits to this result that are useful to consider. For example, in the case where the time to suffer a catastrophe is much larger than $d/v$, the microtubule will hit the chromosome on first try and the average time is $d/v$. To see this limit in practice when $k_c\tau \ll 1$, we Taylor expand the exponential resulting in

$$⟨t⟩\approx \frac{1}{k_c}\left(1+k_c\tau -1\right)\approx \tau .$$ (28)

Figure 7 shows the average search time as a function of the catastrophe rate $k_c$. The minimum in this curve is found when the catastrophe rate goes to zero. We can see this more formally by evaluating

$$\frac{d⟨t⟩}{d{k}_c}=-\frac{1}{k_c^2}\left(e^{k_c\tau }-1\right)+\frac{\tau }{k_c}{e}^{k_c\tau }=0$$ (29)

implying that the minimum occurs at

$$k_c=\frac{1}{\tau }\left(1-e^{-k_c\tau }\right),$$ (30)

which is satisfied when $k_c\tau =0$, meaning that the catastrophe rate itself is zero. The best strategy for this particularly simple model is to have no catastrophes. Let’s now consider the more serious model that allows for a three-dimensional search [34, 36].

Figure 7.

Search time as a function of the catastrophe rate for the one-dimensional model of chromosome search and capture. For this case $\tau =2\mathrm{m}\mathrm{i}\mathrm{n}$.

4.2. Three-Dimensional Chromosome Capture by Exploratory Dynamics

The calculation given above made the unrealistic assumption that every microtubule started out in the right direction, and thus we only needed to find out whether such microtubules lasted long enough to reach the chromosome before suffering a catastrophe. We now examine the case in which every time a microtubule starts anew, it has a probability $f_r$ of going in the right direction to meet the chromosome, and a probability $f_w=1-f_r$ of going in the wrong direction. This concept is illustrated schematically in Figure 5(C).

As with the previous examples, we need to write an expression for the probabilities of all the different kinds of trajectories as indicated in Figure 8. For concreteness, let’s consider those trajectories that involve $n$ unsuccessful polymerization events before a success on the ${(n+1)}^{\mathit{th}}$ try. However, there is a new twist relative to the calculation of the previous section. Now our trajectories can fail either because they are going in the right direction, but don’t last long enough to reach the chromosome or alternatively, they are going in the wrong direction and no matter how long they last, they will not find the chromosome. As a result, we introduce the notation $p_i(n)$ for those trajectories that fail $n$ times before a success, with $i$ of those failures being in the right direction, but of insufficient duration, and $n-i$ of those failures going in the wrong direction. The probability of a trajectory in the right direction, but that does not last long enough to reach the chromosome is given by $f_r\left(1-p_s\right)$ while the probability of a trajectory in the wrong direction is given by $\left(1-f_r\right)$. Given these partial probabilities, we can now write the probability of $p_i(n)$ of this category of trajectories as

$$p_i\left(n\right)=\frac{n!}{i!\left(n-i\right)!}\underset{\text{right}\text{direction}}{\underbrace{f_r^i{\left(1-p_s\right)}^i}}\underset{\text{wrong}\text{direction}}{\underbrace{{\left(1-f_r\right)}^{n-i}}}{f}_r{p}_s,$$ (31)

where we see that out of the $n$ unsuccessful trajectories, $i$ of them went in the right direction and $n-i$ of them went in the wrong direction. The prefactor counts the number of ways that out of these $n$ trajectories, $i$ of them will be in the right direction and $n-i$ of them will be in the wrong direction.

Figure 8.

Trajectories and weights for three-dimensional model of chromosome search and capture. Trajectories are labeled by the label $n$ which tells how many failed trajectories there were and the label $i$ which tells us out of the $n$ failed trajectories, how many of them were in the right direction. Each such trajectory has an associated probability and time until success and to find the average time we sum over the infinite set of trajectories as $n\to \mathrm{\infty }$.

The time associated with a trajectory involving $n$ unsuccessful polymerization events before a final successful polymerization event is given by

$$t_i\left(n\right)=i{t}_r+\left(n-i\right)t_w+\tau .$$ (32)

Here we acknowledge that $i$ of the unsuccessful trajectories go in the right direction and have an average lifetime $t_r,n-i$ of the unsuccessful trajectories go in the wrong direction and have an average lifetime $t_w$ and the final successful trajectory has a time $\tau$. As a reminder, recall that

$$t_w=\frac{1}{k_c},$$ (33)

since these trajectories are of unrestricted length and $t_r$ is given by the expression in eqn. 18 since these trajectories are constrained to be less than $\tau =d/v$.

Now we need to sum over all possible trajectories which we can organize successively by considering all trajectories with 0 failures, all with 1 failure, all with 2 failures ··· and all with $n$ failures and so on, resulting in

$$⟨t⟩=\sum _{n=0}^{\mathrm{\infty }}\sum _{i=0}^n{t}_i\left(n\right)p_i\left(n\right).$$ (34)

More explicitly, this can be written as

$$⟨t⟩=\sum _{n=0}^{\mathrm{\infty }}\sum _{i=0}^n\frac{n!}{i!\left(n-i\right)!}{f}_r^i{\left(1-p_s\right)}^i{\left(1-f_r\right)}^{n-i}{f}_r{p}_s\times \left[i{t}_r+\left(n-i\right)t_w+\tau \right].$$ (35)

Using all of the tools already introduced throughout the paper, all of these sums are of the form $\sum _{i=0}^{\mathrm{\infty }}i{x}^i$ which we can evaluate to yield

$$⟨t⟩=\tau +\frac{1}{p_s{f}_r}\left(t_r\left(1-p_s\right)f_r+t_w\left(1-f_r\right)\right)$$ (36)

which simplifies to the remarkably compact result

$$⟨t⟩=\frac{1}{k_c}\left(\frac{e^{k_c\tau }}{f_r}-1\right),$$ (37)

nearly identical to the earlier result of eqn. 27. This expression is plotted in Figure 9(A) where we see that there is an optimal catastrophe rate to achieve finding the chromosomes in the least time.

Figure 9.

Optimal search time as a function of the catastrophe rate. (A) Time to capture the chromosome as a function of the catastrophe rate. (B) Plot of the two sides of the equation for the optimal search time (eqn. 40) as a function of the catastrophe rate $k_c$. The parameters used here are: $f_r=0.05,d=5\mu \mathrm{m},v=10\mu \mathrm{m}/\mathrm{m}\mathrm{i}\mathrm{n}$ implying $\tau =v/d=2\mathrm{m}\mathrm{i}\mathrm{n}$.

It is of interest to explore this expression in various limits to see if it jibes with our intuition. We note that in the case when $f_r=1$, we recover precisely the same result we found earlier as eqn. 27, meaning that in the case where every microtubule starts off in the right direction, the present expression reduces to the earlier equation based upon the simplified model of Figure 5(B). As we did before, we can examine this equation for the search time to see if we can understand where it takes its minimum with respect to the catastrophe rate. To that end, we evaluate

$$\frac{d⟨t⟩}{d{k}_c}=-\frac{1}{k_c^2}\left[\frac{e^{k_c\tau }}{f_r}-1\right]+\frac{1}{k_c}\left[\frac{\tau }{f_r}{e}^{k_c\tau }\right]=0$$ (38)

which simplifies to the condition

$$\frac{1}{k_c^2}\left[\frac{e^{k_c\tau }}{f_r}-1\right]=\frac{\tau }{k_c{f}_r}{e}^{k_c\tau }.$$ (39)

If we introduce the notation $x=k_c\tau$, then this condition simplifies further to

$$f_r=e^x\left(1-x\right).$$ (40)

In Figure 9(B), we plot both sides of this equation and note that the solution corresponds to the point of intersection of the two curves, the horizontal line representing the left side, $f_r=$ constant and the curve corresponding to the right side $e^x(1-x)$. However, since we know that $f_r\ll 1$ because the microtubules polymerizing in the “right” direction is rare, this licenses defining the small parameter $\epsilon$ via $x=1-\epsilon$ resulting in

$$f_r=e^{(1-\epsilon )}(1-(1-\epsilon ))$$ (41)

Now because of the smallness of the parameter $\epsilon$, we can Taylor expand the exponential resulting in the very simple result

$$\epsilon \approx \frac{f_r}{e}.$$ (42)

Recalling that $x=k_c\tau =1-\epsilon$, we now have an approximate value for the optimal catastrophe rate given by

$$k_c\approx \frac{1}{\tau }\left(1-\frac{f_r}{e}\right).$$ (43)

Given the fact that nearly all directions are “wrong” this implies that $f_r\ll 1$ (e.g. we used $f_r=0.05$ in our plots), we end with the very simple result that the optimal catastrophe rate corresponds to $k_c\approx 1/\tau$.

Note that our treatment of the dynamics thus far has considered trajectories in which the lifetimes associated with the various processes were all average lifetimes. A more careful treatment of the exploratory dynamics would sum over all possible lifetimes for each and every step. This would mean that instead of considering every right or wrong trajectory as having a lifetime, $t_r$ or $t_w$, respectively, we would sum over every allowed lifetime. This would turn our sums over trajectories into integrals over trajectories, but ultimately would not change the results obtained above.

4.3. Molecular Control of Exploratory Dynamics

The description of exploratory dynamics considered thus far in the paper is decidedly incomplete. There is a whole pageant of accessory proteins that are part of the process of chromosomal search and capture that we have not mentioned at all since the primary focus of our paper is to focus on general ideas about exploratory dynamics. Several excellent reviews give a scholarly assessment of the complexities of the process of chromosomal search [37, 67]. Further, several earlier theoretical approaches examined precisely these kinds of elaborations on the most naive version of the search and capture model as exploratory dynamics [34, 36]. The way we think about the many molecular interventions implicated in the low error rate construction of the spindle of dividing cells [68] is that the parameters $k_c$ and $v$ used throughout our discussion are not fixed, but rather are subject to molecular control. For example, we know that there are a variety of proteins that can either speed up or hinder the polymerization of cytoskeletal filaments [67], which the model accounts for by having the rate $v$ be a function of concentration and state of modification of these regulatory proteins.

This brief section will focus on how parameters such as the catastrophe rate $k_c$ and the microtubule growth rate $v$ might be controlled by specific molecular interventions. One reason for our interest in this topic is because we wonder how the search time depends upon the size of the cell itself, a topic that was examined carefully by Wollman et al. [36]. In particular, motivated by the idea that in some organisms during embryonic development there are reductive cell divisions which continually reduce the size of the cells. This then has the effect of changing the distance between chromosomes and the centrosome, which suggests that the search time will change as well, being shorter in smaller cells. Figure 10 illustrates this idea by showing how the solid angle cone that characterizes growth in the “right” direction depends upon cell size. As cells get smaller, the area of the correct solid angle becomes a larger fraction of the total area.

Figure 10.

The geometry of chromosome search and cell size. (A) For a large cell size, the cone of successful directions is characterized by the probability $f_r=a/4\pi R^2$. (B) For a smaller size, $f_r$ is larger.

To estimate how cell size, and hence distance between centrosome and kinetochore, alters the search time, we begin by considering that the size of the kinetochore can be measured by an area $a$ (independent of the cell size) as shown in Figure 10. Hence, the fraction of trajectories which are starting in the right direction is given by

$$f_r=\frac{a}{4\pi R^2},$$ (44)

where $R$ is a measure of the cell size or the distance from centrosome to kinetochore. In light of this parameter, we can rewrite the search time given in eqn. 37 as

$$⟨t⟩=\frac{1}{k_c}\left(\frac{4\pi R^2}{a}{e}^{R/\lambda }-1\right),$$ (45)

where we have introduced the length scale $\lambda =v/k_c$ which emerges naturally as a key parameter in the problem. Specifically, $\lambda$ enjoys the interpretation of the typical length grown by a microtubule before catastrophizing, so $R/\lambda$ is a dimensionless ratio characterizing how large the cell size is compared to typical growth excursions of the microtubule. An immediate and simple consequence of this model is that the search time is exponentially sensitive to changes in cell size, assuming that $R/\lambda$ is large. In the opposite case, when $R/\lambda$ is small, the exponential is roughly equal to one. In this case the search time grows quadratically with cell size. Either way, the simple model for kinetochore search we have introduced predicts that the search time will depend sensitively on the cell size.

Our examination of the simple model of kinetochore search is motivated primarily by the goal of revealing a larger set of problems that we feel are usefully described by the ideas of exploratory dynamics. In the specific context of chromosome capture the inadequacies of this model have been examined previously [36]. In particular, the model described above predicts search times that are significantly longer than those measured experimentally [36]. We now review the approaches that have been taken to account for discrepancies with what is known experimentally. Here to demonstrate the utility of even this simple a model we start from the large body of literature, very well reviewed in [37, 67], which explores the spatial and temporal scaling of the spindles of dividing cells. As we will touch on below, the simplest overarching observation is that the sensitive dependence of search time on cell size described above is not consistent with the observed behavior.

Remarkably, in a variety of experiments on developing embryos and encapsulated Xenopus egg extracts, the search time is roughly constant (or more precisely, the spindle assembly time), independent of cell size [69, 70]. For a concrete example, see Figure 6 of [69]. In the embryo, different cell sizes result from reductive cell divisions, while in the synthetic system the experimentalist controls the size of the encapsulating vesicle, which plays the role of a synthetic cell. These experimental observations on the lack of a cell-size dependence to the search time are at odds with the result of eqn. 45. This fact invites us in the remainder of this section to consider how the two rate parameters, $v$ and $k_c$, should scale with the cell size [71, 34, 36, 67] so that the search time will be independent of $R$. As cogently described by Belmont et al. [71], the dynamic instability model features four rate parameters: the rates of polymerization and depolymerization, and the rates of catastrophe and rescue. In the highly schematized model described here, we only preserved two of those four rates, namely, the catastrophe rate $k_c$ and the polymerization rate associated with the speed of growth $v$, since we ignored rescues altogether.

Based on eqn. 45, to prevent the search time from growing exponentially with cell size (when the cell is big compared to typical microtubule excursions so the dimensionless parameter $R/\lambda$ is large), the typical length the microtubules reach before collapsing must scale linearly with cell size. That is, $\lambda =v/k_c\propto R$, where the $\propto$ symbol represents the idea that if the quantity on the right hand side (i.e. $R$) were to be doubled or halved, the same fate would befall the quantity on the left hand side (i.e. $v/k_c$). We are still left with the term that is quadratic in $R$ and in our simple model we would further require $k_c\propto R^2$ in order for the search time to be independent of $R$. It is interesting to note that the only way to satisfy both scaling requirements is for the speed of polymerization to be proportional to $R^3$, which is the same as saying that it is proportional to the cell volume. This scaling is exactly what was found experimentally in [69], and corresponds to a constant speed of polymerization per cell volume.

Another parameter that we have thus far not touched upon is the number of microtubules. Thus far, the entirety of our discussion has knowingly focused on the single microtubule case. But of course, as is immediately obvious in the most cursory examination of dividing cells, there are many microtubules present in the spindle [72]! More recently a set of beautiful experiments revealed that the scaling of spindle size with cell size during embryonic development is due to a finely-regulated number of microtubules in the spindle [70]. While our simple model does not consider this parameter, the presence of multiple microtubules will decrease the search time in proportion to the number of microtubules, assuming that each one searches independently; this picture results in a revised search time for at least one microtubule to reach the kinetochore of,

$$⟨t⟩=\frac{1}{N{k}_c}\left(\frac{4\pi R^2}{a}{e}^{\frac{R}{\lambda }}-1\right),$$ (46)

where $N$ is the number of microtubules. Interestingly, this new twist allows us to return to the problem of how the search time can be independent of cell size. If the number of microtubules $N$ scales with the cell’s surface area ($N\propto R^2$), the quadratic prefactor’s dependence on $R$ drops out. Indeed, careful measurements have revealed precisely this kind of scaling [70]. The whole prefactor would lose all cell size dependence if the catastrophe rate $k_c$ were constant with cell size, $k_c=$ constant. Next, to further abolish the exponential factor’s variation with cell size in eqn. 46, either the cell is small (giving $R/\lambda \ll 1$), or—as discussed previously—$\lambda =v/k_c\propto R$, which (with constant catastrophe rate) compels the microtubule growth rate $v$ to vary linearly with cell size $R$, namely $v\propto R$. These distinct biophysical relationships make precise predictions amenable to experimental scrutiny.

While we find these evocative comparisons between experimental data and the simple model of exploratory dynamics intriguing [34, 36], much more work needs to be done to specifically measure how the key parameters of models such as these scale with cell size (and other variables of physiological state). Here we only wish to point out how a simple model such as this points to fresh and specific questions about regulation of assembly of microtubules, which can be addressed by experiments.

4.4. Exploratory Power

At first glance, the analogy shown in Figure 2 in which a railroad between Boston and New York is constructed by randomly laying down track seems completely ridiculous. When viewed at human engineering scales, exploratory dynamics leads immediately to questions of efficiency, costs and benefits, words that are likely too imprecise for discussing the molecular and cellular scale events that use this mechanism in living organisms. A fascinating question that arises in the context of these problems in exploratory dynamics is how much they cost energetically. Post-translational modifications imply a steady-state energy flux [73]; adaptation of the chemotaxis receptors a steady-state energy flux in the form of methylation [74]; and the repeated cycles of growth and shrinkage in microtubule systems imply costs both in the form of cytoskeletal polymerization and of the motors that attend this process [75]. Recent calorimetry measurements using microtubule-motor systems found a remarkably large heat dissipation in comparison with the larger scale mechanical motions of these collectives [76]. In the remainder of this section, we perform a succession of calculations as shown in Figure 11 and summarized in Figure 12, beginning with the energetics of the exploratory dynamics of a single microtubule, followed by crude estimates of the cost of all of the microtubules in a spindle and then a series of comparisons of these powers and energies to other key processes in the dynamics of a cell.

Figure 11.

The power of biological processes. (A) Comparison of the bioenergetics of exploratory dynamics in chromosome search and capture with other key cellular processes. In each case, the $\mathrm{\Delta }G$ shows the free energy cost of a unit process such as the addition of a tubulin monomer to a growing microtubule or the cost of transporting a single ion across the cell membrane. The power is given in turn by $J_{\text{process}}\mathrm{\Delta }{G}_{\text{process}}$, where $J_{\text{process}}$ is the flux of the process of interest. (B) Power to maintain cellular processes, as inferred by continuous chemostat measurements, as a function of cell volume; proportional scaling (acceptably capturing these data) is denoted by the gray line [77]. (C) Power densities per volume ensuing from the data in (B), reflecting a highly typical power density of $⟨\rho ⟩\approx 9\times {10}^5\mathrm{A}\mathrm{T}\mathrm{P}/\left(\mu {\mathrm{m}}^3\mathrm{s}\right)$ for overall cellular maintenance.

Figure 12.

Summary of the power required for various cellular processes per volume. The goal of the figure is to compare the cost of microtubule search and capture by exploratory dynamics with the cost of other cellular processes. Note that we use the symbol $f$ for “few” with the arithmetic rule that $f\times f\approx 10$ [84].

For the exploratory dynamics of microtubule search and capture described in previous sections, we can make an elementary estimate of how many ATPs or GTPs have been consumed. The simplest estimate asks us to examine the energetic cost of repeated cycles of microtubule polymerization and depolymerization, ignoring for now the attendant costs associated with the many molecular motors that are part of the process as well. The starting point of the estimate is the recognition that every tubulin dimer added to the growing microtubule costs a GTP as shown in Figure 11(A) and is labeled as $\mathrm{\Delta }{G}_{\text{search}}$. The total power is given as $J_{\text{search}}\mathrm{\Delta }{G}_{\text{search}}$, where $J_{\text{search}}$ is the number of tubulin additions per unit time. As a result, the rate of GTP hydrolysis can be estimated as

$$J_{\text{search}}=\frac{n_{\text{proto}}v}{l_{\text{mono}}}$$ (47)

where $n_{\text{proto}}$ is the number of protofilaments in a microtubule, $v$ is the speed of microtubule growth in units of length/time and $l_{\text{mono}}$ is the length of a tubulin dimer. If we assign a characteristic energy of ${\epsilon }_{GTP}=20k_B\mathrm{T}$ per GTP hydrolysis, this tells us that the power is given by

$$\text{power}\approx J_{\text{search}}{\epsilon }_{GTP}\approx {\epsilon }_{GTP}\frac{n_{\text{proto}}v}{l_{\text{mono}}}.$$ (48)

If we approximate the relevant numbers such as $n_{\text{proto}}\approx 10,v\approx 10\mu \mathrm{m}/\mathrm{m}\mathrm{i}\mathrm{n}$ and $l_{\text{mono}}\approx 5nm$, we find that the power for a single microtubule to perform exploratory behavior is roughly 500 ATP/s.

The results given above examine the energetics of a single microtubule doing repeated instances of exploratory search and capture. But as any microscopy image will reveal, this process involves hundreds to thousands of microtubules [36, 78] and hence our estimates both for the time scale of search and capture, and for the energetics and power need to be amended to account for this effect. As a result, we estimate the power expended during the search and capture process is between 5 × 10³ and 5 × 10⁴ ATP/s.

With this equation in hand, we can also then estimate the total energy expended in the search process given by the power times the elapsed time. If we now recall eqn. 37 which tells us on average how long the search process lasts, we find that the total energy is given by,

$$\text{energy}=\text{power}\times ⟨t⟩={\epsilon }_{GTP}\times J_{\text{search}}\times ⟨t⟩=\frac{n_{\text{proto}}v{\epsilon }_{GTP}}{l_{\text{mono}}}\frac{1}{k_c}\left[\frac{e^{k_c\tau }}{f_r}-1\right].$$ (49)

Given these various estimates, it is of great interest to compare them to other estimates of the energy budgets of cells. A critical question that we come back to over and over in the context of biological problems is how to judge whether a given process is “costly” or not. The starting point for answering that question is to take stock of the entire energy budget of a cell. A systematic study was made of the “biosynthetic cost of a gene” which attempted to work out the energetics of all of the key processes that take place within cells of all types [77]. In their Figure 1 whose data we reproduce here as Figure 11(B), they report on the maintenance cost with a value of roughly f × 10⁹ ATP/(μm³ hr), which translates into roughly 10⁶ ATP/(μm³ s). This characteristic power density, inferred from extrapolated chemostat measurements on bacteria and amoebozoa, also aligns with typical metabolic rates of various human tissue cells, ranging from the slower metabolism of resting erythrocytes (≈ 8.5 × 10⁻¹⁷W/μm³) to the rapid metabolism of T-cell lymphocytes in antigen responses (≈ 6.5 × 10⁻¹³W/μm³) [79], where we invoked the characteristic scale that one ATP hydrolysis liberates 10⁻¹⁹ J. Accordingly, a power density of about a million ATPs consumed per second per cubic micron serves as a very useful baseline power for the entirety of the processes associated with cellular maintenance. To develop specific intuition, we consider several particular processes that are constantly in play in cells of different types and that will give us a sense of the scale of the cost of the search and capture process.

Cells constantly remake proteins to compensate for their active or spontaneous degradation. The energetic cost to resynthesize proteins can be formidable, as a simple estimate reveals. Specifically—assuming that the cellular abundance of proteins is maintained at a steady-state—the stable copy number $N$, overall resynthesis rate $q$, and degradation rate $k$, and typical lifetime $\tau$ can be related by Little’s theorem in queuing theory [80], $N/\tau =q=k$. Since mammalian cells typically have concentrations $c\approx \mathrm{a}$ few million proteins per cubic micron [81]; typical protein turnover rates in bacteria [82] and eukaryotes [83] are both of order $k\approx f\times {10}^{-5}{\mathrm{s}}^{-1}$ (consistent with measured typical half lives of a few hours and where we remind the reader that we adopt the convention that $f=\mathrm{f}\mathrm{e}\mathrm{w}$ [84]); each protein is $\ell \approx \mathrm{a}$ few hundred amino acids long and each peptide bond requires $\eta \approx 4$ ATP equivalents to polymerize [85], protein resynthesis to maintain the standing proteome pool should demand of order $ck\ell \eta \approx {10}^5$ ATP/(μm³ s). This significant cost can be within a factor few of the typical total power maintenance density of the cell we referred to above and is clearly much larger than the cost of the microtubule search and capture dynamics. Proteins have energetic demands in other ways as well. The post-translational modification of proteins can also demand modest to appreciable energy expenditures in a cell. Classic measurements and modeling of cyclic (de)phosphorylation cascades [86] report that typical operating conditions for these cascades demand at most of order 1 mM ATP/min, corresponding to at most a few × 10⁵ ATP/μm³ s.

Another fascinating, crucial, energetically-costly process that must be carried out at all times is the maintenance of the membrane potential as seen in Figure 11(A). In particular, the difference in ionic concentrations across membranes will be lost in the absence of active processes to maintain those gradients. Specifically, despite a very small permeability to such ions, there is a leak current of ions across lipid bilayer membranes. In particular, if we assume a typical permeability of $p={10}^{-5}\mathrm{n}\mathrm{m}/\mathrm{s}$ [87] and a concentration difference across the membrane of 100 mM, then the flux is given by

$$J_{ion}=p\mathrm{\Delta }c\approx 1\frac{\text{ion}}{\mu {\mathrm{m}}^2\mathrm{s}}.$$ (50)

For a eukaryotic cell with a surface area of $f\times {10}^3{\mu \mathrm{m}}^2$, this corresponds to $f\times {10}^3\mathrm{ions}/\mathrm{s}$. The cost to transport one such ion up its gradient is roughly $\mathrm{\Delta }{G}_{\mathit{ion}}=q\mathrm{\Delta }V+k_{B}T\mathrm{l}\mathrm{n}\left(c_{\text{in}}/c_{\text{out}}\right)\approx 1$ ATP as seen in Figure 11(A), indicating that the power of this process is about 10³ ATP/s, a tiny fraction of the overall power of the cell, but quite comparable to the power of a single microtubule engaging in the search and capture process.

Membrane potentials are not the only gradients within cells. Protein gradients of various kinds are present in cells and embryos of all types. A particularly well studied example that might give us intuition for power and energy scales of such gradient formation is offered by the eukaryotic flagellum. Here the process of intraflagellar transport (IFT) is characterized by molecular motors that carry cargo to the tip of the flagellum [88] as seen in Figure 11(A) with the cost of a single motor step given by $\mathrm{\Delta }{G}_{\mathit{step}}={\epsilon }_{ATP}$. This process results in a linear concentration gradient of molecular motors within the flagellum, some large fraction of which are attached to microtubules and thereby consuming ATP [89]. Rough estimates yield a power of the form

$$\text{power}=\frac{c_0+c_{\mathrm{t}\mathrm{i}\mathrm{p}}}{2}L\left(\frac{v}{\delta }\right){\epsilon }_{ATP},$$ (51)

where $L$ is the length of the flagellum separating base from tip; $c_0$ and $c_{\text{tip}}$ are respectively the concentrations of motors at the base and tip; and $\delta$ is the step size of the motor. Using parameters reported by the experiments of ref. [89] yields 5 × 10⁵ ATP/s, or 5 × 10⁴ ATP/μm³ s (on the basis of flagellar volume).

The process of membrane trafficking is constantly rearranging the internal and external membranes of cells in a highly energetically costly way. As seen in Figure 11(A), every time a vesicle is created, a simple estimate implies a cost of $\mathrm{\Delta }{G}_{\text{traffic}}\approx 10-20$ ATPs corresponding to the $250-500k_{B}T$ cost of each such vesicle [90]. As was noted in a review on vesicles and trafficking: “A fibroblast kept in resting conditions in a tissue culture plate internalizes an amount of membrane equivalent to the whole surface area of the cell in one hour” [91]. If we estimate the area of such a cell as $f\times {10}^3{\mu \mathrm{m}}^2$ and consider 100 nm vesicles, the area of each such vesicle is roughly $f\times {10}^4{\mathrm{n}\mathrm{m}}^2$, meaning roughly 6 × 10⁴ vesicles are created every hour, or roughly 20 every second. At a cost of 10–20 ATP per vesicle, this amounts to a power of roughly $f\times {10}^2\mathrm{A}\mathrm{T}\mathrm{P}/\mathrm{s}$.

All of these estimates—collectively summarized in Figure 12—form a useful and fascinating backdrop for trying to better understand the energy budgets of cells and specifically, the costs of exploratory dynamics. For a process such as microtubule search and capture which results in the high-fidelity segregation of chromosomes, even in light of such estimates, it is not clear how to decide if a given power (or energy) is costly since chromosome segregation with errors can be deadly, the highest cost of all.

5. Transcription Factor Search for Binding Sites as Exploratory Dynamics

The problem of how transcription factors search and find their binding sites provides another illuminating example of exploratory dynamics. When we write simple chemical reaction schemes for describing the binding and unbinding of transcription factors to their cognate sites as is shown in Figure 13(A) they mask a much more complicated underlying exploratory reality. As seen in Figure 13(B), for a transcription factor to find its binding site and bind to it, the protein needs to traverse the complex intracellular environment including the complex polymeric geometry of the chromosomal DNA itself. Mechanistically, this exploratory dynamics takes place as shown schematically in Figure 14, where a combination of 1D diffusion along the DNA punctuated by episodes of 3D diffusion result in a search time that, in principle, is faster than either approach by itself. We also note as was said above, that this mechanism shares many features of chromosome search and capture, but unlike that case, does not involve any energy expenditure.

Figure 13.

Searching for a binding site on the chromosome. (A) A particular binding site for a transcription factor controls the expression of some gene of interest and our goal is to replace the simple schematized reaction on the left with a picture of the dynamics that acknowledges how the kinetics depends upon the complex geometry of the chromosome. (B) The transcription factor has to “search” throughout the entire genome.

Figure 14.

Simple model of combined 1D and 3D transcription factor search for a specific binding site. A transcription factor diffuses along the DNA for an approximate time ${\tau }_{1\mathrm{D}}$ after which it falls of the DNA and performs three-dimensional diffusion for a time scale ${\tau }_{3\mathrm{D}}$. The cycle is then repeated until the transcription factor reaches its binding site.

Our approach to this problem essentially imitates a beautiful treatment of the problem by Hachmo and Amir [32] though it is important to note that a broad swath of theoretical works have shed light on this fascinating topic of “facilitated diffusion” [28, 29, 30, 31]. We invoke a model where a transcription factor can “slide” along the DNA and also jump between genomically distant but spatially proximal parts of the genome. We note that, in addition to now classic in vitro bulk studies [29, 30], there have been a variety of beautiful in vitro studies using the tools of single-molecule biophysics to watch proteins as they bind to and translocate along DNA [92, 93, 94, 95]. Similarly, a new generation of single-molecule experiments has made it possible to query this same search process in the much more complex in vivo environment of living cells [96, 97]. For our purposes, all of this work provides a rich canvas for further elaborating the general idea of exploratory dynamics.

In Figure 14 we examine this canonical model of “facilitated diffusion” that combines 3D diffusion and 1D sliding along the DNA. In this model, a transcription factor diffuses along the DNA with a diffusion constant $D_{1\mathrm{D}}$ until it falls off the DNA after a time ${\tau }_{1\mathrm{D}}$. Once in the cytoplasm, the transcription factor undergoes 3D diffusion for a time ${\tau }_{3\mathrm{D}}$ until it lands again on a different part of the DNA. Our goal is to estimate the search time under this composite exploratory dynamics which can be thought of as one-dimensional localized search for the binding site coupled with a “reset” mediated by three-dimensional diffusion and indicated schematically in Figure 15 using the evocative language of variation (diffusion) and selection (binding).

Figure 15.

Exploratory dynamics of transcription factor search as an example of variation and selection. We can think of transcription factor search as a succession of search events along the DNA, punctuated by reset events of three dimensional diffusion. “Selection” occurs when the transcription factor binds to its target site.

We begin by examining the probability of the transcription factor finding its target solely through 1D diffusion. The key idea in formulating the estimate is that during the time the transcription factor engages in 1D diffusion, it covers a distance given approximately by

$$L_{1\mathrm{D}}=\sqrt{D_{1\mathrm{D}}{\tau }_{1\mathrm{D}}}.$$ (52)

If the total length of DNA to be searched is $L_{\text{genome}}$ and the transcription factor lands on a random DNA position, then the probability $p$ that the transcription factor will find its specific binding site during an episode of 1D diffusion that lasts a time ${\tau }_{1\mathrm{D}}$ is given by the fraction of the genome explored during a sliding period. This argument leads to the estimate

$$p=\frac{L_{1\mathrm{D}}}{L_{\text{genome}}}=\sqrt{\frac{D_{1\mathrm{D}}{\tau }_{1\mathrm{D}}}{L_{\text{genome}}^2}}.$$ (53)

Now we want to compute the probability of finding the site during repeated cycles of 1D and 3D diffusion. First, as shown in eqn. 53, the probability of finding the binding site after the first sliding period, corresponding to zero 3D hops, is given by

$$p_0=p=\sqrt{\frac{D_{1\mathrm{D}}{\tau }_{1\mathrm{D}}}{L_{\text{genome}}^2}}.$$ (54)

This takes approximately a time

$$T_0={\tau }_{1\mathrm{D}}.$$ (55)

If no site is found during the first sliding interval, there is a chance that the site will be found during the next sliding period. This probability is given by

$$p_1=\left(1-p\right)p,$$ (56)

where $1-p$ is the probability of the first 1D search being unsuccessful, and $p$ the probability of succeeding in the next 1D search. The time it takes for a binding site to be found in this second attempt, after one 3D hop, is given by

$$T_1={\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}+{\tau }_{1\mathrm{D}},$$ (57)

where the first term corresponds to the time of the first 1D search, the second term to the time of the subsequent 3D search, and the last term to the time of the second 1D search, when the site was found. Similarly, if the binding site is found after the second 3D hop, the transcription factor will have to have failed in its search during the the previous two 1D searches such that

$$p_2=(1-p)(1-p)p$$ (58)

and

$$T_2={\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}+{\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}+{\tau }_{1\mathrm{D}}.$$ (59)

We see a clear pattern emerge, namely, for the case involving $i$ episodes of 3D search the probability of finding the site is given by

$$p_i=(1-p{)}^{i}p$$ (60)

and the corresponding search time is

$$T_i=\left(i+1\right){\tau }_{1\mathrm{D}}+i{\tau }_{3\mathrm{D}}.$$ (61)

As a result, we can rewrite the average time for the transcription factor to find its binding site as

$$T=\sum _{i=0}^{\mathrm{\infty }}{T}_i{p}_i,$$ (62)

where $p_i$ is the probability of the transcription factor finding its site after $i$ 3D hops, and $T_i$ is the time it takes to perform the trajectory with $i$ episodes of 3D diffusion. We can write this all out explicitly as

$$T=\sum _{i=0}^{\mathrm{\infty }}{T}_i{p}_i=\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}p\left(\left(i+1\right){\tau }_{1\mathrm{D}}+i{\tau }_{3\mathrm{D}}\right).$$ (63)

We note that the mathematical structure of this problem is essentially identical to our calculations in the previous section where we also found that our exploratory trajectories have probabilities described by the geometric distribution. This means that we can exploit much of the same algebra we did in the previous section.

To start, we expand the terms in parentheses in eqn. 63 to obtain

$$T={\tau }_{1\mathrm{D}}p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i+{\tau }_{1\mathrm{D}}p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^i+{\tau }_{3\mathrm{D}}p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i.$$ (64)

We now regroup the terms and arrive at

$$T=\left({\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}\right)p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i+{\tau }_{1\mathrm{D}}p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^i.$$ (66)

We note that the second sum is of the form of a geometric series which can be summed once again to yield

$$\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^i=\frac{1}{1-(1-p)}=\frac{1}{p}$$ (66)

As a result, we have

$$T=\left({\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}\right)p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i+{\tau }_{1\mathrm{D}}p\frac{1}{p}=\left({\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}\right)p\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i+{\tau }_{1\mathrm{D}}.$$ (67)

To carry out the remaining sum, we invoke the use of the derivative trick we originally introduced in eqn. 9. Namely, we write the sum as

$$\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^{i}i=-\left(1-p\right)\frac{d}{dp}\left(\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^i\right),$$ (68)

where the extra $-(1-p)$ term takes care of the fact that, after taking the derivative, the $(1-p{)}^i$ term inside the summation would otherwise become $-(1-p{)}^{i-1}$. We again have the familiar geometric sum that can be carried out to yield

$$-\left(1-p\right)\frac{d}{dp}\left(\sum _{i=0}^{\mathrm{\infty }}(1-p{)}^i\right)=-\left(1-p\right)\frac{d}{dp}\frac{1}{p}=-\left(1-p\right)\frac{\left(-1\right)}{p^2}=\frac{1-p}{p^2}.$$ (69)

As a result, eqn. 65 becomes

$$T=\left({\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}\right)p\frac{1-p}{p^2}+{\tau }_{1\mathrm{D}}p\frac{1}{p}=\left({\tau }_{1\mathrm{D}}+{\tau }_{3\mathrm{D}}\right)\frac{1-p}{p}+{\tau }_{1\mathrm{D}}.$$ (70)

This can be simplified to the form

$$T=\frac{{\tau }_{1\mathrm{D}}}{p}+\frac{{\tau }_{3\mathrm{D}}}{p}-{\tau }_{3\mathrm{D}}.$$ (71)

Now, we assume that $p\ll 1$, meaning that it will always take many cycles of 1D and 3D exploration to find the binding site. Mathematically, this means ${\tau }_{3\mathrm{D}}/p\gg {\tau }_{3\mathrm{D}}$ such that we can ignore the last ${\tau }_{3\mathrm{D}}$ term in the expression. In light of this approximation, we find a total search time of the form

$$T=\frac{{\tau }_{1\mathrm{D}}}{p}+\frac{{\tau }_{3\mathrm{D}}}{p}.$$ (72)

To understand the implications of the calculation that led to the total search time shown in eqn. 72, we recall the definition of $p$ introduced in eqn. 54. As a result of this definition, we can use the formula for $p$ to rewrite our result for the search time as

$$T=\sqrt{\frac{{\tau }_{1\mathrm{D}}{L}_{\text{genome}}^2}{D_{1\mathrm{D}}}}+{\tau }_{3\mathrm{D}}\sqrt{\frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}{\tau }_{1\mathrm{D}}}},$$ (73)

Further, we will non-dimensionalize the equation by measuring the search time in units of the 3D exploration time ${\tau }_{3\mathrm{D}}$, leading to

$$\frac{T}{{\tau }_{3\mathrm{D}}}=\bar{T}=\sqrt{\frac{{\tau }_{1\mathrm{D}}}{{\tau }_{3\mathrm{D}}}\frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}{\tau }_{3\mathrm{D}}}}+\sqrt{\frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}{\tau }_{1\mathrm{D}}}},$$ (74)

where we have defined $\bar{T}=T/{\tau }_{3\mathrm{D}}$. In addition, we define the non-dimensional 1D search time ${\bar{\tau }}_{1\mathrm{D}}={\tau }_{1\mathrm{D}}/{\tau }_{3\mathrm{D}}$ measured in units of the 3D search time, to arrive at

$$\bar{T}=\sqrt{{\bar{\tau }}_1\frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}{\tau }_{3\mathrm{D}}}}+\sqrt{\frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}{\tau }_{3\mathrm{D}}{\bar{\tau }}_{1\mathrm{D}}}}.$$ (75)

We see that both square roots in the equation have a factor

$$\alpha =\sqrt{\frac{D_{1\mathrm{D}}{\tau }_{3\mathrm{D}}}{L_{\text{genome}}^2}},$$ (76)

corresponding to the fraction of the genome the transcription factor would have explored by 1D diffusion during the time it was actually diffusing in 3D. As a result, the non-dimensionalized time can be written as

$$\bar{T}=\frac{1}{\alpha }\left(\sqrt{{\bar{\tau }}_{1\mathrm{D}}}+\sqrt{\frac{1}{{\bar{\tau }}_{1\mathrm{D}}}}\right).$$ (77)

This expression for the search time makes the striking prediction that there is an optimal balance between the time the transcription factor spends searching in one dimension and three dimensions. Specifically, when the one-dimensional search time is relatively long (${\bar{\tau }}_{1\mathrm{D}}\gg 1$), the first term $\sqrt{{\bar{\tau }}_{1\mathrm{D}}}$ dominates the overall search time, implying that the overall search time grows as $\bar{T}\sim {\bar{\tau }}_{1\mathrm{D}}^{1/2}$. Otherwise (for short one-dimensional search times ${\bar{\tau }}_{1\mathrm{D}}\ll 1$), the second term $1/{\bar{\tau }}_{1\mathrm{D}}\gg 1$ dominates, implying in turn that the overall search time shrinks as $\bar{T}\sim {\bar{\tau }}_{1\mathrm{D}}^{-1/2}$. The crossover between these behaviors gives a minimum overall search time, when these terms are comparable.

We can quickly foresee that the value of the one-dimensional search time ${\bar{\tau }}_{1\mathrm{D}}$ that minimizes the overall search time is located at ${\bar{\tau }}_1\mathrm{D}=1$ by remembering the arithmetic mean-geometric mean inequality, which states that for two nonnegative values $a,b,\frac{a+b}{2}\ge \sqrt{ab}$; taking $b=1/a$ gives $a+\frac{1}{a}\ge 2$, where the optimal value of equality requires $a=1/a$, or $a=1$. (Amusingly, the same optimization problem emerges when asking for the side lengths of a rectangle of unit area that has the smallest perimeter.) Letting $a$ be $\sqrt{{\bar{\tau }}_{1\mathrm{D}}}$ translates this result to mean that smallest overall search time occurs when $\sqrt{{\bar{\tau }}_{1\mathrm{D}}}=1$. This optimum may also be seen by taking the derivative of $\bar{T}$ with respect to ${\bar{\tau }}_{1\mathrm{D}}$

$$\frac{d\bar{T}}{d{\bar{\tau }}_{1\mathrm{D}}}=\frac{1}{\alpha }\frac{d}{d{\bar{\tau }}_{1\mathrm{D}}}\left(\sqrt{{\bar{\tau }}_{1\mathrm{D}}}+\sqrt{\frac{1}{{\bar{\tau }}_{1\mathrm{D}}}}\right)=\frac{1}{\alpha }\left(\frac{1}{2}\frac{1}{{\bar{\tau }}_{1\mathrm{D}}^{1/2}}-\frac{1}{2}\frac{1}{{\bar{\tau }}_{1\mathrm{D}}^{3/2}}\right)=0,$$ (78)

which is satisfied if ${\bar{\tau }}_{1\mathrm{D}}=1$, regardless of the numerical value of the parameter $\alpha$. This result carries the physical meaning that ${\tau }_{1\mathrm{D}}={\tau }_{3\mathrm{D}}$ : the shortest overall search time is reached when the times spent by the transcription factor exploring in one and three dimensions are equal. Equivalently, this optimum corresponds to the case where the ratio of the distance traveled in 1D ($L_{1D}$) and that in 3D ($L_{3D}\equiv \left.\sqrt{D_{3D}{\tau }_{3D}}\right)$ are matched to an appropriate balance of the respective diffusion coefficients, $\frac{L_{1D}}{L_{3D}}=\sqrt{\frac{D_{1D}}{D_{3D}}}$.

Armed with this analytical understanding of the tradeoff between one dimensional searches and three dimensional resets, we now explore where typical cells operate parametrically. Before we can make progress on exploring all of these results numerically, we first need to estimate some of the key parameters that are present in the model. In particular, we need an estimate of ${\tau }_{3\mathrm{D}}$, the time scale for 3D exploration between successive episodes of 1D diffusion along the DNA. Figure 16 offers a naive way to make such an estimate. In Figure 16(B), we consider a transcription factor in a “gas” of binding sites. When the transcription factor is not bound to the DNA, we imagine these binding sites as freely diffusing with diffusion constant $D_{\mathrm{b}\mathrm{s}}$ in the vicinity of the transcription factor, and the transcription factor itself diffuses with a diffusion constant $D_{\mathrm{T}\mathrm{F}}$. In this simple model, the time ${\tau }_{3\mathrm{D}}$ corresponds to the average time for one of these binding sites to reach the transcription factor. To estimate the concentration of this gas of binding sites, we use the fact that there are of order 10⁶ such binding sites in a bacterial genome in a volume of $1{\mu \mathrm{m}}^3$, each one corresponding to a base pair in the genome. We can use this concentration in conjunction with the diffusion limited on-rate [38, 98, to estimate the diffusion time as

$${\tau }_{3\mathrm{D}}=\frac{1}{4\pi \left(D_{\mathrm{b}\mathrm{s}}+D_{\mathrm{T}\mathrm{F}}\right)a{c}_0}.$$ (79)

Figure 16.

Estimating time scales of encounters between a transcription factor and the genome. (A) During episodes of 3D diffusion, the transcription factor diffuses through the cellular environment before encountering the genome. (B) Estimating the time scale of diffusion by using the diffusion-limited on rate idea by pretending that the binding sites themselves are diffusing around. (C) Estimating the time scale of diffusion by pretending that the diffusion takes place within a discrete lattice of binding sites in 3D space.

We note, however, that base pairs do not diffuse independently of each other. Because base pairs are part of the huge polymer that makes up the chromosome, their effective diffusion coefficient is expected to be much lower than that of a free base pair. Indeed, recent in vivo studies in a broad swath of organisms have reported an effective diffusion coefficient of chromosomal loci of the order of 10⁻² μm²/s [99, 100, 101]. As a result, if we take the diffusion constant of a base pair as $D_{\text{bs}}\approx {10}^{-2}{\mu \mathrm{m}}^2/s$ we see that it is much smaller than the diffusion constant of the protein itself ($D_{\mathrm{T}\mathrm{F}}\approx 10{\mu \mathrm{m}}^2/s$) and hence we can safely ignore the contribution of base pair diffusion. If we now impose the size of the target site to be $a\approx f\mathrm{n}\mathrm{m}$ and the concentration of base pairs is approximated by $c_0={10}^6/\mu {\mathrm{m}}^3\approx 1\mathrm{m}\mathrm{M}$, we arrive at ${\tau }_{3\mathrm{D}}\approx f\times {10}^{-6}\mathrm{s}$. Finally, in order to estimate the numerical value of $\alpha$ defined in eqn. 76, we must also determine the 1D diffusion constant describing the sliding of the transcription factor along the DNA. Single-molecule studies have found 1D diffusion constants of the order of $D_{1\mathrm{D}}\approx {10}^5-{10}^6{\mathrm{b}\mathrm{p}}^2/\mathrm{s}\approx {10}^{-2}-{10}^{-1}{\mu m}^2/\mathrm{s}$ [92, 93]. In light of all of these estimated parameters, we find $\alpha \approx f\times {10}^{-7}-{10}^{-6}$.

Using the parameters we have estimated, Figure 17 shows the total search time as a function of the time spent searching in 1D in units of the time spent searching in 3D. The figure affirms the appearance and location of the optimal 1D search time, namely, when the time spent searching in 1D is equal to the time searching in 3D.

Figure 17.

Time for a transcription factor to find its binding site in a model of combined 1D and 3D search. The search time is measured in units of the time spent searching in 3D as a function of the relative time searching in 1D to 3D. The parameter $\alpha$ was chosen to have the value $\alpha ={10}^{-6}$ corresponding to the fraction of the genome that would have been explored by 1D diffusion during the time of a 3D exploration. Note that as the time spent exploring in 1D approaches zero, ${\bar{\tau }}_{1D}\to 0$, clearly the total overall search time $\bar{T}$ diverges, as encountering the correct binding site requires at least some time spent exploring in 1D.

Despite how crude our estimates were, we can now examine whether the transcription factor binding site search dynamics takes a time consonant with the optimal search time which is found when the 1D and 3D search times are equal. To make that determination, we need the 1D exploration time which we can then compare to our estimate of the 3D exploration time that we performed earlier on in this section. In vivo experiments in bacteria have estimated the sliding length of a transcription factor to be of order $L_{1\mathrm{D}}\approx 100$ bp [96]. Given our estimated 1D diffusion constant, this translates into a 1D search time of roughly ${\tau }_{1\mathrm{D}}\approx {10}^{-2}-{10}^{-1}\mathrm{s}$. This value for ${\tau }_{1\mathrm{D}}$ differs by several orders of magnitude from our estimate of ${\tau }_{3\mathrm{D}}$ shown in Figure 16 (B). However, it is important to note that neither the 1D nor the 3D exploration times have been directly measured in vivo and that our numerical estimates depend upon a succession of poorly known parameters. Indeed, recent experiments in mammalian cells paint a different picture. These experiments estimated that a transcription factor spends a similar amount of time in the 1D and 3D exploration modes (75% of the time doing 3D exploration, to be more precise) [97].

A second question we can explore guided by the estimate resulting in Figure 17 is to ask whether this facilitated diffusion mechanism results in faster search times of transcription factors for their specific binding sites than explorations based on a 1D or 3D search alone. To make this possible, we first assign absolute units to the optimal non-dimensional total search time $\bar{T}$ shown in the figure. Specifically, we compute

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}={\bar{T}}_{\mathrm{m}\mathrm{i}\mathrm{n}}{\tau }_{3\mathrm{D}}\approx f\mathrm{s}.$$ (80)

Given in vitro measurements of the 1D diffusion constant of $D_{1\mathrm{D}}\approx {10}^5-{10}^6{\mathrm{b}\mathrm{p}}^2/\mathrm{s}$ already described above, we can estimate a search time for a specific binding site through a genome composed of $L_{\text{genome}}={10}^6$ bp as

$$T_{1\mathrm{D}}\approx \frac{L_{\text{genome}}^2}{D_{1\mathrm{D}}}\approx {10}^6-{10}^7\mathrm{s},$$ (81)

which is clearly several orders of magnitude slower than the search time associated with the facilitated diffusion mechanism. Finally, we estimate the time for a search mechanism based on pure 3D diffusion as

$$T_{3\mathrm{D}}=\frac{1}{4\pi D_{\mathrm{T}\mathrm{F}}^{\mathrm{e}\mathrm{f}\mathrm{f}}a{c}_0},$$ (82)

where $c_0\approx 1\mathrm{n}\mathrm{M}$, the concentration of an individual base pair in a 1μm³ nuclear volume, and where $a\approx f\mathrm{n}\mathrm{m}$. To make this estimate, we also acknowledge that, in this case, $D_{\text{TF}}^{\text{eff}}$ is an effective diffusion TF constant that needs to account for the interactions between the transcription factor and the milieu of the nucleus such as those leading to crowding and to non-specific DNA binding during the search process. Measurements and estimates of this effective diffusion constant remain an open subject. We choose values of $D_{\text{TF}}^{\text{eff}}={10}^{-1}-1{\mu \mathrm{m}}^2/\mathrm{s}$ based on recent in vivo measurements [97]. As a result, we get

$$T_{3\mathrm{D}}=\frac{1}{4\pi D_{\mathrm{T}\mathrm{F}}^{\mathrm{e}\mathrm{f}\mathrm{f}}a{c}_0}\approx f\left(10-{10}^2\right)\mathrm{s}.$$ (83)

This search time is up to two orders of magnitude slower than the optimum search time due to facilitated diffusion $T_{\text{min}}$, suggesting that facilitated diffusion could be at play to speed up the transcription factor binding search process. Regardless of the specific numerical values, which are clearly still a subject of active investigation in the field, we posit that the model shown in this section is yet another intriguing example of the power of exploratory dynamics—in this case in the form of facilitated diffusion—to accomplish and potentially speed up specific biological functions.

6. Discussion

The work of Prof. Erich Sackmann was characterized by a principled approach to understanding the huge diversity of biological processes. In the spirit of his many successes, we used this opportunity to playfully engage with the idea that many biological processes can be thought of as case studies in exploratory dynamics. In contrast with the conventional physics paradigms for dynamics in which, given some initial conditions, the equations of dynamics are used to find the future evolution of the system, we have argued that biology’s unique and necessary “solution” to achieving some target function is through a quite distinct alternative: exploratory dynamics. From this point of view, the classic ideas of variation and selection in the context of biological evolution is hypothesized to have a much broader reach.

To provide some quantitative substance to the hypothesis of exploratory dynamics, we considered several important case studies that have been studied rigorously and successfully for decades, namely, chromosome search and capture, and the search of transcription factors for their target DNA binding sites. Though our paper focused primarily on these particular case studies, we argue that the idea of exploratory dynamics has much broader biological reach, both in terms of the phenomena of living organisms and in the diversity of possible theoretical approaches to these problems.

Ultimately, from a theoretical perspective, we think of questions in exploratory dynamics as falling within the broad province of statistical physics. More specifically, there are a growing number of examples of different approaches to considering the statistical mechanics of microscopic trajectories as opposed to the more traditional statistical mechanics of states. The underlying statistical mechanics described here forced us to consider a number of problems involving the geometric distribution. However, we suspect that there are other mathematical paradigms that will be useful, or even necessary, for considering the broad class of problems in exploratory dynamics. For example, there is a growing literature on the subject of random walks with reset in which the random walker is forced to return to the origin [62].

Clearly, molecular structures, cells, and organisms are often steered by external cues that vary in time, space, and history—cues that themselves are often sculpted by feedbacks from exploratory trajectories, as with ants leaving odor trails. Further, exploratory success probabilities (setting the “life expectancies” of exploratory trajectories) may manifest rich memory effects, for instance when having explored for a while itself increases or decreases the probability of impending success. These complexities urge new mathematical frameworks to be developed and applied that accurately capture the exquisite and complex resulting tradeoffs and consequences of exploration. While it may prove productive to build metaphors between the biological phenomena showing these complexities and algorithmic phenomena such as reinforcement learning or stochastic gradient descent, our central contention is that the unique ingredients, constraints, and goals of biology demand altogether new classes of exploratory behavior whose structure and contrasting examples may also shed light on these more traditional domains.

In summary, we view a fascinating dichotomy between the many distinct and impressive dynamical laws that have formed the backdrop of physics and mathematics for centuries and the unique and necessary exploratory dynamics adopted by many biological processes. Though we often adopt perspectives in which systems move deterministically from some initial condition, from the examples of exploratory dynamics, perhaps one learns that the best plan can be no plan at all.