Abstract. Chromatography is often introduced through smooth peak shapes, retention factors, and plate numbers. Behind those familiar signals, however, individual molecules follow irregular histories: they move with the mobile phase, pause at binding sites, release, rebind, diffuse, and sometimes interconvert while travelling through the column. Stochastic theory provides a natural language for these particle-level processes. In this feature-style article, we outline how classical stochastic chromatography can be extended from simple exponential waiting times to multistep and multipathway processes, including hyper-Erlang residence-time models and mixtures of lognormal binding-site populations. The main emphasis is visual and computational: Markov Monte Carlo simulations can generate individual molecular trajectories that are then animated, making invisible retention mechanisms easier to see. We illustrate this idea with animations produced using chromanim, an open-source GitHub workflow for particle-level chromatography animation. The final example introduces on-column interconversion, where the same stochastic framework produces the characteristic coalesced, plateau-like “Batman” peak shape of dynamic chromatography. These animations are useful not only for modeling and experimental design, but also as teaching tools for explaining why chromatographic peaks have the shapes they do.

Key points

  • Stochastic chromatography interprets retention as an ensemble of random particle histories rather than only as a macroscopic band-broadening equation.
  • Multistep residence processes can be represented by Erlang, hyper-Erlang, or phase-type waiting-time distributions.
  • Heterogeneous columns can be modeled as mixtures of binding-site populations, for example with lognormally distributed residence times.
  • Markov Monte Carlo trajectories can be animated to connect particle-level behavior with detector-level chromatograms.
  • Interconversion during migration provides an intuitive route to visualizing dynamic chromatography and Batman peaks.

Why animate chromatography?

Chromatography is a visual science, but most of what matters is invisible. We see an injected band become one or more detector peaks. We infer retention, selectivity, efficiency, and resolution from those peaks. Yet the underlying events happen at the molecular scale: a molecule alternates between mobile and stationary environments, experiences local heterogeneity, and samples many possible paths before it reaches the detector.

That gap between what we measure and what molecules do is usually bridged by equations. The plate model, rate theory, random-walk models, and stochastic theory all describe different aspects of the same story (Guiochon et al. 2006). For day-to-day chromatography, the macroscopic language is usually enough. A peak has a retention time, a width, an asymmetry, and an area. But when retention becomes dynamic, heterogeneous, overloaded, or chemically coupled, the conventional picture can become less intuitive.

Animations offer a different route. Instead of asking readers to imagine a molecule alternating between phases, we can simulate many such molecules and watch their paths. Some particles fly through the column with only brief stationary-phase visits. Others pause for longer. Some repeatedly bind and release. In dynamic chromatography, some even change identity while moving. The detector peak is then no longer an abstract curve: it is the accumulated arrival-time distribution of many individual histories.

This article introduces a practical animated view of stochastic chromatography. The examples are built around a Markov Monte Carlo simulation workflow and the accompanying GitHub repository chromanim. The aim is not to replace established chromatographic theory, but to make its particle-level interpretation more tangible.

The stochastic view: retention as a random clock

In the simplest stochastic picture (Giddings and Eyring 1955; Felinger 2008), a solute molecule alternates between two states. In the mobile state, it is transported along the column. In the stationary state, it is temporarily immobilized. Retention emerges because each molecule spends only a fraction of its total time in the mobile phase.

A very simple two-state model can be written as:

\[ M \underset{\lambda_S}{\overset{\lambda_M}{\rightleftharpoons}} S \]

where \(M\) is the mobile state, \(S\) is the stationary state, \(\lambda_M\) is the rate of entering the stationary state (i.e. association rate constant), and \(\lambda_S\) is the rate of returning to the mobile state (i.e. dissociation rate constant), as depicted in Figure 1. If the waiting times are exponential, the process is memoryless: a molecule that has already spent a long time in the stationary phase is no closer to release than a molecule that just arrived there.

Figure 1. Schematic view of one particle during migration. When in mobile phase the particle either travels in direction of flow (black arrow) or it adsorbes into the stationary phase (red arrow).

Figure 1. Schematic view of one particle during migration. When in mobile phase the particle either travels in direction of flow (black arrow) or it adsorbes into the stationary phase (red arrow).

This memoryless assumption is mathematically convenient and often surprisingly useful. It leads to tractable first-passage-time distributions and connects naturally with classical stochastic theory. The detector signal is the distribution of times at which particles first reach the end of the column.

However, real retention is not always a single-step process. A molecule may enter a pore, reorient, sample several sub-sites, or pass through a sequence of microstates before it returns to the mobile phase. In those cases, an exponential clock can be too simple.

From one-step to multistep residence times

A straightforward extension is to replace a single exponential stationary-phase residence time with an Erlang waiting time. An Erlang distribution can be interpreted as a sequence of \(n\) exponential substeps:

\[ S_1 \rightarrow S_2 \rightarrow \cdots \rightarrow S_n \rightarrow M \]

The molecule is stationary throughout these substeps, but release is only possible after the final substep. This is still Markovian if the hidden substate is included, but the total stationary residence time is no longer exponential. As \(n\) increases, the residence time becomes more regular and less memoryless.

This is useful because it gives chromatographers a physically interpretable way to tune peak shape. A one-step residence process produces broad stochastic variability. A multistep residence process can represent more structured retention, such as diffusion into and out of pores, sequential binding events, or conformational accommodation at a chiral selector.

Multipathway retention: hyper-Erlang waiting times

Columns are rarely homogeneous. Even a chemically well-defined stationary phase can contain a distribution of local environments: accessible and less accessible sites, shallow and deep pores, strongly and weakly interacting domains, or multiple binding geometries. One molecule may experience a short residence pathway, another a long one.

A hyper-Erlang model captures this idea by mixing several Erlang pathways. At the moment of binding, a particle is assigned to one of several residence-time routes:

\[ S^{(1)}_1 \rightarrow \cdots \rightarrow S^{(1)}_{n_1} \rightarrow M \]

\[ S^{(2)}_1 \rightarrow \cdots \rightarrow S^{(2)}_{n_2} \rightarrow M \]

\[ \cdots \]

Each route has its own number of substeps, rate constants, and probability weight. The total residence-time distribution is therefore a mixture of structured waiting times. This is a compact way to represent multipathway retention without explicitly modeling every microscopic detail of the stationary phase.

In practical terms, hyper-Erlang models can reproduce peak shapes that are difficult to obtain from a single exponential residence process. They can generate shoulders, extended tails, and broader-than-expected arrival-time distributions while still retaining a clear particle-level interpretation.

Binding-site heterogeneity as a lognormal mixture

Another useful extension is to model binding-site heterogeneity continuously rather than as a small number of discrete pathways. For example, stationary-phase residence times or release rates can be drawn from a lognormal distribution. This is attractive because many microscopic physical quantities are positive and multiplicative in origin.

A lognormal mixture can be interpreted as follows: each local binding environment has its own characteristic residence time \(\tau\), and these residence times vary across sites according to

\[ \log(\tau) \sim \mathcal{N}(\mu_\tau, \sigma_\tau^2). \]

Small values of \(\sigma_\tau\) represent a nearly uniform stationary phase. Larger values represent stronger heterogeneity, with a long tail of rare but strongly retaining sites. At the detector level, this can appear as tailing or as delayed subpopulations. At the particle level, it appears as a few molecules becoming temporarily trapped while most continue to migrate.

This distinction is important pedagogically. A chromatographic tail is often described as a nuisance feature of the peak. In the stochastic view, the tail is the visible trace of rare histories.

Markov Monte Carlo simulation at the particle level

The most direct way to connect stochastic theory and animation is to simulate individual particles. In a Markov Monte Carlo model (Montroll and Weiss 1965), each particle carries a current state and position. At each step, the model decides whether the particle remains mobile, binds, releases, changes substate, or, in dynamic chromatography, interconverts to another chemical form.

A minimal simulation cycle is:

  1. Initialize many particles at the column inlet. Usually particles all start at the same position, but some initial distribution may be given to model injection profile.
  2. Assign each particle an initial chemical identity and mobile/stationary state. Usually all particles start in the mobile phase.
  3. Advance time in small increments or by event-driven jumps.
  4. Move mobile particles along the column according to flow and dispersion.
  5. Hold stationary particles in place while their residence clocks evolve. Minor jitter may be given to stationary particles.
  6. Record the first time each particle reaches the detector.
  7. Convert arrival times into a chromatographic signal.
  8. Save particle positions over time for animation.

The resulting detector trace and the animation are two views of the same simulation. The chromatogram summarizes the ensemble. The animation shows the individual histories that produced it.

The chromanim workflow

The animations in this article are generated using the open-source chromanim workflow, available at:

https://github.com/mirzahosseini-arash-semmelweis/chrom_anim

The repository is intended as a practical bridge between stochastic simulation and visual explanation. The code generates particle-level trajectories, converts them into animation frames, and exports video files that can be embedded in HTML manuscripts, dashboards, lectures, or online teaching material.

The goal is deliberately modest: to create simulations that are transparent enough to teach from, but flexible enough to explore mechanisms. Rather than treating the chromatographic peak as the starting point, the workflow starts from particle rules and lets the peak emerge.

Animation 1: ordinary stochastic retention

The first animation shows a simple stochastic retention process. Particles enter the column together, but they do not arrive together. Some remain mostly mobile and progress rapidly. Others spend more time immobilized and lag behind. The detector peak is the histogram-like accumulation of these first-passage events.

Animation 1. Particle-level stochastic retention. Individual particles alternate between mobile migration and temporary stationary-phase residence. The detector peak emerges from the distribution of first-passage times.
Figure 2. Empirical first-passage-time histogram obtained from the animated particle ensemble. This detector-level signal is the chromatogram corresponding to the simulated stochastic particle trajectories shown in Animation 1.

Figure 2. Empirical first-passage-time histogram obtained from the animated particle ensemble. This detector-level signal is the chromatogram corresponding to the simulated stochastic particle trajectories shown in Animation 1.

Figure 2 shows the empirical first-passage-time histogram of the animated particles, which is the detector-level chromatogram generated by the same simulation. In other words, the video visualizes the individual particle histories, while the figure summarizes those histories as the observed peak shape.

This type of animation is especially useful for explaining why a peak has a finite width even when all molecules are injected at the same time. The particles are chemically identical, but their stochastic histories are not identical. Retention time is therefore an ensemble property, not a single deterministic travel time.

What the animation adds beyond the chromatogram

A chromatogram compresses many molecular histories into one curve. That compression is useful, but it can also hide mechanism. In the animation, several familiar chromatographic concepts become visually explicit:

  • Retention appears as intermittent pauses.
  • Band broadening appears as progressive spreading of particle positions.
  • Tailing appears as a small number of delayed particles.
  • Heterogeneity appears as unequal residence behavior among otherwise similar particles.
  • Selectivity appears as different particle populations separating in space and time.

This is why the approach is useful for teaching. Students often learn equations for plate number, retention factor, and resolution before they have a physical picture of what those quantities summarize. Particle animation reverses that order: first watch the process, then interpret the equations.

From retention to reaction: adding interconversion

The same simulation framework can be extended to dynamic chromatography by allowing particles to change identity during migration. This is the key step toward Batman peaks.

Consider two interconverting species, \(A\) and \(B\), such as enantiomers undergoing on-column interconversion:

\[ A \underset{k_{BA}}{\overset{k_{AB}}{\rightleftharpoons}} B \]

Each species may have its own retention behavior. For example, \(A\) and \(B\) may bind differently to a chiral stationary phase and therefore migrate at different average speeds, as depicted in Figure 3. If interconversion is slow relative to separation, two resolved peaks are observed. If interconversion is very fast, the system may collapse toward a single averaged peak. Between these limits, the detector signal can form a plateau or coalesced double peak, often described as a Batman peak.

Figure 3. Schematic view of one enantiomer pair during migration coupled with interconversion.

Figure 3. Schematic view of one enantiomer pair during migration coupled with interconversion.

In a particle-level simulation, this behavior is natural. Each particle has both a position and an identity. During migration, it may switch from \(A\) to \(B\) or from \(B\) to \(A\). After switching, it follows the retention rules of its new identity. The observed chromatogram is therefore not just a separation pattern; it is the outcome of coupled migration and reaction.

Animation 2: interconversion and the Batman peak

The second animation (and its titular Batman chromatogram in Figure 4) shows the same particle-level idea with interconversion turned on. Particles are no longer merely delayed by binding events. They may also switch chemical identity during the run. The resulting signal contains the visual signature of dynamic chromatography: two migrating populations that are coupled by exchange (Sepsey et al. 2018).

Animation 2. Particle-level interconversion during chromatographic migration. Particles can switch between two species with different retention behavior. In the intermediate kinetic regime, the detector signal develops the characteristic plateau-like Batman peak.
Figure 4. Detector-level view of the animated interconversion experiment. The Batman-shaped peak is the empirical first-passage-time distribution of particles that not only undergo stochastic retention, but also switch between two interconverting forms while travelling through the column.

Figure 4. Detector-level view of the animated interconversion experiment. The Batman-shaped peak is the empirical first-passage-time distribution of particles that not only undergo stochastic retention, but also switch between two interconverting forms while travelling through the column.

The value of the animation is that the Batman shape becomes less mysterious. The plateau is not simply an odd peak-fitting problem. It is a kinetic trace. Molecules injected as one species can be detected after spending part of the run as the other species. Some particles switch early, some late, and some more than once. The detector records the resulting distribution of coupled histories.

Why Batman peaks are a good test case

Batman peaks are visually striking, but they are also mechanistically rich. They require a model to account for at least three processes at once:

  1. migration along the column,
  2. reversible retention in the chromatographic system, and
  3. chemical interconversion during migration.

This makes them a useful stress test for stochastic modeling. A purely empirical peak model may reproduce the shape, but it does not necessarily explain how the shape arises. A particle-level stochastic model can show the mechanism directly.

For experimental design, this matters because the shape of a dynamic chromatogram depends on the relative time scales of separation and interconversion. Changing temperature, flow rate, column chemistry, or mobile-phase composition can move the system between resolved, coalesced, and averaged regimes. A simulation can help predict which experimental changes are likely to reveal the kinetics most clearly.

Modeling choices and practical compromises

The animated approach is intentionally mechanistic, but it is still a model. Several choices affect realism and computational cost:

  • Time discretization versus event-driven simulation. Small time steps are simple and animation-friendly, but event-driven methods can be more efficient for rare transitions.
  • Residence-time model. Exponential waiting times are simple; Erlang, hyper-Erlang, and phase-type models can represent multistep mechanisms.
  • Heterogeneity model. Discrete site classes are interpretable; continuous lognormal mixtures can represent broad site distributions.
  • Transport model. A minimal animation may use constant mobile velocity, whereas more detailed models can include axial diffusion, extra-column dispersion, or nonlinear adsorption.
  • Reaction model. Interconversion may occur only in the mobile phase, only in the stationary phase, or in both, depending on the chemical system, and they may entail multistep or more complex reaction mechanisms.

For educational use, simpler models are often preferable because they are easier to interpret. For experimental design or parameter estimation, additional realism may be needed.

Where this approach can be useful

Animated stochastic chromatography is not meant to replace established workflows for quantitative method development. Its value lies in connecting mechanism, simulation, and observation. Several applications are especially natural.

Teaching

Particle-level animation can help students understand retention, band broadening, selectivity, and dynamic exchange. It provides a bridge between textbook diagrams and real chromatograms.

Method development

Simulations can be used to explore how peak shapes respond to flow rate, temperature, residence-time distributions, or interconversion rates. This can guide experimental design before time-consuming measurements are performed.

Communication

Complex chromatographic phenomena are often difficult to explain in static figures. Short animations can make mechanisms clearer in presentations, online articles, and supplementary material.

Model checking

When a model produces a chromatogram, the corresponding particle trajectories can reveal whether the mechanism is plausible. A good fit to a detector trace is more convincing when the underlying simulated histories also make physical sense.

Limitations

Several limitations should be kept in view. First, visually compelling simulations are not automatically correct simulations. Animation can make a model easier to understand, but it can also make assumptions look more concrete than they are. Second, particle-level Monte Carlo methods can be computationally demanding when many particles, long columns, fine time resolution, or complex state networks are used. Third, parameter identifiability remains a challenge. Different combinations of residence-time heterogeneity, dispersion, and kinetic exchange can produce similar detector-level signals.

For this reason, animations should be treated as mechanistic illustrations and diagnostic tools, not as standalone proof. Their strongest role is alongside quantitative fitting, independent experimental constraints, and sensitivity analysis.

Conclusion

Stochastic theory gives chromatography a molecular narrative. Instead of viewing a peak only as a curve to be integrated or fitted, we can view it as the endpoint of many random particle histories. Extending the model to multistep and multipathway residence times, heterogeneous binding-site populations, and interconversion makes that narrative rich enough to describe phenomena such as tailing, coalescence, and Batman peaks.

The central message is simple: if we can simulate the particle histories, we can animate them. Those animations make chromatography easier to reason about, easier to teach, and potentially easier to design. For field professionals, this provides a practical way to connect stochastic theory with everyday chromatographic intuition. For students, it turns invisible molecular events into something they can watch.

References

Felinger, Attila. 2008. “Molecular Dynamic Theories in Chromatography.” Journal of Chromatography A 1184 (1): 20–41. https://doi.org/10.1016/j.chroma.2007.12.066.
Giddings, J. Calvin, and Henry Eyring. 1955. “A Molecular Dynamic Theory of Chromatography.” The Journal of Physical Chemistry 59 (5): 416–21. https://doi.org/10.1021/j150527a009.
Guiochon, Georges, Attila Felinger, Daryoush G. Shirazi, and Anant M. Katti. 2006. Fundamentals of Preparative and Nonlinear Chromatography. 2nd ed. Academic Press.
Montroll, Elliott W., and George H. Weiss. 1965. “Random Walks on Lattices. II.” Journal of Mathematical Physics 6 (2): 167–81. https://doi.org/10.1063/1.1704269.
Sepsey, Annamária, Dóra Rita Németh, Gábor Németh, and Attila Felinger. 2018. “Rate Constant Determination of Interconverting Enantiomers by Chiral Chromatography Using a Stochastic Model.” Journal of Chromatography A 1564: 155–62. https://doi.org/10.1016/j.chroma.2018.06.014.