How Many Unknown Reactions Does The System Have Figure 1

You're staring at a reaction network diagram — maybe in a paper, maybe in your own modeling work — and Figure 1 shows a system with some reactions labeled, some dashed, some marked with question marks. In practice, the caption says something like "Putative reaction network. " And the question hits you: **how many unknown reactions does this system actually have?

It sounds like a counting problem. It's not. It's an identifiability problem dressed up as a diagram.

What Is an "Unknown Reaction" in a System Context

Let's get the terminology straight first. When modelers talk about unknown reactions in a system, they usually mean one of three things — and confusing them is where the trouble starts.

Reactions missing from the mechanism entirely

These are the ghost reactions. They happen in reality but nobody put them in the model. That said, maybe they're slow. Here's the thing — maybe they're catalyzed by an impurity nobody checked for. Maybe they only matter at high temperature or low pH. In Figure 1 of most papers, these don't appear at all — not even as dashed lines. They're the "unknown unknowns.

Reactions drawn but unparameterized

You see the arrow. Plus, the rate constant? The activation energy? But the rate law? In a figure, these often show up as dashed arrows or arrows with a "k?You know the stoichiometry. " label. Blank. Or fitted with huge confidence intervals. They're "known unknowns" — you know the reaction exists, but you don't know its kinetics.

Reactions that are structurally unidentifiable

This is the subtle one. The reaction is in the model. Plus, it has a rate constant. But no experiment you can run — given your measured species, your time resolution, your noise floor — will ever let you distinguish its effect from some other reaction's effect. Even so, the parameter is mathematically unidentifiable. The reaction might as well not exist for all the information your data carries.

Figure 1 rarely distinguishes these. It just shows arrows.

Why the Count in Figure 1 Is Almost Certainly Wrong

Here's the thing about that figure you're looking at: the number of unknown reactions it depicts is almost certainly a lower bound. Sometimes a dramatic one.

The lumping illusion

Most published networks are lumped. "A → B" might represent five elementary steps. On the flip side, the figure shows one arrow. Consider this: the system has five unknown reactions (or more, if some steps branch). But the figure counts one. This isn't dishonesty — it's necessary simplification. But it means **the figure's reaction count is not the system's reaction count.

The "inert" species trap

Species that don't appear in the measured data — intermediates too short-lived to detect, surface sites you can't probe, radical pools you infer but never quantify — can participate in arbitrarily many reactions without changing any observable. Each of those reactions is unknown in the practical sense. Figure 1 might show zero of them.

The concentration-dependent pathway switch

A reaction that's negligible at low concentration might dominate at high concentration. If your Figure 1 was drawn for one condition, it misses the reactions that only matter elsewhere. The system has more unknown reactions than the figure shows — they're just conditionally unknown.

How to Actually Estimate the Number

You don't count arrows. You do structural and practical identifiability analysis. Here's how that works in practice.

Step 1: Write the stoichiometric matrix

List every species you think* might exist. Plus, every elementary step you think* might happen. Build the S matrix (species × reactions). Don't worry about rate laws yet — just topology.

Step 2: Compute the null space of S transpose

The left null space of S gives you the conservation laws. The right null space gives you the reaction invariants — linear combinations of rates that leave all species concentrations unchanged. Each independent vector in the right null space represents a set of reactions that can vary together without affecting any concentration trajectory.

The dimension of that null space is the number of structurally unidentifiable reaction combinations.

If you have 20 reactions and the null space dimension is 7, then at most 13 independent reaction rates can ever be determined from concentration data — even with perfect, noise-free, continuous measurements of every species.* The other 7 degrees of freedom are mathematically invisible.

Step 3: Add measurement constraints

Now ask: which species do you actually measure? Build the measurement matrix C (measured species × all species). The observable subspace is the intersection of the column space of S with the row space of C. Reactions that only affect unmeasured species? Invisible. Reactions whose effects are perfectly correlated in the measured species? Indistinguishable.

Want to learn more? We recommend what is the function of xylem and can a rectangle be a parallelogram for further reading.

Step 4: Practical identifiability — the Fisher Information Matrix

Structural identifiability assumes perfect data. Real data has noise, sparse time points, detection limits. Compute the Fisher Information Matrix for your parameterized model (you need rate law forms now, not just topology). Still, small eigenvalues = parameters you can't estimate well. Here's the thing — the number of eigenvalues below your noise threshold? That's your practical* unknown reaction count.

It's almost always larger than the structural count.

Common Mistakes People Make Reading Figure 1

Counting dashed arrows as "the unknowns"

Dashed arrows are the author's hypotheses* about unknown reactions. Here's the thing — the real unknowns include everything they didn't think to draw. The dashed arrows are a subset — often a small one.

Assuming measured species = all species

If the figure shows 6 species and 10 reactions, people assume 4 unknown reactions (10 - 6 = 4 degrees of freedom). But if 3 of those 6 species are never measured, and 2 reactions only interconvert unmeasured species, the real number is higher. The figure doesn't tell you what's measured.

Confusing "reactions" with "parameters"

A single reaction with a complex rate law (e.g., Langmuir-Hinshelwood with 3 adsorption constants) contributes multiple parameters. And figure 1 counts it as one reaction. In real terms, identifiability analysis counts parameters. These are different numbers.

Ignoring the time dimension

A reaction might be identifiable in principle but only if you measure at microsecond resolution — and your data is hourly. That reaction is practically* unknown. Figure 1 has no time axis.

What Actually Works: Strategies to Reduce the Unknown Count

You can't eliminate unknown reactions. You can only shrink the set until the remaining ones don't matter for your predictions.

Targeted perturbation experiments

Don't just watch the system. Think about it: poke it. Pulse a reactant. Think about it: jump the temperature. Now, add an inhibitor. Because of that, each perturbation changes the excitation of different reaction pathways. Plus, the Fisher Information Matrix grows rank. Unknown reactions become known — or at least constrained.

Measure intermediates, not just endpoints

If you only measure final products, you'll never distinguish parallel pathways. Day to day, add in-situ spectroscopy (IR, Raman, UV-vis, MS). Even one intermediate measurement can break a correlation that made two reactions indistinguishable.

Use isotopic labeling

13C, 15N, 18O, 2H — tracers let you track atom flows that concentration data alone can't resolve. A reaction that's invisible in natural abundance data might light up beautifully in a labeling experiment. This

This allows researchers to track specific atoms through reactions, making previously hidden steps detectable. By strategically introducing and monitoring labeled species, experimenters can "tag" individual reaction pathways, bypassing the limitations of bulk concentration measurements. To give you an idea, if a reaction involves a carbon atom that’s labeled with 13C, its presence in a product can confirm the reaction’s occurrence, even if it’s not otherwise obvious. This not only reduces the effective number of unknowns but also provides direct evidence of reaction mechanisms that might otherwise remain obscured.

Conclusion

The challenge of unknown reactions in chemical systems is not insurmountable. While structural models may inherently lack information about certain steps, the tools discussed—Fisher Information Matrix analysis, targeted perturbations, intermediate measurements, and isotopic labeling—offer practical pathways to constrain and even eliminate uncertainty. These methods shift the focus from passive observation to active interrogation of the system, transforming what was once an insurmountable "unknown" into a manageable variable. The key lies in recognizing that identifiability is not just a theoretical exercise but a dynamic process shaped by experimental design and data quality. By integrating these strategies, researchers can achieve models that are both parsimonious and predictive, even in complex, poorly understood systems. The bottom line: the goal is not to eliminate all unknowns but to reduce them to a level where their impact on predictions becomes negligible, ensuring that the model remains a useful tool for scientific inquiry and application.