Simulation: Model Building
First, I built a simple sequential system with three nodes, N1, N2, and N3: Model 1 (Fig. 1). A simple circular system contains only one node that undergoes self-rotation (Model 2). Therefore, an amplification cycle was added for node 1 in Model 2. In the third model, the cycle was moved from node 1 to include both nodes 1 and 2. If n denotes the number of cycles designed in the models, the three models above represent a pure sequential system, self-rotating system (Model 2), and mutually regulated circular system (Model 3).
For connections, simple wave functions for sequential events (N1 to N2 and N2 to N3) were selected because these are consistent with either direct light transmission, indirect reflection or resonator7,8. A simple step function was used instead of a linear function, as the latter denotes a switch for later events (Fig. 1). Furthermore, random sampling was conducted to account for observer effects, which may be due to instrument limitations or potential interference.
Simulation: Circular Effects
Model 1 is a pure sequential event and node 1 is a constant (Fig. 2). Node 2 primarily depends on the connection or wave function between the two nodes. Node 3 is influenced by both the connection between N1 and N2, as well as the connection between N2 and N3 (Fig. 2).
Model 2 is a simple circular system, in which N1 self-rotates. At a certain frequency, it alternates between the two states. These two distinct states would influence the entire system (Fig. 3). In Model 3 of Fig. 4, the increase in N1 may conflict with N2 so they are mutually regulating each other. In this case, theoretical distribution is almost smooth but we may still observe the two states upon appropriate sampling.
Simulation: Sampling Effect
For such simple models, as outlined above, we would expect empirical data to be self-explanatory. After random sampling, the observed data, however, are all over the places, even for a simple sequential system in which N1 is a constant (Fig. 2, the right panels show n2 after random sampling at the top and n3 at the bottom).
For model 2, sampling data would outlined the changes in N1 for both N2 and N3, as such an effect is dominant (Fig. 3, right panels), which is consistent with the theoretical values (Fig. 3, left panels). Interestingly, for Model 3, even though the connection between N1 and N2 theoretically would cancel out the rhythmic variation from N1 (Fig. 4, left panels). Such effects were maintained following sampling (Fig. 4, right panels).
In summary, sampling plays a significant role in observations that may completely hide or show the effects of current or previous connection(s) depending on the circumstances. Nevertheless, the effects of circular systems may be easier to detect than those of sequential systems. Model 2 and Model 3 represent two types of circular system. The self-rotating one would alter the theoretical waves while the other model would only be observed with appropriate sampling.
Simulation: Synchronization Effects
In sequential model 1, nodes 2 and 3 have separate wave functions. When they are in sync, their effects are amplified. However, when they are discordant, their effects are reduced. This can be observed in Fig. 2. This effect is consistent for all models (Figs. 3 and 4).
Even when the sampling data failed to illustrate the actual waves, they may still shift the overall distributions depending on the synchronization effects (Fig. 2, 3, and 4).
Note that the function itself may not matter, for example, a step function (f3) or a wave function (f1, f2). The major deciding factors are the frequency and direction. It is possible for nodes within a circular system to adjust any synchronization effects with the rest by establishing specific connections among them.
Empirical Validation for Basic Models: Unbiased Analysis Detected Circular Systems
The previous simulation outlined a few characteristics of a circular love system compared to a sequential one. However, whether such a system exists in biological systems remains unclear. Because the basic unit of biological systems is cells, I investigated empirical data from natural sources: oocytes from humans.
For transcriptome data, I first examined the distributions of the transcriptome for each sample separately and identified the threshold to separate the background for the bulk to discover any major circular systems. This is similar to the observations made in models 2 and 3 (Figs. 3 and 4, right panels) that observations would shift from the background after a circular function was introduced. If a sequential function was in-place for the oocyte, we would expect that data points measured in a single-cell analysis should all be shifted up or down and there should not be additional elevation at that specific time-cut. If we could observe any elevation, repeatedly, from a completely unbiased data, it must be a circular systemic event.
The three samples from oocytes and the pro-nucleus stage separately all looked similar, and the common genes elevated in all three samples were identified. The newly identified genes were subjected to enrichment pathway analysis9,10,11. The most enriched pathway contained ribosome proteins, which are involved in RNA translation, indicating sufficient ribosomes in these cells to hold transcripts in place and delay translation (Fig. 5). The second enriched pathway was circadian rhythm (Fig. 5). This cycle is well known in most biological systems. The surprising results from this simple and straightforward new analysis demonstrated that it is very likely that the circular system we modeled above does exist in biological systems, as we could detect them easily from completely unbiased high-throughput empirical observations.
To ensure that these observations are not specific to a data set or technology, I identified another proteomic dataset from an independent group using a completely different technology.
The dataset contained proteomic data from a total of 100 oocytes. A similar analysis was conducted using the proteome by setting a threshold above the baseline background (Fig. 6). Newly identified genes were analyzed for over-represented or enriched pathways. Even though circular rhythm did not appear in this dataset, another well-known circular pathway, TCA, was easily identified from the unbiased set (Fig. 6). This pathway plays an important role in cellular metabolism and is present in every cell in the body.
Transcription and translation are separate cellular events and are subjected to distinct regulatory pathways. The new analysis, guided by our simulation experiments above, surprisingly, albeit successfully identified circular pathways from the rest of the genes, verifying that the simple love circular system modeled previously, as simulated above, does exist in the basic unit of all biological systems: a cell.
Simulation Data For Complex Models
Basic circular systems can be extended to a high-dimensional space to simulate complex models. In complex biological systems such as humans, one fertilized egg develops into thousands of different cells and tissues. Each cell was slightly different from the other cells. Furthermore, some cells would perform similar functions and define different classes. This indicates that there are thousands of ways to divide these cells into various groups. Labeling each individual cell to simulate the system is a daunting task. Interestingly, recent studies have shown that almost all cell types can revert to a stem-like state and re-differentiate into another cell type. This may be achieved by the addition of four transfection factors, Oct4, Klf4, IRES-Sox2, c-Myc (OKSM), or various combinations of chemicals12,13,14,15,16. In other words, even if the number of cell types is large, they can be converted into five types: one stem-like and the other four cell types, each of which differs from one major transfection factor. Although these may not appear in real biological systems, this simplified model is sufficient to capture all cell types with various phenotypes.
I first extended the simulation experiments based on Model 3 with 10000 independent nodes from different cell types and different group effects. Figure 7 showed that different cell types could influence (nodes 2 and 3) model trajectories (Model 4), which may be further expanded in the development of complex biological systems. Another potential factor is the overall decay or aging factor of the system, which is achieved through group/age effects. When the system ages faster, the general rhythms of the nodes (Model 5) would slow down and the interval for the unstable system elongates compared with the one that ages slower. When this occurs, local changes are more likely to occur (Model 6). In other words, when the local system ages faster, it is slower than the others (Model 5) and needs to jump to catch up (Model 6). However, when it jumps too fast (Model 7), the system may become more disordered, and this is how cancer is initiated. In summary, extension of the simple circular system could observe a systemic effect similar in cancer initiation.
Similar experiments with Model 2 (self-rotating) showed different profiles vs. Model 3 (mutually-regulation). The step function in Node 1 is much more evident in Node 2 and Node 3 compared with Model3 (mostly invisible) in normal circumstances (Supplemental Model 2 extension Top panel). Looking at another single-cell transcriptome data of oocytes with SMART-seq2 protocol18, which in general captures more transcripts compared with other technologies, we clearly observed data displayed two distinct phases in the raw data but these two phases disappeared after sorting the data according to chromosomal locations, indicating that either this is likely due to some technical nuances or a self-rotating circular model with extended phases may not be prevalent in oocyte transcriptome.
In summary, even though the model is simple by nature, the basic unit of love system could be extended to understand basic mechanisms of disease: e.g. cancer initiation.