Mathematical Conditions for Induced Cell Differentiation and Trans-differentiation in Adult Cells

Abstract

We present a theoretical framework for the analysis of the effect of a fully differentiated cell population on a neighboring stem cell population in Multi-Cellular Organisms (MCOs). Such an organism is constituted by a set of different cell populations, each set of which converges to a different cycle from all possible options, of the same Boolean network. Cells communicate via a subset of the nodes called signals. We show that generic dynamic properties of cycles and nodes in random Boolean networks can induce cell differentiation.

Specifically we propose algorithms, conditions and methods to examine if a set of signaling nodes enabling these conversions can be found. Surprisingly we find that robust conversions can be obtained even with a very small number of signals. The proposed conversions can occur in multiple spatial organizations and can be used as a model for regeneration in MCOs, where an islet of cells of one type (representing stem cells) is surrounded by cells of another type (representing differentiated cells). The cells at the outer layer of the islet function like progenitor cells (i.e. dividing asymmetrically and differentiating). To the best of our knowledge, this is the first work showing a tissue-like regeneration in MCO simulations based on random Boolean networks.

We show that the probability to obtain a conversion decreases with the log of the node number in the network, showing that the model is relevant for large networks as well. We have further checked that the conversions are not trivial, i.e. conversions do not occur due to irregular structures of the Boolean network, and the converting cycle undergoes a respectable change in its behavior.

Finally we show that the model can also be applied to a realistic genetic regulatory network, showing that the basic mathematical insight from regular networks holds in more complex experiment-based networks.

Appendices

Appendix A

In this appendix, we explore the option of using a slightly different set of candidate nodes to induce UQC. The algorithm for detecting candidate nodes first eliminates nodes that behave similarly in the two cycles, and then eliminates nodes that cannot close a directed circle or that do not link between directed circles in the remaining graph. Finally, the algorithm selects the nodes that are fixed in both cycles. However, if the decision which cycle is going to be the inducing one (potentially) is taken in advance, an alternative algorithm can be proposed, with the small difference that at the last step of the algorithm, nodes fixed only in the potentially inducing cycle are chosen.

Such nodes actually enable UQC to occur in MCOs as well. However, such nodes are problematic. We will refer to UQC obtained from such nodes as Raw UQC. The focus of the work is not on Raw UQC. Since tissue stem cells are located in small stable groups surrounded by differentiated tissue cells, it is important that the conversions will take place layer by layer with only external conversion of stem cells by differentiated cells and no natural differentiation of stem cells induced by other stem cells, as would have happened if nodes twinkling in the induced set of cells are used as signaling nodes. We have tested that Raw UQC properties in MCOs are similar to the ones of UQC in general (Table 3).

Table 3 Projection from the two cell system to MCOs when different kinds of sets of nodes are used as signals. The columns and rows are as in Table 1

We have checked whether all possible unsynchronized conversions in MCOs are the result of Raw UQC. There were a few cases (0.5 % of cycle pairs), where a conversion occurred in MCO although there was no Raw UQC (the conversion occurred using the default choice of using nodes with different patterns as signals). In these cases, the cell populations first synchronized themselves and then conversion occurred. However, those cases under a little different MCO dimensions lose the ability to self synchronize as well as the ability to convert.

Thus, while Raw UQC does not cover all the possible unsynchronized conversions between any two cycles, the results indicate that there are no unsynchronized conversions that are not sensitive to the MCOs’ dimensions that do not follow Raw UQC.

From another perspective, in order to determine whether there will be conversion in cases that have potential inducing nodes but do not induce UQC in the two cell system, 1000 pairs of such cycles were tested and no MCO conversion was found for any of them.

Appendix B

In this appendix, we validate that the UQC conversions are not trivial in terms of the locations of the nodes that enable the conversions in the Boolean network and also in terms of validating that the induced cycle undergoes a significant change in its character, otherwise, its conversion is meaningless. For each network size, UQC was checked in 10,000 random Boolean networks.

2.1 B.1 Relation Between UQC and the Giant Connectivity Component

We have here studied K=2 homogeneous networks. Such networks have a giant strongly connected component¹ (GSCC) (Tarjan et al. 1972), but they still may have other small strongly connected components. Note that the small SCCs can be connected to the GSCC in the equivalent undirected network. It is easier to induce UQC in a small network than in a large one (Fig. 4). Thus one may wonder if the results in the large networks are not an artifact of the presence of small strongly connected components. What happens in real genomes is unclear; however, we do know that there are strongly connected components (SCCs) in real genomes containing hundreds of genes, so it is important to measure the fraction of cases where all nodes inducing UQC belong to the GSCC. This fraction was found to be constant at around 40 % and was not sensitive to the network size (Fig. 6A). For clarity, in the following sections, the UQC with all the inducing nodes in the GSCC will be referred as case C, where the regular UQC cycle pairs will be referred as case B and a random pair of cycles will be referred as case A.

Fig. 6

Properties of UQC cycles of different categories. In all subplots, the X axis is the number of nodes. Glossary for all subplots: Dashed black line—random pairs of cycles (case A). Black line—regular UQC pairs (case B). Dark gray line—cycles where all nodes belong to the GSCC (case C). Dashed dark gray line—induced and target cycles have lengths equal or bigger than the median (case D). Bright gray line—induced and target cycles have different lengths (case E). Dashed bright gray line—the combination of the three previously described categories (case F). Upper left drawing (A) Categories of UQC. The Y axis denotes the percentage of categories out of the regular UQC pairs. Note that in this subplot case B represents the base line of 1 (i.e. 100 %). Upper right drawing (B) the mean of the fractions of nodes with different pattern. The Y axis denotes the fraction of nodes in the induced and target cycles that have different patterns. The fraction is out of all nodes in the network. Lower left drawing (C) the mean of the correlations between cycles. The Y axis denotes the correlation between cycles. In contrast with the fraction of different nodes, here we compute “how different are the patterns of the nodes in the different cycles”. Values close to 1 indicate that two cycles have almost the same cycling patterns. The correlation is generally decreasing for the cases in the following order: D, A, B, C, F, and E. The lower the correlation is, the more different the cycles are. Lower right drawing (D) the mean of the lengths of the cycles. The Y axis denotes the mean of the lengths of cycle pairs. This is the mean over the converting and converted cycles. There was an irregular result for case F in the networks of 700 nodes, since there are a few cycles in this category and large fluctuations can occur

2.2 B.2 Lengths of the Converting and Converted Cycles

The lengths of the cycles are also important. Very long cycles are obviously non-biological. On the other hand, too short cycles might be meaningless as well. Very short cycles may not reflect the dynamic character of living cells. We ensured that there were pairs of cycles that follow the UQC, where the lengths of both the induced and the target cycles were equal or bigger than the median cycle length. This fraction was between 25–35 % (Fig. 6A). Thus, UQC is not an artifact of short cycles. Those cycle pairs will be further referred to as case D.

2.3 B.3 Difference Between the Length of the Inducing and Induced Cycle

An interesting case is pairs of UQC cycles that have different lengths. Mathematically these cases are interesting, since they must require that a set of fixed nodes will become unfixed or the opposite—all this being done via signaling fixed nodes (mostly bimodal). Biologically these cases are interesting since they may reflect arousal/shutting of some dynamical cellular activity. This fraction was between 5–15 % (Fig. 6A). Those cases will be referred as case E. Finally, we introduce one more category (case F), which is the combination of the three previously described categories: (A) all required nodes belong to the GSCC, (B) both the induced and target cycles have lengths equal or bigger than the median, and (C) their cycle length is different. This fraction is shown in Fig. 6A, and was found to be very small.

2.4 B.4 Similarity Between the Converting and Converted Cycles

Conversion between two cycles is meaningless if the induced and target cycles are practically the same. We have thus checked if conversion is indeed limited to the practically meaningless conversion of almost equal cycles. Two measures of similarity were proposed. The first is the fraction of nodes with a different pattern between the cycles. The second measure is the correlation between the average value vectors of each cycle (i.e. the fraction of time each node receives a value of 1 in each cycle (see Sect. 4)). Note that we do expect some correlation, since even if 10 % of the nodes are changed, this may represent a significant pattern change.

From a biological point of view, such a correlation is equivalent to the correlation between the average mRNA expression levels in two types of asynchronized cell population. We have tested the correlation for all the cycle pair groups mentioned above. One can clearly observe (Fig. 6B and 6C) that as the complexity requests for the cycles increases, the fraction of different nodes increases and the similarity decreases. On the other hand, as the network size increases, the similarity increases.

To summarize, when comparing the length of the cycles in all the above mentioned categories, the cycles in groups E and F are the longest and are the most different from one another. This was consistent over all network sizes (number of nodes) tested.