Data. The text8 dataset [25] was derived from a subset of the enwik9 Wikipedia
dataset by removing punctuation and restricting the text to lowercase Latin
letters and spaces, giving an alphabet of size 27. For clarity, we represent
the space character with an underscore in figures.

(a) Test Data

(b) Generated Data

Figure 18: text8 real and generated data. Samples generated with 1000 steps.

Setup. The network architecture was a Transformer similar to the small model
($d_{\text{model}}=768$) used by Radford et al. [31] except that it uses the
GELU activation function [10] and the depth was increased to 24 layers. The
input and output of the Transformer were concatenated and then projected back
to the output size to produce the final output. The standard
training/validation/test split of 90M/5M/5M consecutive characters was used,
and the network was trained with a batch size of 3328 sequences of length 256,
randomly cropped from the training set, for 1.2 M weight updates using the
AdamW optimizer[22]. The learning rate was set to $10^{-4}$, weight decay to
0.1 and $(\beta_{1},\beta_{2})$ to $(0.9,0.98)$. An exponential moving average
of model parameters with a decay rate of 0.9999 was used for evaluation and
sample generation. Dropout was not used, but overfitting was observed towards
the end of training indicating that regularization may further improve
results. $\beta(1)$ was 0.75. The total number of learnable parameters was
approximately 170M. Note that the batch size and number of layers were larger
than prior results from diffusion models. The first choice increases model
capacity while the second tends to make overfitting more likely. These choices
were made to maximize the utilization of available resources while achieving
results in reasonable time.

Figure 19: text8 Input and Output Distributions. The heatmaps show the
character probability distributions across part of a test sequence at various
times during the flow process. Whereas the expected entropy for each letter
decreases independently in the input distribution, the entropy of the output
distribution tends to chunk into words and phrases — e.g. the date
“one_five_six_one” is confidently predicted early in the process.

Results. Table 4 shows that BFN yielded a 1.41 BPC on the text8 test set,
which is better than all discrete diffusion models we found in the literature,
and close to the best order-agnostic model, MAC at 1.40 BPC. We note however
that both a standard autoregressive baseline and a discrete flow model perform
substantially better at 1.23 BPC. Table 5 shows that performance is reasonably
robust to decreased $n$, with only 100 steps required to reach 1.43 BPC. This
result could probably be improved by training with the discrete-time loss.

Figure 20: text8 Input and Output Distributions. An alternative visualisation
with the character sizes scaled in proportion to their probability.

##  8 Conclusion

This paper introduced Bayesian Flow Networks, a new class of generative model
that combines Bayesian inference with neural networks in an iterative
modelling process. Discrete and continuous-time loss functions were derived
along with sampling procedures, and the model was succesfully applied to
continuous, discretised and discrete data. We hope this work will inspire
fresh perspectives and new directions for generative modelling research.

## Ackowledgements

We would like to thank Vojtech Micka for his invaluable engineering and
infrastructure support.

## References

  * Austin et al. [2021] Jacob Austin, Daniel D. Johnson, Jonathan Ho, Daniel Tarlow, and Rianne van den Berg.  Structured Denoising Diffusion Models in Discrete State-Spaces.  _arXiv preprint arXiv:2107.03006_ , July 2021.
  * Chen et al. [2022] Ting Chen, Ruixiang Zhang, and Geoffrey Hinton.  Analog bits: Generating discrete data using diffusion models with self-conditioning.  _arXiv preprint arXiv:2208.04202_ , 2022.
  * Child [2020] Rewon Child.  Very deep vaes generalize autoregressive models and can outperform them on images.  _arXiv preprint arXiv:2011.10650_ , 2020.
  * Child et al. [2019] Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever.  Generating long sequences with sparse transformers.  _arXiv preprint arXiv:1904.10509_ , 2019.
  * Dhariwal and Nichol [2021] Prafulla Dhariwal and Alexander Nichol.  Diffusion models beat gans on image synthesis.  _Advances in neural information processing systems_ , 34:8780–8794, 2021.
  * Dieleman et al. [2022] Sander Dieleman, Laurent Sartran, Arman Roshannai, Nikolay Savinov, Yaroslav Ganin, Pierre H Richemond, Arnaud Doucet, Robin Strudel, Chris Dyer, Conor Durkan, et al.  Continuous diffusion for categorical data.  _arXiv preprint arXiv:2211.15089_ , 2022.
  * Duda [2009] Jarek Duda.  Asymmetric numeral systems.  _arXiv preprint arXiv:0902.0271_ , 2009.
  * Georgii [2008] H.O. Georgii.  _Stochastics: Introduction to Probability and Statistics_.  De Gruyter textbook. Walter De Gruyter, 2008.  ISBN 9783110191455.  URL https://books.google.co.uk/books?id=ttJ5xpQX2MgC.
  * Graves [2013] Alex Graves.  Generating sequences with recurrent neural networks.  _arXiv preprint arXiv:1308.0850_ , 2013.
  * Hendrycks and Gimpel [2016] Dan Hendrycks and Kevin Gimpel.  Gaussian error linear units (gelus).  _arXiv preprint arXiv:1606.08415_ , 2016.
  * Hinton and Van Camp [1993] Geoffrey E Hinton and Drew Van Camp.  Keeping the neural networks simple by minimizing the description length of the weights.  In _Proceedings of the sixth annual conference on Computational learning theory_ , pages 5–13, 1993.
  * Ho et al. [2020] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  _Advances in neural information processing systems_ , 33:6840–6851, 2020.
  * Hoogeboom et al. [2021a] Emiel Hoogeboom, Alexey A Gritsenko, Jasmijn Bastings, Ben Poole, Rianne van den Berg, and Tim Salimans.  Autoregressive diffusion models.  _arXiv preprint arXiv:2110.02037_ , 2021a.
  * Hoogeboom et al. [2021b] Emiel Hoogeboom, Didrik Nielsen, Priyank Jaini, Patrick Forré, and Max Welling.  Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions.  In _Advances in Neural Information Processing Systems_ , volume 34, pages 12454–12465. Curran Associates, Inc., 2021b.
  * Jain et al. [2020] Ajay Jain, Pieter Abbeel, and Deepak Pathak.  Locally masked convolution for autoregressive models.  In _Conference on Uncertainty in Artificial Intelligence_ , pages 1358–1367. PMLR, 2020.
  * Kim et al. [2021] Dongjun Kim, Seungjae Shin, Kyungwoo Song, Wanmo Kang, and Il-Chul Moon.  Soft truncation: A universal training technique of score-based diffusion model for high precision score estimation.  _arXiv preprint arXiv:2106.05527_ , 2021.
  * Kingma et al. [2021] Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho.  Variational diffusion models.  _Advances in neural information processing systems_ , 34:21696–21707, 2021.
  * Kingma and Welling [2013] Diederik P Kingma and Max Welling.  Auto-encoding variational bayes.  _arXiv preprint arXiv:1312.6114_ , 2013.
  * Krizhevsky [2009] Alex Krizhevsky.  Learning multiple layers of features from tiny images.  Technical report, University of Toronto, 2009.
  * LeCun and Cortes [2010] Yann LeCun and Corinna Cortes.  MNIST handwritten digit database, 2010.  URL http://yann.lecun.com/exdb/mnist/.
  * Li et al. [2022] Xiang Lisa Li, John Thickstun, Ishaan Gulrajani, Percy Liang, and Tatsunori B. Hashimoto.  Diffusion-lm improves controllable text generation.  _arXiv preprint arXiv:2205.14217_ , 2022.
  * Loshchilov and Hutter [2017] Ilya Loshchilov and Frank Hutter.  Decoupled weight decay regularization.  _arXiv preprint arXiv:1711.05101_ , 2017.
  * Lou and Ermon [2023] Aaron Lou and Stefano Ermon.  Reflected diffusion models.  _arXiv preprint arXiv:2304.04740_ , 2023.
  * Mahabadi et al. [2023] Rabeeh Karimi Mahabadi, Jaesung Tae, Hamish Ivison, James Henderson, Iz Beltagy, Matthew E. Peters, and Arman Cohan.  Tess: Text-to-text self-conditioned simplex diffusion.  _arXiv preprint arXiv:2305.08379_ , 2023.
  * Mahoney [2009] Matt Mahoney.  Large text compression benchmark., 2009.  URL http://mattmahoney.net/dc/textdata.html.
  * Menick and Kalchbrenner [2018] Jacob Menick and Nal Kalchbrenner.  Generating high fidelity images with subscale pixel networks and multidimensional upscaling.  _arXiv preprint arXiv:1812.01608_ , 2018.
  * Murphy [2007] Kevin Murphy.  Conjugate bayesian analysis of the gaussian distribution.  Technical report, University of British Columbia, 2007.
  * Nichol and Dhariwal [2021] Alexander Quinn Nichol and Prafulla Dhariwal.  Improved denoising diffusion probabilistic models.  In _International Conference on Machine Learning_ , pages 8162–8171. PMLR, 2021.
  * OpenAI [2023] OpenAI.  Gpt-4 technical report.  _arXiv preprint arXiv:2303.08774_ , 2023.
  * Parmar et al. [2018] Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran.  Image transformer.  In _International conference on machine learning_ , pages 4055–4064. PMLR, 2018.
  * Radford et al. [2019] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.  Language models are unsupervised multitask learners.  Technical report, OpenAI, 2019.
  * Rezende and Mohamed [2015] Danilo Rezende and Shakir Mohamed.  Variational inference with normalizing flows.  In _International conference on machine learning_ , pages 1530–1538. PMLR, 2015.
  * Richemond et al. [2022] Pierre H. Richemond, Sander Dieleman, and Arnaud Doucet.  Categorical SDEs with simplex diffusion.  _arXiv preprint arXiv:2210.14784_ , 2022.
  * Rombach et al. [2022] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 10684–10695, 2022.
  * Sadeghi et al. [2019] Hossein Sadeghi, Evgeny Andriyash, Walter Vinci, Lorenzo Buffoni, and Mohammad H Amin.  Pixelvae++: Improved pixelvae with discrete prior.  _arXiv preprint arXiv:1908.09948_ , 2019.
  * Salakhutdinov and Murray [2008] Ruslan Salakhutdinov and Iain Murray.  On the quantitative analysis of deep belief networks.  In _Proceedings of the 25th international conference on Machine learning_ , pages 872–879. ACM, 2008.
  * Salimans and Ho [2022] Tim Salimans and Jonathan Ho.  Progressive distillation for fast sampling of diffusion models.  _arXiv preprint arXiv:2202.00512_ , 2022.
  * Salimans et al. [2017] Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma.  Pixelcnn++: Improving the PixelCNN with discretized logistic mixture likelihood and other modifications.  _arXiv preprint arXiv:1701.05517_ , 2017.
  * Shih et al. [2022] Andy Shih, Dorsa Sadigh, and Stefano Ermon.  Training and inference on any-order autoregressive models the right way.  _Advances in Neural Information Processing Systems_ , 35:2762–2775, 2022.
  * Sinha and Dieng [2021] Samarth Sinha and Adji Bousso Dieng.  Consistency regularization for variational auto-encoders.  _Advances in Neural Information Processing Systems_ , 34:12943–12954, 2021.
  * Sohl-Dickstein et al. [2015] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.  Deep unsupervised learning using nonequilibrium thermodynamics.  In _International conference on machine learning_ , pages 2256–2265. PMLR, 2015.
  * Song et al. [2020] Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole.  Score-based generative modeling through stochastic differential equations.  _arXiv preprint arXiv:2011.13456_ , 2020.
  * Song et al. [2023] Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever.  Consistency models.  _arXiv preprint arXiv:2303.01469_ , 2023.
  * Strudel et al. [2022] Robin Strudel, Corentin Tallec, Florent Altché, Yilun Du, Yaroslav Ganin, Arthur Mensch, Will Grathwohl, Nikolay Savinov, Sander Dieleman, Laurent Sifre, et al.  Self-conditioned embedding diffusion for text generation.  _arXiv preprint arXiv:2211.04236_ , 2022.
  * Sukhbaatar et al. [2019] Sainbayar Sukhbaatar, Edouard Grave, Piotr Bojanowski, and Armand Joulin.  Adaptive Attention Span in Transformers.  _arXiv preprint arXiv:1905.07799_ , August 2019.
  * Sutskever et al. [2011] Ilya Sutskever, James Martens, and Geoffrey E Hinton.  Generating text with recurrent neural networks.  In _Proceedings of the 28th international conference on machine learning (ICML-11)_ , pages 1017–1024, 2011.
  * Townsend et al. [2019] James Townsend, Tom Bird, and David Barber.  Practical lossless compression with latent variables using bits back coding.  _arXiv preprint arXiv:1901.04866_ , 2019.
  * Tran et al. [2019] Dustin Tran, Keyon Vafa, Kumar Agrawal, Laurent Dinh, and Ben Poole.  Discrete flows: Invertible generative models of discrete data.  _Advances in Neural Information Processing Systems_ , 32, 2019.
  * Vahdat and Kautz [2020] Arash Vahdat and Jan Kautz.  Nvae: A deep hierarchical variational autoencoder.  _Advances in neural information processing systems_ , 33:19667–19679, 2020.
  * Vahdat et al. [2021] Arash Vahdat, Karsten Kreis, and Jan Kautz.  Score-based generative modeling in latent space.  _Advances in Neural Information Processing Systems_ , 34:11287–11302, 2021.
  * Wallace [1991] Chris S. Wallace.  Classification by minimum-message-length inference.  In _International Conference on Computing and Information_ , 1991\.
  * Watson et al. [2022] Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi.  Learning fast samplers for diffusion models by differentiating through sample quality.  _arXiv preprint arXiv:2202.05830_ , 2022.
  * Witten et al. [1987] Ian H Witten, Radford M Neal, and John G Cleary.  Arithmetic coding for data compression.  _Communications of the ACM_ , 30(6):520–540, 1987\.
  * Ziegler and Rush [2019] Zachary Ziegler and Alexander Rush.  Latent Normalizing Flows for Discrete Sequences.  In _Proceedings of the 36th International Conference on Machine Learning_ , pages 7673–7682. PMLR, May 2019.


#  Universal Transitions between Growth and Dormancy via Intermediate Complex
Formation

Jumpei F. Yamagishi  Department of Basic Science, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan  Kunihiko Kaneko  Center for
Complex Systems Biology, Universal Biology Institute, University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-0041, Japan  Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

A simple cell model consisting of a catalytic reaction network with
intermediate complex formation is numerically studied. As nutrients are
depleted, the transition from the exponential growth phase to the growth-
arrested dormant phase occurs along with hysteresis and a lag time for growth
recovery. This transition is caused by the accumulation of intermediate
complexes, leading to the jamming of reactions and the diversification of
components. These properties are generic in random reaction networks, as
supported by dynamical systems analyses of corresponding mean-field models.

##  I Introduction

As microbial cells proliferate, they are crowded and nutrients in the
environment are depleted. The cells then enter the dormant phase (or so-called
stationary phase), in which cell growth is significantly arrested 111Such
growth-arrested states are often termed dormant, non-growing, or quiescent
phases/states, as well as stationary phase for an ensemble of cells. Herein,
we adopt the term dormant phase. This behavior is commonly observed across
microbial species and even mammalian cells under a variety of environmental
conditions [2]. In fact, most microbial cells in natural ecosystems are in the
growth-arrested dormant phase as they are under resource limitation [3, 4, 5,
6, 7]. Once cells enter the dormant phase, the intracellular metabolic
phenotypes drastically change, and hysteresis and bistability between
phenotypes with exponential and arrested growth are observed [8, 9, 10]; in
addition, a certain time is required to recover growth even after the resource
supply has resumed, and it is known as the lag time [11, 12, 13].

Despite the importance of such universal and mundane behavior, the theoretical
understanding of dormancy is still in its infancy compared with that of the
exponential growth phase, for which well-established quantitative theories are
available [14, 15, 16]. Although specific molecular mechanisms of dormancy
have been extensively studied [4, 17], little attention has been paid to
establishing a theory for universal characteristics of the dormant phase and
transitions to it. Refs. [18] and [19] represent a few early exceptions. In
Ref. [18], by assuming that nutrient limitation leads to the accumulation of
waste chemicals, a phenomenological model for the growth-dormant transition
was proposed and quantitative laws of the lag time were derived. In Ref. [19],
an abstract spin glass model for aging dynamics was proposed. However, the
universality of their results and the origin(s) of these specific mechanisms
have not been fully explored. Therefore, a better understanding of the growth-
dormant transition as a universal behavior of cells growing through
intracellular reactions with many components is required.

In this Letter, by considering a simple cell model consisting of catalytic
reactions of many components, we demonstrate that such a transition between
growth and dormant phases generally appears without specifically tuning the
intracellular reactions as long as intermediate complexes between substrates
and catalysts have sufficient lifetimes. The transition is aused by the
accumulation of complexes under the depletion of nutrients, and it is
characterized as a cusp bifurcation in dynamical systems theory. The
transition observed in random reaction networks is then analyzed using the
“mean-field” theory of catalytic reaction dynamics, which also implies that
the transition to a dormant phase does not require any special mechanism and
is a universal feature of cells that grow by intracellular catalytic
reactions.

##  II Model

In this study, we adopt a simple model of cellular dynamics that captures only
the basic features of these dynamics. It consists of intracellular reaction
networks and transport reactions of externally supplied nutrient(s).
Complicated intracellular metabolic reactions are simplified as randomly
connected catalytic reaction networks. Although such models with catalytic
reaction networks have reproduced the statistics of cells in the exponential
growth phase [20, 21], they do not demonstrate the growth-dormant transition.
One possible drawback in these models is that catalytic reactions progress
immediately. In reality, each chemical reaction progresses after the formation
of an intermediate complex between the substrate and catalyst is formed.

Here we introduce a model that includes the formation of intermediate
complexes in reactions and examine whether and how the growth-dormant
transition is exhibited by the model. Then, each catalytic reaction $\rho$, in
which substrate $X_{\rho_{s}}$ is converted into product $X_{\rho_{p}}$ by
catalyst $X_{\rho_{c}}$, consists of two-step elementary reaction processes
with the formation of an intermediate complex $Y_{\rho}$ as follows:

$\displaystyle
X_{\rho_{s}}+X_{\rho_{c}}\rightleftharpoons^{k^{+}_{\rho}}_{k^{-}_{\rho}}Y_{\rho}\overset{v_{\rho}}{\to}X_{\rho_{p}}+X_{\rho_{c}}.$

Here, each elementary process proceeds according to the law of mass action
with the labeled coefficient, and $\rho_{s}$, $\rho_{p}$, and $\rho_{c}$
denote the indices of the substrate, product, and catalyst for reaction
$\rho$, respectively. For the adiabatic limit $v_{\rho}\to\infty$, the above
reaction processes are reduced to the single mass action kinetics without
intermediate complex formation, $X_{\rho_{s}}+X_{\rho_{c}}\to
X_{\rho_{p}}+X_{\rho_{c}}$, and the model is reduced to those studied earlier
[20, 21]. In contrast, when $v_{\rho}$ is small, the intermediate complex
$Y_{\rho}$ can accumulate, leading to a decrease in free reactants that are
not bound into complexes, which can hinder the reaction processes.

Considering a cell consisting of $n$ chemicals and $N_{r}$ reactions (and
corresponding intermediate complexes), its state is represented by a set of
concentrations $(\textbf{x},\textbf{y})$ of free reactants (that are not bound
into complexes) $X_{i}$ and complexes $Y_{\rho}$. The time change of the
cellular state $(\textbf{x},\textbf{y})$ is then given as follows:

$\displaystyle\dot{x}_{i}$ $\displaystyle=$
$\displaystyle\sum_{\rho}(\delta_{i,\rho_{p}}+\delta_{i,\rho_{c}})v_{\rho}y_{\rho}-(\delta_{i,\rho_{s}}+\delta_{i,\rho_{c}})f_{\rho}({\bf
x},{\bf y})$ (1) $\displaystyle+F_{i}(\textbf{x};S_{\mathrm{ext}},\alpha)-\mu
x_{i},$ $\displaystyle\dot{y}_{\rho}$ $\displaystyle=$ $\displaystyle
f_{\rho}({\bf x},{\bf y})-v_{\rho}y_{\rho}-\mu y_{\rho},$ (2)

where $f_{\rho}({\bf x},{\bf
y}):=k^{+}_{\rho}x_{\rho_{s}}x_{\rho_{c}}-k^{-}_{\rho}y_{\rho}$ is the total
consumption rate of substrate $\rho_{s}$ by reaction $\rho$, and $\delta$ is
Kronecker’s delta. The term $F_{i}(\textbf{x};S_{\mathrm{ext}},\alpha)$ in Eq.
(1) represents the intake of chemical $X_{i}$ ($i=0,1,\cdots,n-1$), which can
be non-zero if $X_{i}$ is a nutrient but is zero otherwise. The last terms in
Eqs. (1-2), $-\mu x_{i}$ and $-\mu y_{\rho}$, represent the dilution of each
concentration due to cellular volume growth. The growth rate $\mu$ is given by
$\mu(\textbf{x},\textbf{y}):=\sum_{i}F_{i}({\bf x})$, because for simplicity,
we assumed that the contribution of each chemical $X_{i}$ to volume or weight
is uniform regardless of $i$. $\sum_{i}x_{i}+2\sum_{\rho}y_{\rho}=1$ is then
constant in the dynamics (1-2) based on the law of mass conservation.

Below, for simplification purposes, the reaction rate constants
$k^{+}_{\rho}$, $k^{-}_{\rho}$, and $v_{\rho}$ are set as independent of
$\rho$, and they are denoted by $k^{+}=1$, $k^{-}=0$, and $v$, respectively.
For simplicity, we also assumed that there is only a single nutrient chemical
$X_{0}$. Its intake is mediated by transporter chemical $X_{1}$ with
$\alpha=2$, i.e.,
$F_{i}(\textbf{x};S_{\mathrm{ext}},\alpha)=\delta_{i0}S_{\mathrm{ext}}x_{1}^{\alpha}$,
where $S_{\mathrm{ext}}$ denotes the environmental concentration of nutrient
chemical $X_{0}$, and the transport coefficient for $F_{i}$ is normalized as
unity. Note that the following results and arguments hold independent of the
details of settings, such as parameter values and specific functional forms of
nutrient intake $F_{i}$ (see also Supplemental Material (SM), Sec. A).

Figure 1:  Example of a growth-dormant transition in randomly generated
networks. (a) Reaction network ($n=10,N_{r}=30$). Chemicals at arrowtails are
transformed into those at arrowheads, catalyzed by the chemicals labeled on
the edges. Nutrient $X_{0}$ is taken up via active transport by transporter
$X_{1}$ in proportion to $x_{1}^{2}$. (b) Dependence of $\mu^{\ast}$ (black
points) and $\textbf{x}^{\ast}$ (colored lines) on $S_{\mathrm{ext}}$.
$v=0.01$. (inset) Hysteresis and bistability for $\mu^{\ast}$. $v=0.03$. (c)
Dominant pathways for the growth phase ($S_{\mathrm{ext}}=10$; top) and
dormant phase ($S_{\mathrm{ext}}=0.0016$; bottom). $v=0.01$. The edge colors
represent reaction fluxes in the log scale. (d) Dependence of composition
entropy $H:=-\sum_{i}x_{i}\log x_{i}-\sum_{\rho}2y_{\rho}\log(2y_{\rho})$
(black) and $Y:=\sum_{\rho}y_{\rho}$ (gray) on $S_{\mathrm{ext}}$. $v=0.01$.
(e) Dependence of $\mu^{\ast}$ on $(S_{\mathrm{ext}},v)$. $\mu^{\ast}$ is
numerically calculated by decreasing $S_{\mathrm{ext}}$ for each $v$, and
hysteresis is observed in the area surrounded by the gray line.

##  III Results

###  III.1 Randomly generated networks

To understand the behaviors of the above model, we first randomly generated
hundreds of intracellular reaction networks and numerically investigated the
networks 222The reaction networks are randomly generated such that nutrient
chemical $X_{0}$ can only be a substrate of intracellular reactions,
transporter chemical $X_{1}$ cannot catalyze any intracellular reactions, and
autocatalytic reactions do not occur. We then observed discontinuous
transitions between growth and dormant phases against external nutrient
abundance $S_{\mathrm{ext}}$.

As an example, we consider the reaction network in Fig. 1(a). In Fig. 1(b),
the steady growth rate $\mu^{\ast}$, numerically obtained by solving the
dynamics (1-2), is plotted against the environmental nutrient concentration
$S_{\mathrm{ext}}$. As shown, $\mu^{\ast}$ decreases by orders of magnitude at
$S_{\mathrm{ext}}={}^{\exists}S_{\mathrm{ext}}^{c}$, thus demonstrating the
transition from growth to growth-arrested dormant phase. In addition, when
$S_{\mathrm{ext}}$ is increased starting from the dormant phase, the
transition occurs at a larger $S_{\mathrm{ext}}$, thus demonstrating
hysteresis and bistability between the growth and dormant phases with
intermediate levels of nutrient supply $S_{\mathrm{ext}}$ (Fig. 1(b)) 333 In
this Letter, we basically plot the steady states that cells reach when
$S_{\mathrm{ext}}$ is lowered, i.e., the initial state for each set of
parameters is given as the steady state with slightly higher
$S_{\mathrm{ext}}$. In actual cells, the fluctuation of internal
concentrations leads to attractor selection [41]; thus, the branch with the
higher growth rate would be more likely to be observed., as is observed for
real microbes [8, 9, 10].

Through this growth-dormant transition, the intracellular chemical
compositions and dominant reactions at work also change drastically (Fig.
1(b,c)). In the growth phase with larger $S_{\mathrm{ext}}$, the nutrient
influx is concentrated on an autocatalytic growth subnetwork [24, 25, 26, 27,
28] consisting of a few chemicals and reactions that connect the nutrient to
the transporter (and associated byproducts). In contrast, in the dormant
phase, fluxes spread over many chemicals in a subnetwork that cannot sustain
growth by itself, which we term the non-growing subnetwork. Here, a subnetwork
is referred to as a closed set of chemicals and reactions whose catalysts,
substrates, and products are included therein and in which no other chemicals
are included (see also SM, Sec. A for details). In contrast, in the dormant
phase, fluxes spread over many chemicals in a subnetwork that cannot sustain
growth by itself, which we term the non-growing subnetwork. Here, a subnetwork
is referred to as a closed set of chemicals and reactions whose catalysts and
substrates are included therein (see also SM, Sec. A for details). These
subnetworks compete with each other while also overlapping: the activation of
the autocatalytic growth subnetwork suppresses the non-growing subnetwork via
growth-induced dilution, whereas the latter inhibits the former by the
accumulation of complexes because reactants that are combined into some
complexes in the non-growing subnetwork cannot be used for the reactions in
the autocatalytic growth subnetwork. Consistently, the total concentration of
complexes $Y:=\sum_{\rho}y_{\rho}$ increases across the growth-dormant
transition, as shown in Fig. 1(d). Due to the competition between the
autocatalytic and non-growing subnetworks, this transition exhibits
discontinuity, hysteresis, and bistability (Fig. 1(b)).

To measure such competition between autocatalytic and non-growing subnetworks,
we defined composition entropy, $H(\textbf{x},\textbf{y}):=-\sum_{i}x_{i}\log
x_{i}-\sum_{\rho}2y_{\rho}\log(2y_{\rho})$. In general, in the growth phase,
an autocatalytic subnetwork with the largest growth rate should be dominant
and $H$ should be small, whereas in the dormant phase, many reactions and
chemicals in the non-growing subnetwork could be engaged to the same degree
and $H$ can be relatively large. As both subnetworks are comparably active
near the critical nutrient concentration $S_{\mathrm{ext}}^{c}$, the
composition entropy $H$, or the diversity of the intracellular chemical
composition, reaches a maximum near the transition point (Fig. 1(d)). Notably,
such a trend is common among randomly generated networks (Fig. 2). From a
biological perspective, this prediction would be consistent with the
observations that stringent responses increase the diversity of the cellular
components during the transition and in the dormant phase [4, 17, 29].

The suppression of growth at the transition can be understood as a type of
jamming caused by the accumulation of intermediate complexes: when the total
concentration of complexes is larger, the free catalysts necessary for
reactions in the autocatalytic growth subnetwork are limited [30].
Consistently, with $v\,{>}\,^{\exists}v^{c}$, discontinuous transition and
hysteresis are not observed against changes in $S_{\mathrm{ext}}$ (Fig. 1(e)).
Moreover, the dependence of the steady growth rate $\mu^{\ast}$ on
$(S_{\mathrm{ext}},v)$ in Fig. 1(e) suggests a cusp bifurcation in the
dynamical systems theory (as is also confirmed by the following mean-field
analysis). We also found that as $v$ is smaller, both $S_{\mathrm{ext}}^{c}$
and $\mu_{\max}$ are smaller; in other words, when $v$ varies, a trade-off
occurs between maximum growth rate $\mu_{\max}$ and minimal nutrient
concentration for the growth phase, $S_{\mathrm{ext}}^{c}$. Such a trade-off
has historically been considered a result of evolution leading to adaptations
to either abundant or scarce nutrient environments [31, 32], whereas our
results suggest that this trade-off is a universal feature of growing cells
with complex reaction networks.

Statistically, sufficiently large reaction networks are expected to include
non-growing subnetwork(s) in addition to autocatalytic growth subnetwork(s).
Indeed, even with $n=10\sim 30$, about half of the randomly generated networks
exhibited growth-dormant transitions (Fig. S1(a) in SM). In addition, the
proportion of networks that exhibit transitions is maximal for relatively
sparse reaction networks, and the peak value gradually increases as the number
$n$ of chemicals increases. The following characteristics are also common to
such networks: (i) growth-dormant transition against changes in
$S_{\mathrm{ext}}$ requires small $v$; (ii) hysteresis against changes in
$S_{\mathrm{ext}}$; (iii) increases in composition entropy $H$ around the
transitions; and (iv) a trade-off between maximum growth rates $\mu_{\max}$
and minimal nutrient concentrations $S_{\mathrm{ext}}^{c}$ to sustain growth.
These results suggest the universality of growth-dormant transitions due to
reactant competition via complex formation in complicated reaction networks,
as is the case for metabolic networks in actual cells.

Figure 2:  The steady growth rate $\mu^{\ast}$, the total concentration of
complexes $Y$, and composition entropy $H$ are plotted against
$S_{\mathrm{ext}}/S_{\mathrm{ext}}^{c}$ for several randomly generated
networks that exhibit growth-dormant transitions in different colors.
$n=10,N_{r}=30$, $v=0.01$.

We also numerically calculated the time for growth recovery after starvation
as follows: First, up to $t=0$, cells are set in nutrient-rich conditions with
sufficiently large $S_{\mathrm{ext}}$, and they remain in steady states with
exponential growth. Second, the external nutrient supply is instantaneously
depleted to $S_{\mathrm{ext}}=0$ until $t=T_{\mathrm{stv}}$. Finally,
$S_{\mathrm{ext}}$ is instantaneously increased to the original value. Then, a
certain period $T_{\mathrm{lag}}\gg 1/\mu_{\max}$, known as the lag time, is
required for the cell to recover the original exponential growth, if the non-
growing subnetwork is not a cycle and the amount of transporter chemical is
sufficiently reduced therein; the lag time $T_{\mathrm{lag}}$ increases with
starvation time $T_{\mathrm{stv}}$ in the form $T_{\mathrm{lag}}\propto
T_{\mathrm{stv}}^{\beta}$ for a certain range (up to some saturation time), as
observed in real microbes [33, 34] (see SM, Fig. S2 for an example). Here, the
transporter’s concentration gradually decreases under starvation; thus, the
time for growth recovery, which requires the regain of the transporter,
increases with starvation time 444If cells reach fixed points that completely
lose the transporter(s), cells cannot recover growth. We term such steady
states death phases (see SM, Sec. A for details).. The exponent $\beta$ ranges
from approximately $0.1$ to $0.5$ depending on the network structures that
alter the intracellular reaction dynamics.

Figure 3: Mean-field models. (a) Mean-field model with $S$, $X$, and $Y$ only.
(left) Network structure. (right) Dependence of $\mu^{\ast}$ and steady states
on $S_{\mathrm{ext}}$. $\alpha=3,v=0.001$. (b-e) Mean-field model with the
distinction between transporter $T$ and the remaining chemicals $X$. Unless
otherwise stated, $v=0.01,n_{X}=2$. (b) (left) Network structure. (right)
Dependence of $\mu^{\ast}$ and steady states on $S_{\mathrm{ext}}$. (c)
Bifurcation diagram: Dependence of $T^{\ast}$ on $(S_{\mathrm{ext}},v)$. (d)
Flow diagram in the phase space $(T,X)$. Red and blue lines represent
$T$-nullcline and $X$-nullcline, respectively. Arrows with brighter colors
correspond to faster flows. (e) Self-consistent equation for $T$ and $\mu$.
Black and orange lines depict $T=T^{\ast}(\mu;v,n_{X})$ (Eq. (B.2) in SM) and
$T=\left(\mu/S_{\mathrm{ext}}\right)^{1/\alpha}$ with
$S_{\mathrm{ext}}=10,25,50$, respectively.

###  III.2 Mean-field analysis

To further investigate the mechanism underlying the growth-dormant transition
in terms of dynamical systems theory, we constructed mean-field models. First,
we considered a model with one effective concentration variable $X$ and
associated complexes $Y$ in addition to the nutrient $S$ (Fig. 3(a)). This
model exhibits the growth-dormant transition, although it requires extremely
small values of the parameter $v$, $v<\mu_{\max}$, and $\alpha>2$.

Then, we considered another mean-field model that incorporates another
variable $T$ representing the mean-field for the concentration of the
transporter(s) in addition to $X$ representing the remaining non-nutrient
chemicals (Fig. 3(b)). The number of chemicals represented by $X$ and $T$ are
denoted by $n_{X}$ and $n_{T}$, respectively. As only the complex $Y$ between
$X$ and $T$ is considered for simplicity in this model 555Note that there are
several possibilities for the choice of mean-field models with effective
reactions among $X$ and $T$ that exhibit the growth-dormant transition (see
SM, Sec. B for details)., it includes the autocatalytic growth subnetwork,
$S+X\to T+X$ and $S+X\to 2X$, and the single non-growing subnetwork,
$T+X\rightleftharpoons Y$. This mean-field model reproduces common behaviors
observed for randomly generated networks, including discontinuous growth-
dormant transitions with $v>\mu_{\max}$ (Fig. 3(b)). The transition occurs
when $n_{X}>n_{T}=1$, and a larger number $n_{X}$ of $X$ leads to a larger
$S_{\mathrm{ext}}^{c}$ (see SM, Fig. S5).

From the bifurcation analysis (Fig. 3(c-d)), we found that the growth-dormant
transition occurs as a cusp bifurcation against changes in $S_{\mathrm{ext}}$
and $v$ [37]. This observation can explain some of the above-mentioned
properties of randomly generated networks, i.e., discontinuous transitions and
hysteresis. Notably, although both transporter $T$ and the remaining chemicals
$X$ are essential for cell growth, their competition leads to a flow field
with mutual inhibition as in the toggle switch at the intermediate value of
$S_{\mathrm{ext}}$. Furthermore, from the self-consistent equation for the
steady growth rate $\mu^{\ast}$, we can determine where and how the growth-
dormant transition occurs (Fig. 3(e)).

##  IV Discussion

In this Letter, we studied a model of catalytic reaction networks wherein a
variety of components react via the formation of intermediate complexes. This
model exhibits discontinuous growth-dormant transitions against nutrient
conditions as long as the formed complexes have sufficient lifetimes (i.e.,
with small $v$). This transition to growth-arrested dormant phases is caused
by the accumulation of intermediate complexes under nutrient-poor conditions,
which results in the jamming of reactions in the autocatalytic growth
subnetwork and the relatively even distribution of chemical concentrations
over diverse components in the non-growing subnetwork. Remarkably, other basic
characteristics of dormancy, i.e., hysteresis between the exponential growth
and dormant phases, the lag time for growth recovery after starvation, and a
trade-off between maximum growth rate $\mu_{\max}$ and minimal nutrient
concentration $S_{\mathrm{ext}}^{c}$ to sustain growth (or a sort of
sensitivity to nutrient scarcity) are also reproduced. These results indicate
that growth-dormant transitions and dormancy might be inevitable for cells
that grow via complex-forming catalytic reaction networks and likely emerge
without tuning by evolution or adaptation; thus, even protocells at the
primitive stage of life [38, 39] are expected to exhibit such transitions to
dormancy, which would be relevant to their survival under environmental
stresses. On the other hand, further studies of detailed realistic models,
such as those including distributed parameters and more realistic network
structures, will be necessary to reveal how the above fundamental
characteristics of dormancy are preserved or changed by evolution.

Moreover, the composition entropy $H$ is predicted to increase towards the
transition point as a result of the competition between the autocatalytic and
non-growing subnetworks. Biologically, it would correspond to the stringent
responses that increase the diversity of the intracellular components [4, 17,
29]. It will be crucial to experimentally verify the predicted increases in
the composition entropy and jamming of reactions due to the accumulation of
complexes.

We also analyzed the dynamics of mean-field models and thereby demonstrate
that the growth-dormant transition occurs as a cusp bifurcation [37], which
supports the discontinuity of the transitions as well as hysteresis. The
validity of the coarse-grained mean-field models suggests the universality of
the growth-dormant transition across many-body reaction systems. Note that,
although the mean-field models in Fig. 3 capture the mechanism of the dormancy
or growth arrest due to the accumulation of complexes, the diversification of
components in nutrient-poor conditions owing to competition between
subnetworks [40] is not addressed. To consider such diversification and
improve the mean-field theory, the incorporation of two-body correlations
among the concentrations beyond their mean will be required.

In conclusion, our study explains the ubiquity and fundamental characteristics
of dormancy as general properties in reaction networks with complex formation,
by offering a coherent view of cell growth and dormancy.

###### Acknowledgements.

The authors would like to thank Yuichi Wakamoto, Yusuke Himeoka, Tetsuhiro S.
Hatakeyama, and Shuji Ishihara for stimulating discussions. J. F. Y. is
supported by Grant-in-Aid for JSPS Fellows (21J22920). K. K. is supported by
Grant-in-Aid for Scientific Research (A) (20H00123) from the Ministry of
Education, Culture, Sports, Science, and Technology (MEXT) of Japan and the
Novo Nordisk Foundation.

## References

  * Note [1] Such growth-arrested states are often termed dormant, non-growing, or quiescent phases/states, as well as stationary phase for an ensemble of cells. Herein, we adopt the term dormant phase.
  * Gray  _et al._ [2004] J. V. Gray, G. A. Petsko, G. C. Johnston, D. Ringe, R. A. Singer, and M. Werner-Washburne, Microbiology and molecular biology reviews 68, 187 (2004).
  * Finkel [2006] S. E. Finkel, Nature Reviews Microbiology 4, 113 (2006).
  * Navarro Llorens  _et al._ [2010] J. M. Navarro Llorens, A. Tormo, and E. Martínez-García, FEMS microbiology reviews 34, 476 (2010).
  * Del Giorgio and Gasol [2008] P. A. Del Giorgio and J. M. Gasol, Microbial ecology of the oceans 2, 243 (2008).
  * Gefen  _et al._ [2014] O. Gefen, O. Fridman, I. Ronin, and N. Q. Balaban, Proceedings of the National Academy of Sciences 111, 556 (2014).
  * Kirchman [2018] D. L. Kirchman, Processes in microbial ecology (Oxford University Press, 2018) Chap. 6.
  * Kotte  _et al._ [2014] O. Kotte, B. Volkmer, J. L. Radzikowski, and M. Heinemann, Molecular systems biology 10, 736 (2014).
  * Veening  _et al._ [2008] J.-W. Veening, W. K. Smits, and O. P. Kuipers, Annual review of microbiology 62, 193 (2008).
  * Krishna and Laxman [2018] S. Krishna and S. Laxman, Molecular biology of the cell 29, 2243 (2018).
  * Levin-Reisman  _et al._ [2010] I. Levin-Reisman, O. Gefen, O. Fridman, I. Ronin, D. Shwa, H. Sheftel, and N. Q. Balaban, Nature methods 7, 737 (2010).
  * Jõers and Tenson [2016] A. Jõers and T. Tenson, Scientific reports 6, 1 (2016).
  * Kaplan  _et al._ [2021] Y. Kaplan, S. Reich, E. Oster, S. Maoz, I. Levin-Reisman, I. Ronin, O. Gefen, O. Agam, and N. Q. Balaban, Nature 600, 290 (2021).
  * Scott and Hwa [2011] M. Scott and T. Hwa, Current opinion in biotechnology 22, 559 (2011).
  * Kaneko  _et al._ [2015] K. Kaneko, C. Furusawa, and T. Yomo, Physical Review X 5, 011014 (2015).
  * Jun  _et al._ [2018] S. Jun, F. Si, R. Pugatch, and M. Scott, Reports on Progress in Physics 81, 056601 (2018).
  * Jaishankar and Srivastava [2017] J. Jaishankar and P. Srivastava, Frontiers in microbiology 8, 2000 (2017).
  * Himeoka and Kaneko [2017] Y. Himeoka and K. Kaneko, Physical Review X 7, 021049 (2017).
  * Reich  _et al._ [2022] S. Reich, S. Maoz, Y. Kaplan, H. Rappeport, N. Balaban, and O. Agam, Physical Review Research 4, 033127 (2022).
  * Furusawa and Kaneko [2003] C. Furusawa and K. Kaneko, Physical Review Letters 90, 088102 (2003).
  * Furusawa and Kaneko [2012] C. Furusawa and K. Kaneko, Physical Review Letters 108, 208103 (2012).
  * Note [2] The reaction networks are randomly generated such that nutrient chemical $X_{0}$ can only be a substrate of intracellular reactions, transporter chemical $X_{1}$ cannot catalyze any intracellular reactions, and autocatalytic reactions do not occur.
  * Note [3] In this Letter, we basically plot the steady states that cells reach when $S_{\mathrm{ext}}$ is lowered, i.e., the initial state for each set of parameters is given as the steady state with slightly higher $S_{\mathrm{ext}}$. In actual cells, the fluctuation of internal concentrations leads to attractor selection [41]; thus, the branch with the higher growth rate would be more likely to be observed.
  * Eigen and Schuster [2012] M. Eigen and P. Schuster,  _The hypercycle: a principle of natural self-organization_ (Springer Science & Business Media, 2012).
  * Kauffman  _et al._ [1993] S. A. Kauffman _et al._ ,  _The origins of order: Self-organization and selection in evolution_ (Oxford University Press, USA, 1993).
  * Jain and Krishna [1998] S. Jain and S. Krishna, Physical Review Letters 81, 5684 (1998).
  * Kaneko [2005] K. Kaneko, Advances in Chemical Physics 130, 543 (2005).
  * Blokhuis  _et al._ [2020] A. Blokhuis, D. Lacoste, and P. Nghe, Proceedings of the National Academy of Sciences 117, 25230 (2020).
  * Nirupama  _et al._ [2018] R. Nirupama, B. Rajaraman, and H. Yajurvedi, Imaging J Clin Med Sci 5, 8 (2018).
  * Hatakeyama and Furusawa [2017] T. S. Hatakeyama and C. Furusawa, PLoS computational biology 13, e1005847 (2017).
  * Shipley and Keddy [1988] B. Shipley and P. Keddy, The Journal of Ecology , 1101 (1988).
  * Fink  _et al._ [2023] J. W. Fink, N. A. Held, and M. Manhart, Proceedings of the National Academy of Sciences 120, e2207295120 (2023).
  * Augustin  _et al._ [2000] J.-C. Augustin, L. Rosso, and V. Carlier, International Journal of food microbiology 57, 169 (2000).
  * Pin and Baranyi [2008] C. Pin and J. Baranyi, Applied and environmental microbiology 74, 2534 (2008).
  * Note [4] If cells reach fixed points that completely lose the transporter(s), cells cannot recover growth. We term such steady states death phases (see SM, Sec. A for details).
  * Note [5] Note that there are several possibilities for the choice of mean-field models with effective reactions among $X$ and $T$ that exhibit the growth-dormant transition (see SM, Sec. B for details).
  * Arnold [2012] V. I. Arnold,  _Geometrical methods in the theory of ordinary differential equations_ , Vol. 250 (Springer Science & Business Media, 2012).
  * Luisi [2016] P. L. Luisi,  _The emergence of life: from chemical origins to synthetic biology_ (Cambridge University Press, 2016).
  * Ameta  _et al._ [2021] S. Ameta, Y. J. Matsubara, N. Chakraborty, S. Krishna, and S. Thutupalli, Life 11, 308 (2021).
  * Kamimura and Kaneko [2016] A. Kamimura and K. Kaneko, Physical Review E 93, 062419 (2016).
  * Kashiwagi  _et al._ [2006] A. Kashiwagi, I. Urabe, K. Kaneko, and T. Yomo, PloS One 1, e49 (2006).
  * de Groot  _et al._ [2020] D. H. de Groot, J. Hulshof, B. Teusink, F. J. Bruggeman, and R. Planqué, PLoS computational biology 16, e1007559 (2020).
  * Müller [2021] S. Müller, bioRxiv (2021).
  * Schilling  _et al._ [2000] C. H. Schilling, D. Letscher, and B. Ø. Palsson, Journal of theoretical biology 203, 229 (2000).

Supplemental Material for
Universal Transitions between Growth and Dormancy via Intermediate Complex
Formation

Figure S1:  Statistics for randomly-generated networks. (a) Fraction of
randomly-generated networks exhibiting growth-dormant(-death) transition,
growth-death transition, and no transition. Networks are randomly generated
for each set of parameters (i.e., given $N_{r}$ and $n$ shown below). (b)
$\mu$, $Y$, and $H$ are plotted against
$S_{\mathrm{ext}}/S_{\mathrm{ext}}^{c}$ for networks showing growth-dormant
transitions with non-growing subnetworks (II) that cannot be active exactly
with $S_{\mathrm{ext}}=0$. $n=10,N_{r}=30$, $v=0.01$. Different colors
correspond to different networks. See Fig. 2 in the main text for networks
with non-growing subnetworks (I) that are active even with
$S_{\mathrm{ext}}=0$. See also Sec. A for details.

##  Appendix A Details about complex-formation reaction-network model and
numerical simulations

###  A.1 Details about phases and subnetworks

Below we discuss the definition and characteristics of the growth, dormant,
and death phases and the subnetworks working in those phases. Here, we
referred to a subnetwork as a closed set of reactions and their catalysts,
substrates, and products.

Growth phase.— Under the growth phase, a minimal autocatalytic subnetwork [24,
25, 26, 27, 28] (and its byproducts) becomes dominant, which we term a
autocatalytic growth subnetwork. It consumes nutrient chemical $X_{0}$ and
produces transporter chemical $X_{1}$ (i.e., one of so-called elementary
growth modes [42, 43]).

Dormant phase.— Under the dormant phase, a closed set of chemicals and
reactions that does not sustain growth by itself is dominantly active, which
we term a non-growing subnetwork. The non-growing subnetworks are classified
into two types, depending on whether they are cycles or not.

Non-growing subnetworks (I) are cycles within which the set of chemicals
remains conserved in number (see Figs. 1 and 3 in the main text and Fig. S5
for other examples). Those subnetworks can be active even when
$S_{\mathrm{ext}}=0$. Such cycles are also termed allocatalytic cycles (e.g.,
$A+E\leftrightarrows B+E$) [28].

Non-growing subnetworks (II) are non-cyclic subnetworks “parasitic” to the
autocatalytic growth subnetwork: For the reactions in those subnetworks to be
sustained, some chemical(s) in the autocatalytic growth subnetwork are
required as substrate(s) (see Fig. S2 for an example); thus, those subnetworks
cannot be active when $S_{\mathrm{ext}}=0$. They include some elementary flux
modes [44] (i.e., elementary growth modes with zero growth rate) that consume
the nutrient(s) and produce only non-transporter chemicals.

In the main text, the results of networks with non-growing subnetworks (I) are
mainly presented. However, the results are basically valid even with non-
growing subnetworks (II), except that $H$ gets small in $S_{\mathrm{ext}}\to
0$ because the components are concentrated on a few free reactants without any
reactions occurring to form the intermediate complexes. Notablye, even with a
non-growing subnetwork (II), there is a peak of $H$ near the transition point
(see Figs. S2(c) and S1(b)).

Death phase.— With some reaction networks, there exists a death phase, in
addition to the growth and dormant phases. Thus, the dependence on
$S_{\mathrm{ext}}$ in our model can be classified into growth-dormant
transition, growth-dormant-death transition, growth-death transition, and no
transition. The death phase is defined as a stable steady state with zero
growth rate with $S_{\mathrm{ext}}>0$ (even though the degradation of internal
components is not included in our model). When the nutrient concentration is
increased from $S_{\mathrm{ext}}=0$ to sufficiently large $S_{\mathrm{ext}}$,
cells can return to the growth phase from the dormant phase after a certain
lag time, while they cannot from the death phase.

Figure S2:  Example of randomly-generated networks with non-growing subnetwork
(II). $n=10,N_{r}=30$. (a) Reaction network. Chemicals at arrowtails are
transformed to those at arrowheads, catalyzed by those labeled on edges. (b)
Dependence of $\mu^{\ast}$ and $\textbf{x}^{\ast}$ on $S_{\mathrm{ext}}$. (c)
Dependence of $Y:=\sum_{\rho}y_{\rho}$ and $H:=-\sum_{i}x_{i}\log
x_{i}-\sum_{\rho}2y_{\rho}\log(2y_{\rho})$ on $S_{\mathrm{ext}}$. (d) Fluxes
for the dormant phase ($S_{\mathrm{ext}}=0.112$; left) and growth phase
($S_{\mathrm{ext}}=10.0$; right). The edge colors represent the log scale of
reaction fluxes. (e) Dependence of the lag time $T_{\mathrm{lag}}$ on the
starvation time $T_{\mathrm{stv}}$.

###  A.2 Model with degradation

In the main text, the degradation or leakage of intracellular chemical
components is not considered, while one can include it. Then, the degradation
terms, $-d_{i}x_{i}$ and $-d_{\rho}y_{\rho}$, should be added to the right-
hand side of Eqs. (1-2) in the main text as:

$\displaystyle\dot{x}_{i}$ $\displaystyle=$
$\displaystyle\sum_{\rho}\left[(\delta_{i,\rho_{p}}+\delta_{i,\rho_{c}})v_{\rho}y_{\rho}-(\delta_{i,\rho_{s}}+\delta_{i,\rho_{c}})f_{\rho}({\bf
x},{\bf
y})\right]+F_{i}(\textbf{x};S_{\mathrm{ext}},\alpha)-(\mu+d_{i})x_{i},$ (3)
$\displaystyle\dot{y}_{\rho}$ $\displaystyle=$ $\displaystyle f_{\rho}({\bf
x},{\bf y})-v_{\rho}y_{\rho}-(\mu+d_{\rho})y_{\rho},$ (4)

where the growth rate is defined as
$\mu(\textbf{x},\textbf{y}):=\sum_{i}F_{i}-\sum_{i}d_{i}x_{i}-2\sum_{\rho}d_{\rho}y_{\rho}$.

The model considered in the main text corresponds to the case with
$d_{i}=d_{\rho}=0$; whereas, if $d_{i},d_{\rho}>0$, cells reach fixed points
with non-positive growth rate, which correspond to death phases, with finite
$S_{\mathrm{ext}}$.

###  A.3 Forms of nutrient transport

In the main text, we assumed that the intake of nutrient $X_{0}$ is mediated
by $\alpha$-th order transporter $X_{1}$, as
$F_{0}(\textbf{x};S_{\mathrm{ext}},\alpha)=S_{\mathrm{ext}}x_{1}^{\alpha}$.
However, other forms of nutrient transport can be considered as well: e.g.,
$\alpha$-th order channel as
$F_{i}(\textbf{x};S_{\mathrm{ext}},\alpha,D_{i})=D_{i}x_{\sigma_{i}}^{\alpha}(S_{\mathrm{ext}}-x_{i})$
and passive diffusion without transporters as
$F_{i}(\textbf{x};S_{\mathrm{ext}},D_{i})=D_{i}(S_{\mathrm{ext}}-x_{i})$. Even
with these forms of nutrient transport, the growth-dormant transition can
occur due to a similar mechanism, namely, jamming of reactions due to the
accumulation of complexes (see e.g., Fig. S3).

Figure S3:  Example of randomly-generated networks with nutrient transport via
the $\alpha$-th order channel $X_{1}$. $n=10,N_{r}=20$. Unless otherwise
stated, $\alpha=1$ and $v=0.1$. (a) Reaction network. Chemicals at arrowtails
are transformed to those at arrowheads, catalyzed by those labeled on edges.
(b) Dependence of $\mu^{\ast}$ and $\textbf{x}^{\ast}$ on $S_{\mathrm{ext}}$.
(c) Dependence of $Y:=\sum_{\rho}y_{\rho}$ and $H:=-\sum_{i}x_{i}\log
x_{i}-\sum_{\rho}2y_{\rho}\log(2y_{\rho})$ on $S_{\mathrm{ext}}$. (d) Dominant
fluxes for the dormant phase ($S_{\mathrm{ext}}=0.112$; left) and growth phase
($S_{\mathrm{ext}}=10.0$; right). The edge colors represent the log scale of
reaction fluxes, while the node colors represent the concentrations. (e)
Dependence of the lag time $T_{\mathrm{lag}}$ on the starvation time
$T_{\mathrm{stv}}$ with different $\alpha$.

###  A.4 Details about the lag time

As also described in the main text, when the intracellular composition in the
non-cyclic non-growing subnetwork (II) (working in the dormant or death phase)
includes a sufficiently small amount of the transporter chemical, the cell in
our model exhibits the lag time $T_{\mathrm{lag}}$. In addition,
$T_{\mathrm{lag}}$ increases with starvation time $T_{\mathrm{stv}}$ in the
form $T_{\mathrm{lag}}\propto T_{\mathrm{stv}}^{\beta}$ (see e.g., Figs.S2 and
S3).

Intuitively, this is understood as follows. Under nutrient scarcity, the
chemical concentrations are gradually concentrated toward the non-growing
subnetwork in the dormant phase or death phase; since that absorbing state
contains only a small amount or none of some component(s) of the autocatalytic
subnetwork of the autocatalytic growth subnetwork, their concentration
decreases exponentially. Then, the cell takes a long time to recover the
growth by regaining the intracellular composition in the growth phase after
the nutrient supply is recovered.

The exponent $\beta$, ranging from approximately $0.1$ to $0.5$, seems to be
mainly dependent on the network structure which alters the intracellular
reaction dynamics: e.g., for the network of Fig. S3, $\beta\sim 1/3$ holds,
almost independently of the exponent $\alpha$ for the nutrient transport.

##  Appendix B Details about mean-field analysis

###  B.1 Mean-field model with $S$, $X$, and $Y$

In Fig. 3(a) in the main text, a fully-connected mean-filed model including
one mean-filed variable $X$ in addition to the nutrient $S$ is considered. In
this model, $X$ represents all $n_{X}$ non-nutrient chemicals one of which is
the transporter; thus, the growth rate is given as
$\mu(X/n_{X};S_{\mathrm{ext}}):=S_{\mathrm{ext}}(X/n_{X})^{\alpha}$.

Then, its time evolution is given as

$\displaystyle\dot{S}$ $\displaystyle=$
$\displaystyle\mu(X/n_{X};S_{\mathrm{ext}})-\phi SX-\mu S,$
$\displaystyle\dot{X}$ $\displaystyle=$ $\displaystyle\phi
SX-2\phi(X^{2}-vY)-\mu X,$ $\displaystyle\dot{Y}$ $\displaystyle=$
$\displaystyle\phi(X^{2}-vY)-\mu Y,$

where $Y$ denotes the complexes of non-nutrient chemicals and $\phi$ denotes
the reaction path density. Here, the complexes between $S$ and $X$ are not
explicitly incorporated, in other words, $v_{\rho}$ for the reactions between
$S$ and $X$ are assumed sufficiently large, for the sake of simplicity.

Even such a mean-field model consisting of only nutrient chemical $S$ and non-
nutrient chemical $X$ exhibits the growth-dormant transition with sufficiently
small $v\ll\mu_{\max}$ and sufficient non-linearity $\alpha>2$ (Fig. S4).

Even with smaller $\alpha=2$, a transition-like behavior can be observed but
it is not a discontinuous transition, as can be seen from the following self-
consistent equation for $\mu$ (Fig. S4(c)). From the steady state condition
$\dot{S}=0,\dot{Y}=0$,

$\displaystyle S^{\ast}=\frac{\mu}{\phi_{S}X+\mu},\quad
Y^{\ast}=\frac{\phi_{X}X^{2}}{\phi_{X}v+\mu}.$

Then, the steady state $X^{\ast}$ must satisfy

$\displaystyle
0=\dot{X}=\phi_{S}\frac{\mu}{\phi_{S}X+\mu}X-2\phi_{X}\frac{\mu}{\phi_{X}v+\mu}X^{2}-\mu
X.$

By solving $X^{\ast}$ as a function of $\mu$, we obtain

$\displaystyle
X^{\ast}(\mu;\alpha,v,\phi_{S},\phi_{X})=\frac{-\phi_{S}\phi_{X}v-\phi_{S}\mu-2\phi_{X}\mu+\sqrt{(-\phi_{S}\phi_{X}v-\phi_{S}\mu-2\phi_{X}\mu)^{2}+8\phi_{S}\phi_{X}(\phi_{S}\phi_{X}v+\phi_{S}\mu-\phi_{X}v\mu-\mu^{2})}}{4\phi_{S}\phi_{X}}.$

Figure S4: Mean-field model with $S$, $X$, and $Y$. (a) Reaction network
structure. (b) Dependence of $\mu^{\ast}$, $S^{\ast}$, $X^{\ast}$, and
$Y^{\ast}$ on $S_{\mathrm{ext}}$. $\alpha=2,v=10^{-4}$. (c) Self-consistent
equation for $\mu$ with $\alpha=2,v=10^{-4}$. Orange lines correspond to
$S_{\mathrm{ext}}=8,32,128$, respectively. (d) Dependence of $\mu^{\ast}$,
$S^{\ast}$, $X^{\ast}$, and $Y^{\ast}$ on $S_{\mathrm{ext}}$.
$\alpha=2,v=10^{-4}$. (e) Self-consistent equation for $\mu$ with
$\alpha=3,v=10^{-3}$. Orange lines correspond to $S_{\mathrm{ext}}=8,32,128$,
respectively.

###  B.2 Simple mean-field model with $S$, $T$, $X$, and $Y$ in the main text

The dynamics for the simplest “mean-field” model with $S$, $T$, $X$, and $Y$
in the main text is given as

$\displaystyle\dot{S}$ $\displaystyle=$
$\displaystyle\mu(T;S_{\mathrm{ext}})-\phi\frac{n_{T}+n_{X}}{n_{X}}SX-\mu S,$
$\displaystyle\dot{T}$ $\displaystyle=$
$\displaystyle\phi\frac{n_{T}}{n_{X}}SX-\phi(XT-vY)-\mu T,$
$\displaystyle\dot{X}$ $\displaystyle=$ $\displaystyle\phi SX-\phi(XT-vY)-\mu
X,$ $\displaystyle\dot{Y}$ $\displaystyle=$ $\displaystyle\phi(XT-vY)-\mu Y.$

Note that, in this model, the intermediate complex between $X$ and $T$ is
considered for simplicity. Even in such a case, the growth-dormant transition
is observed (Fig. 3(b) in the main text).

Self-consistent equation.— The self-consistent equation in Fig. 3(e) in the
main text is calculated as follows.

First, from the definition of growth rate
$\mu(\textbf{x}):=S_{\mathrm{ext}}(T/n_{T})^{\alpha}$,
$T=n_{T}(\mu/S_{\mathrm{ext}})^{1/\alpha}$ holds.

In contrast, from the steady state condition, we can explicitly solve
$T^{\ast}$ in the steady state as a function of $\mu$. From $\dot{S}=0$ and
$\dot{Y}=0$, we immediately obtain
$S^{\ast}(X^{\ast},\mu)=\frac{\mu}{\mu+\phi(n_{T}+n_{X})X^{\ast}/n_{X}}$ and
$Y^{\ast}(T^{\ast},X^{\ast},\mu)=\phi\frac{X^{\ast}T^{\ast}}{\mu+\phi v}$,
respectively. Then, by solving $\dot{X}=0\;\Leftrightarrow\;\phi
S^{\ast}(X^{\ast},\mu)X^{\ast}-\phi(X^{\ast}T^{\ast}-vY^{\ast}(T^{\ast},X^{\ast},\mu))-\mu
X^{\ast}=0$ in the case $n_{T}=1,\phi=1,\alpha=2$,

$\displaystyle
X^{\ast}(T^{\ast},\mu;v,n_{X})=\frac{n_{X}}{1+n_{X}}\frac{v+\mu-v\mu-T^{\ast}\mu-\mu^{2}}{v+T^{\ast}+\mu}.$

Finally, from $\dot{T}=0$ and $T^{\ast}\geq 0$, we obtain

$\displaystyle T^{\ast}(\mu;v,n_{X})=$
$\displaystyle\frac{-v+n_{X}v+v^{2}+n_{X}v^{2}-\mu+n_{X}\mu+4v\mu+n_{X}v\mu+3\mu^{2}}{2(-v-n_{X}v-2\mu)}$
$\displaystyle-\frac{(v+\mu)\sqrt{1-2n_{X}+n_{X}^{2}+2v+4n_{X}v+2n_{X}^{2}v+v^{2}+2n_{X}v^{2}+n_{X}^{2}v^{2}+2\mu+6n_{X}\mu+2v\mu+2n_{X}v\mu+\mu^{2}}}{2(-v-n_{X}v-2\mu)}.$

Notably, this is formally independent of $S_{\mathrm{ext}}$.

Therefore, the steady states are given as the intersections of
$T=\sqrt{\mu/S_{\mathrm{ext}}}$ and $T=T^{\ast}(\mu;v,n_{X})$ (Fig. 3(e) in
the main text).

Figure S5:  Additional data on the simple mean-field model with $S$, $T$, $X$,
and $Y$ (see also Fig. 3(b)-(e) in the main text). (a) Flow diagram and
nullclines with $v=0.1$. (b) Self-consistent equations for $T$ and $\mu$ with
$n_{X}=1,b=0.01$ (left; no transition) and $n_{X}=2,v=0.1$ (right; no
transition).

###  B.3 Fully-connected mean-field model with $S$, $T$, $X$, and their
complexes

We can also consider a fully-connected mean-field model with reactants. Here,
all the reactions between $S$, $T$, and $X$ are considered, and the
intermediate complex formation (i.e., finite $v_{\rho}$) for all internal
reactions is assumed as in the randomly-generated networks. As shown below,
this more “symmetric” model also exhibits the growth-dormant transition.

The dynamics for the fully-connected mean-filed model is given as

$\displaystyle\dot{S}$ $\displaystyle=$ $\displaystyle
S_{\mathrm{ext}}(\frac{T}{n_{T}})^{\alpha}-\phi(n_{T}+n_{X})S\frac{X}{n_{X}}-\mu
S,$ $\displaystyle\dot{T}$ $\displaystyle=$ $\displaystyle\phi
n_{T}v\frac{Y_{SX}}{n_{T}+n_{X}}-\phi
n_{T}n_{X}\frac{T}{n_{T}}\frac{X}{n_{X}}+\phi
n_{T}n_{X}v\frac{Y_{XX}}{(n_{T}+n_{X})n_{X}}-\mu T,$ $\displaystyle\dot{X}$
$\displaystyle=$
$\displaystyle-\phi(n_{T}+n_{X})S\frac{X}{n_{X}}+\phi(n_{T}+2n_{X})v\frac{Y_{SX}}{n_{T}+n_{X}}-\phi
n_{T}n_{X}\frac{T}{n_{T}}\frac{X}{n_{X}}+2\phi
n_{T}n_{X}v\frac{Y_{TX}}{n_{T}n_{X}}$
$\displaystyle-2\phi(n_{T}+n_{X})n_{X}\left(\frac{X}{n_{X}}\right)^{2}+\phi(n_{T}+2n_{X})n_{X}v\frac{Y_{XX}}{(n_{T}+n_{X})n_{X}}-\mu
X,$ $\displaystyle\dot{Y}_{SX}$ $\displaystyle=$
$\displaystyle\phi(n_{T}+n_{X})S\frac{X}{n_{X}}-\phi(n_{T}+n_{X})v\frac{Y_{SX}}{n_{T}+n_{X}}-\mu
Y_{SX},$ $\displaystyle\dot{Y}_{TX}$ $\displaystyle=$ $\displaystyle\phi
n_{T}n_{X}\frac{T}{n_{T}}\frac{X}{n_{X}}-\phi
n_{T}n_{X}v\frac{Y_{TX}}{n_{T}n_{X}}-\mu Y_{TX},$ $\displaystyle\dot{Y}_{XX}$
$\displaystyle=$
$\displaystyle\phi(n_{T}+n_{X})n_{X}\left(\frac{X}{n_{X}}\right)^{2}-\phi(n_{T}+n_{X})n_{X}v\frac{Y_{XX}}{(n_{T}+n_{X})n_{X}}-\mu
Y_{XX}.$

Here, the number of reactions between chemicals $i$ and $j$ are proportional
to $n_{i}n_{j}$ ($i,j=S,T,X$). When $Y_{\rho}$s are adiabatically eliminated
as

$\displaystyle Y_{SX}^{\ast}=\frac{\phi(n_{T}+n_{X})/n_{X}}{\phi
v+\mu}SX,\quad Y_{TX}^{\ast}=\frac{\phi}{\phi v+\mu}TX,\quad
Y_{XX}^{\ast}=\frac{\phi(n_{T}+n_{X})/n_{X}}{\phi v+\mu}X^{2},$

we obtain:

$\displaystyle\dot{S}$ $\displaystyle=$ $\displaystyle
S_{\mathrm{ext}}(\frac{T}{n_{T}})^{\alpha}-\phi(n_{T}+n_{X})S\frac{X}{n_{X}}-\mu
S,$ $\displaystyle\dot{T}$ $\displaystyle=$
$\displaystyle\phi_{S}n_{T}v\frac{\phi_{S}/n_{X}}{\phi_{S}v+\mu}SX-\phi
TX+\phi n_{T}v\frac{\phi/n_{X}}{\phi v+\mu}X^{2}-\mu T,$
$\displaystyle\dot{X}$ $\displaystyle=$
$\displaystyle\frac{n_{X}\phi_{S}v-(n_{T}+n_{X})\mu}{\phi_{S}v+\mu}\phi_{S}S\frac{X}{n_{X}}+\frac{\phi
v-\mu}{\phi v+\mu}\phi TX-\phi\frac{2(n_{T}+n_{X})\mu+n_{T}v\phi}{\phi
v+\mu}X^{2}/n_{X}-\mu X.$

This fully-connected mean-field model exhibits the growth-dormant transition,
(Fig. S6).

Figure S6: Fully-connected mean-field model with $S$, $T$, $X$, and their
complexes. (a) Reaction network structure. (b) Dependence of $\mu^{\ast}$,
$S^{\ast}$, $X^{\ast}$, $T^{\ast}$, and
$Y:=Y^{\ast}_{SX}+Y^{\ast}_{XX}+Y^{\ast}_{TX}$ on $S_{\mathrm{ext}}$.
$v=0.01$.